THE VECTOR FIELD METHOD AND ITS APPLICATIONS TO NONLINEAR

EVOLUTION EQUATIONS

By

Leonardo Enrique Abbrescia

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mathematics – Doctor of Philosophy

2020

ABSTRACT

THE VECTOR FIELD METHOD AND ITS APPLICATIONS TO NONLINEAR

EVOLUTION EQUATIONS

By

Leonardo Enrique Abbrescia

The vector field method was introduced in the 1980s by Sergiu Klainerman to analyze the

decay properties of the linear wave equation. Since its historical debut, the vector field

method has been at the forefront of several breakthrough results including the global sta-

bility of Minkowski space, the dynamical formation of black holes, and shock formation

in 3D compressible fluids.

This work showcases how the vector field method can be used in a systematic way

to derive a priori estimates for nonlinear evolution equations. For nonlinear dispersive

equations, these estimates can be married to the decay properties enjoyed by the solu-

tions to derive quantitative asymptotics. This is done in this work through the lens of

three concrete problems: a nonlocal kinetic model, the wave maps equation, and the rel-

ativistic membrane equation. For the kinetic model, the vector field method is paired

with dispersive decay properties of the spatial density to prove global wellposedness of

small data. This can be interpreted physically as “stability” of the trivial solution. For

the wave maps equation, a stability result is proven for a “non-trivial” ODE geodesic so-

lution. For the relativistic membrane equation, the vector field method is used to prove

stability of large simple-traveling-waves. For the wave map and membrane equations,

we intimately use several structural properties known as null conditions that preclude

singular behavior.

Copyright by
LEONARDO ENRIQUE ABBRESCIA
2020

To Britta and Little Leo, for your endless love and support.

A mis padres, por sus sacrificios.

To the families of those who were wrongfully taken from them. Black lives matter.

To Esteban, I hope you find a righteous future.

To all of the lives lost due to COVID-19.

iv

ACKNOWLEDGEMENTS

It took a village to raise me as a mathematician, and Willie Wong was its leader for the

past four years. I extend my wholehearted gratitude to him for arming me with the tools

to become an independent mathematician, for posing tangible and interesting problems,

for preparing me to tackle the job market, and for putting up with my dramatic life in

the past four years. As his first student, I hope all of those who follow in my footsteps

acknowledge how lucky they are for having a such a nurturing advisor. I will miss our

weekly meetings and discussions about math, gossip, and life. I also extend my thanks to

Suki, Samara, and Edmond for letting me borrow Willie so much during these years. All

the best to your lovely family.

I extend my thanks to Jared Speck, for providing me with a wonderful job opportunity

for the next three years.

There are many people at Michigan State University that need to be appropriately

thanked. I am forever indebted to Jon Wolfson and his 2015 graduate committee for

making the math department at MSU take a chance on me, especially since it was the

only one to do so. I would also like to thank Jon and his team for spearheading some

much-needed changes in the graduate program; it was a pleasure to be here. I would like

to thank my graduate committee. Many thanks to Jun Kitagawa for teaching and rein-

forcing the foundations of PDEs, and for making my first year at MSU quite enjoyable. I
owe a lot to Jeff Schenker for teaching me so many of the analysis tools that I will need in

my career, and for being so patient when I had to miss lectures when Little Leo was born.

And of course, many thanks to Tom Parker for being very welcoming during my first year,

for teaching me geometry, and for preparing me to handle a hard audience. Other fac-

ulty members who helped me become who I am are Ben Schmidt, Ignacio Uriarte-Tuero,

Gabor Francsics, Peter Bates, Baisheng Yan, Keith Promislow, Olga Turanova, and Russell

Schwab. Special thanks go to Olga for doing such a close reading of my NSF application.

v

I would also like to thank Rodrigo Matos for being such an awesome officemate, it made

every day a little bit easier. I would also like to thank the graduate students who came

to my PDE seminar: Seonghyeon Jeong, Wenchuan Tian, Shih-Fang Yeh, Woongbae Park,

Chamila Malagodagamage, Reshma Menon, Keshav Sutrave, Hayriye Guckircakir, and

Arman Tavakoli. It was a lot of fun to create an AMS Student Chapter with Rachel Do-

magalski, Franciska Domokos, Tristan Wellsfilbert, and Rob Mcconkey. Other graduate
students that made the department more pleasant to me were ´Akos Nagy, Sanjay Kumar,

Zak Tilocco, Danika Van Niel, Charlotte Ure, Abhishek Mallick, Gora Bera, Joe Melby,

Hitesh Gakhar, Craig Gross, and Andr´es Galindo Olarte. Mollee and Michael, it was fun

to have a child of a similar age during graduate school with you. Last and definitely not
least, my gratitude also extends to the staff of the math department. Particularly, Carolyn

Wemple, Estrella Starn, and of course, Britta Abbrescia, were always extremely helpful.

My time at Columbia University was also very important in making me a mathemati-

cian. I owe everything to Professors Daniela De Silva and Ovidiu Savin, for taking a

chance on a student who was obviously nothing special. Thank you both for teaching me

how to integrate by parts, a technique I will use forever! I would also like to thank Jo

Nelson for showing me how fun research can be. And of course, many thanks go to my

math friends who encouraged me to take more math courses: Yifei Zhao, Matei Ionita,

Nilay Kumar, Ryan Contreras.

For their kindness and support, I extend my gratitude to the Rajadhyaksha family.

I would not have been half as prepared as I was to study mathematics at Columbia
University if it were not for the extremely effective STEM Program at CCNY. My thanks

go to Mr. And Mrs. Marte for maintaining a program that has and will continue to

change the lives of so many students from underrepresented groups in STEM. In particu-

lar, thank you both for making the program free for all students who would, me included,

otherwise never had such an opportunity. The STEM Program is so successful because of

its fantastic teachers. Special acknowledgments are owed to Ms. Pichardo and Mr. Car-

vi

denas, who were the first to really show me how fun challenging problems can be.

I extend my thanks to Dr. Edwing Medina, whose eloquent words of encouragement

motivated me to pursue academia. I would also like to thank Dr. Walcott for his patience

when I first learned how to traverse the world of research.

I would not be where I am today if it were not for my parents, who risked everything

to give me and my brother a better life than they had. I will never take for granted all of

the sacrifices they made to make sure that we could succeed. Thank you.

I extend my love and thanks to Britta Abbrescia and Leonardo Rafael Abbrescia. I

could write an entire dissertation to show how grateful I am for your support and it

would still not be enough. Thank you both for giving me a purpose in life like no other.

Britta, being your husband is the greatest achievement I will have ever met. Thank you

for being the best friend I didn’t know I needed. Giving you the life and love that you

deserve is the biggest source of motivation. And to Leo, thank you for completing our

family.

I was partially supported by an NSF Graduate Research Fellowship. Some material of

this work is adapted from a paper accepted for publication in Forum of Mathematics, Pi.

vii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

xi

KEY TO SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
1.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.1 Brief outline of the present work . . . . . . . . . . . . . . . . . . . .
4
1.2 Prelude to the vector field method . . . . . . . . . . . . . . . . . . . . . . . .
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 The model
6
1.2.2 Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Approximate-conservation laws . . . . . . . . . . . . . . . . . . . . .
9
1.2.5 Wellposedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.6 Vector field method in action . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.6.1 The weighted vectorfields . . . . . . . . . . . . . . . . . . . 22
1.2.7 Global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3 Main results of the current work . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.3.2 Totally geodesic wave maps
. . . . . . . . . . . . . . . . . . . . . . . 33
1.3.3 Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

CHAPTER 2 THE LINEAR WAVE AND KLEIN–GORDON EQUATIONS . . . . . 40
2.1 Sobolev embeddings adapted to hyperboloidal foliations . . . . . . . . . . . 41
2.1.1 The basic global GNS inequalities . . . . . . . . . . . . . . . . . . . . 44
2.1.2 Interpolating inequalities: non-borderline case . . . . . . . . . . . . 46
2.1.3 Interpolating inequalities: borderline case . . . . . . . . . . . . . . . 47
2.2 A priori estimates via the vector field method . . . . . . . . . . . . . . . . . 48
2.2.1 Energy formalism and pointwise estimates . . . . . . . . . . . . . . . 48
2.2.2 Wave equation, d = 2,3,4 . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.2.3 Klein–Gordon equation, d = 2,3,4 . . . . . . . . . . . . . . . . . . . . 63

CHAPTER 3 TOTALLY GEODESIC WAVE MAPS . . . . . . . . . . . . . . . . . . . 75
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2 Explanation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3 Geodesic normal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4 Perturbed system and reduction to wave-Klein–Gordon system . . . . . . . 87
3.4.1 Reductions when M is a spaceform . . . . . . . . . . . . . . . . . . . 92
3.4.1.1 Negatively curved case . . . . . . . . . . . . . . . . . . . . . 94
. . . . . . . . . . . . . . . . . . . . . 95
3.4.1.2 Positively curved case
3.5 Preliminary L2 analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

viii

3.6 Global stability in the setting of TL . . . . . . . . . . . . . . . . . . . . . . . 98
3.7 Global stability in the setting of SL . . . . . . . . . . . . . . . . . . . . . . . 110

4.4 A semilinear model

CHAPTER 4 THE MEMBRANE EQUATION . . . . . . . . . . . . . . . . . . . . . . 115
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1.1 Our main result and discussions . . . . . . . . . . . . . . . . . . . . . 118
4.1.2 Outline of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2 The background solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2.1 Simple planewave solutions to the membrane equation . . . . . . . . 126
4.2.2 The gauge choice and the perturbed system . . . . . . . . . . . . . . 128
4.3 Preliminary L2 analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.3.1 Global Sobolev estimates on the double-null coordinates . . . . . . . 133
4.3.2 A weighted vector field algebra . . . . . . . . . . . . . . . . . . . . . 135
4.3.3 Generalized energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.4.2 Bootstrap for d ≥ 6 even . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.4.3 Bootstrap for d ≥ 5 odd . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.4.4 Bootstrap for d = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.4.5 Bootstrap for d = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.5 Commuted equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.5.1 The perturbed system, restated . . . . . . . . . . . . . . . . . . . . . 160
4.5.2 Commutator relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.5.3 Statement of the main theorem . . . . . . . . . . . . . . . . . . . . . 166
4.6 Energy quantities and bootstrap assumptions . . . . . . . . . . . . . . . . . 167
4.6.1 The energy quantities defined; bootstrap assumptions . . . . . . . . 168
4.6.2 Inequalities on that we use frequently . . . . . . . . . . . . . . . . . 170
4.6.3 Some first consequences of (BA∞) . . . . . . . . . . . . . . . . . . . . 171
∞
4.6.4 Improved L
. . . . . . . . . . . . . . . . . . . . 175
4.6.5 Controlling the deformation tensor term . . . . . . . . . . . . . . . . 176
4.7 Controlling the inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . 177
. . . . . . . . . . . . . . . . . . . . . . 181
4.7.1 Higher order nonlinear terms
4.7.2 Quadratic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.7.2.1 Zeroth order case . . . . . . . . . . . . . . . . . . . . . . . . 183
4.7.2.2 First order, ψ = L1φ . . . . . . . . . . . . . . . . . . . . . . . 183
4.7.2.3 First order, ψ = Liφ . . . . . . . . . . . . . . . . . . . . . . . 184
. . . . . . . . . . . . . . . . . . . . . . . 184
4.7.2.4 Higher order cases
4.8 Closing the bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

bounds from (BA2)

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

ix

LIST OF TABLES

Table 4.1: (d ≥ 6, even) List of admissible (cid:96)0, (cid:96)1, (cid:96)2 values as well as the corre-
sponding upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2. The value of “—” means any
value compatible with the prescribed columns. The shaded rows are
those with non-integrable upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2.

Table 4.2: (d ≥ 5, odd) List of admissible (cid:96)0, (cid:96)1, (cid:96)2 values as well as the corre-
sponding upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2. The value of “—” means any
value compatible with the prescribed columns. The shaded rows are
those with non-integrable upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2.

. . . . . . . . . . . 151

. . . . . . . . . . . 153

Table 4.3: (d = 4) List of admissible (cid:96)0, (cid:96)1, (cid:96)2 values as well as the corresponding
upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2. The value of “—” means any value com-
patible with the prescribed columns. The shaded rows are those with
non-integrable upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2. (Recall that 3γ < 1 by fiat.)

Table 4.4: (d = 3) List of admissible (cid:96)0, (cid:96)1, (cid:96)2 values as well as the corresponding
upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2. The value of “—” means any value com-
patible with the prescribed columns. The shaded rows are those with
non-integrable upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2. (Recall that 3γ < 1 by fiat.)

. 154

. 156

x

LIST OF FIGURES

Figure 3.1: The two different background solutions.

. . . . . . . . . . . . . . . . . . 79

Figure 3.2: The perturbation as a section of the normal bundle about ϕI.

. . . . . . 88

xi

KEY TO SYMBOLS

Minkowski metric with signature (−1,1, . . . ,1).
(1 + d)-dimensional Minkowski space equipped with m. We will alternate
freely between the notations R1,d and R1+d. We denote the standard Eu-
clidean coordinates on R1,d by {xµ}µ∈{0,...,d}. We will often write t for the time
coordinate x0.
Covariant wave operator on a pseudo-Riemannian manifold (M, g). In any
local coordinate system, for a scalar function f ,

(cid:3)gf = |det g|−1/2∂µ(|det g|1/2gµν∂νf ).

The d’Alembertian on R1,d. In rectangular coordinates, (cid:3) = −∂2
Weighted Lebesgue spaces. See notation in Section 2.1.

t + ∆x.

α Weighted homogeneous and inhomogeneous and Sobolev spaces. See nota-

m
R1,d

(cid:3)g

α

(cid:3), (cid:3)m
Lp
˚W k,p
α ,W k,p
∇M
LXT
(cid:98)u

tion in Section 2.1.
Levi–Civita connection on a pseudo-Riemannian manifold (M, g).
Lie differentiation of a scalar or tensor T in the direction of the vector field
X.
−ix·ξu(x) dx.

Fourier transform of a function, scaled by(cid:98)u(ξ) = (2π)

−d/2(cid:82)

Rd e

S (Rd)
◦
g
g
Υ
u, u, ˆx

yi
I +
τ, ρ, Στ

T , Li
Q
E
W∗

Schwartz space of functions.
Linear part of the dynamical metric; see (4.2.11) in §4.2.2.
Dynamical metric; see (4.2.9) in §4.2.2.
planewave background; see (4.2.1) in §4.2.1.
Null coordinates adapted to planewave background; see (4.2.2) and the dis-
cussion after (4.2.5) in §4.2.1.
Rectangular coordinates adapted to planewave background; see (4.3.8) in
§4.3.2.
Forward light cone; see start of §4.3.1.
Hyperboloidal foliation and related parameters; see (4.3.1) and Notation 4.3.1
in §4.3.1.
Vector fields; see (4.3.5), (4.3.6), and (4.3.7) in §4.3.1.
Stress-energy tensors; see (4.3.10) in §4.3.1.
Background energy integrals; see (4.3.13) and (4.3.14) in §??.
Weight functions; see Definition 4.3.5 in §4.3.2.

xii

W∗
P∗
A∗
∗,∗∗
B∗,∗∗

B

G

Elements of W∗; see Notation 4.5.2 in §4.5.1.
planewave like weights; see Notation 4.5.2 in §4.5.1.
Weighted commutator algebra; see discussion surrounding Proposition 4.3.8
in §4.3.2.
Weighted differential operators; see discussion surrounding (4.3.9) in §4.3.2,
as well as Definition 4.3.10.
Elements of B
Smooth functions representing bounded terms; see Notation 4.5.6 in §4.5.1.

∗,∗∗ ; see Notation 4.5.4 in §4.5.1.

xiii

CHAPTER 1

INTRODUCTION

1.1 Prologue

Decay properties of solutions to nonlinear dispersive equations play a central role in

their long-time behavior in the small data regime. This regime is important as it can be

interpreted as a kind of “stability” of the trivial solution. One expects to be able to solve a
nonlinear equation for a small amount of time so long as the nonlinearities are sufficiently

“weak”, which is typically a consequence of the small size of the initial data, and hence

the dynamics are essentially linear. A global-in-time solution can then be constructed so
long as the nonlinearities can be shown to decay sufficiently fast.

Classically, pointwise L1–L

decay for solutions to linear dispersive equations (such

∞

as Schr¨odinger, Airy, wave) is established by either estimating directly explicit repre-

sentation formulas for their fundamental solutions, or by oscillatory integral techniques

applied to the Fourier representations. These methods are either infeasible or impractical

when the equations are perturbed in a way that changes the principal part (i.e. the part

with the highest derivatives).

In 1985, Sergiu Klainerman developed a technique [Kla85b], now referred to as the

vector field method (VFM), to systematically analyze the decay properties of the lin-

ear wave equation. The appeal of Klainerman’s technique is that it is both a robust

method that is well suited to handle small perturbations of the principal part of lin-

ear waves, and that the general mechanism is applicable to settings of other dispersive

equations [Smu16, FJS17, Won18a]. The general strategy of the vector field method for

linear equations can be summarized as follows:

I: Analyze the inherent symmetries of the equation by looking for commuting vector

fields X that are weighted in time. That is, if φ solves the equation so does Xφ.

1

II: Analyze the inherent symmetries of the equation through its conservation laws.

III: Develop space-time weighted Sobolev inequalities that combine the previous two

points to get lower derivative pointwise estimates from higher derivative integral

norms. The temporal decay will be a consequence of these L

the temporal weight of X.

∞

estimates and from

Even though exact conservation laws are typically not present for nonlinear equa-

tions, one can derive “approximate conservation laws” via energy estimates. In the spirit

of Duhamel’s formula, these estimates control an appropriately defined “size” of the solu-

tion at a later time by the initial size and an integral measuring interactions between the

linear flow and the nonlinearities. In order to counteract the positive feedback of these

nonlinearities, which would make these integrals diverge, we rely on both the linear de-

cay extracted from the vector field method and smallness of the initial data to prevent the

nonlinearities from growing too large.

This dissertation showcases the balancing act between growth and decay in the con-

crete settings of some wave equations which arise in Lagrangian field theories: the wave

maps equation on Minkowski space and the relativistic membrane equation (see [SS98,

Kri07, Hop13] for an overview of the physical and mathematical history). This will be

done by using the vector field method as the technical skeleton to study decay, regularity,

stability properties of solutions to these equations.

1.1.1 Brief outline of the present work

This introductory chapter is split into two halves. In Section 1.2, we provide a hands-on

exposition of the VFM by using it to analyzing a toy model of nonlinear kinetic theory.

The reader should keep in mind that the analytical themes in Section 1.2 will be the

technical skeleton for the rest of the dissertation, see Subsection 1.2.6 for more details.

In Section 1.3, we introduce the main results of this dissertation and explain how they

2

pertain to a novel research program inspired by recent work. We highlight the important

role the decay estimates, which are derived using the vector field method, have in the

analysis.

Chapter 2 serves to establish necessary results for the linear wave equation. In Sec-

tion 2.1 we begin by setting up the geometric formalism of a hyperboloidal foliation of

Minkowski space. The presentation is adapted from the foundational work of Lefloch and

Ma [LM14] and the recent work of [Won17b]. This hyperboloidal viewpoint is taken to

exploit the fact that the symmetries of Minkowski space are Lorentzian and so “space” and

“time” are on equal footing, as opposed to Galilean symmetries of Newtonian mechan-

ics. The main results of this section are space-time weighted versions of the Gagliardo-

Nirenberg-Sobolev (GNS) interpolation inequalities adapted to this hyperboloidal foli-

ation, and are based on original joint work with Wong [AW19b]. These interpolation

inequalities allow one to save derivatives in Lebesgue product estimates compared to the
∞
Morrey L2–L

Sobolev embeddings. In Section 2.2 we marry the vector field method and

these weighted Sobolev estimates to derive a priori estimates for solutions to the linear

wave and Klein–Gordon equations.

In Chapter 3, we show that a totally geodesic map from Minkowski space into a space-

form, under a sign condition, is globally nonlinearly stable as a solution to the wave maps
equation under sufficiently small compactly supported perturbations. This is based on

original joint work with Chen [AC19]. We begin by constructing the perturbation as a

section of the normal bundle of the totally geodesic background. Using the spaceform as-

sumption on the target, we prove that the equations of motion for the perturbation reduce

to a semilinear system of wave-Klein–Gordon equations. Global existence and uniform

decay estimates are proved for this perturbation using the vector field method and the

GNS interpolation estimates of Chapter 2.

The main result of Chapter 4 is to show that planar traveling wave solutions to the
membrane equation are globally nonlinearly stable under sufficiently small compactly

3

supported perturbations. This is based on original joint work with Wong [AW19a]. We

begin by writing the perturbation equations in a convenient gauge which reveals that the

perturbed system can be described by a quasilinear perturbation of the linear wave equa-
tion on Minkowski space, with the background solution only appearing as coefficients

of the nonlinearity. In view of this special geometric feature, we do not need to develop

special methods to perform the linear analysis and can in large rely on the vector field

method approach introduced for the linear wave equation in Chapter 2. The focus is
almost entirely on the nonlinearity, with the main difficulty arising precisely from the

non-decaying background contribution.

1.2 Prelude to the vector field method

In this section, we illustrate the vector field method through the lens of a concrete ex-

ample. The notions of local and global wellposedness are also introduced and made rigor-

ous. Moreover, the analysis of this concrete example also serves to introduce the bootstrap

principle, a fundamental technical tool used in our analysis of the nonlinear kinetic model

of this section, wave map equations of Chapter 3 and the membrane equation of Chapter

4. Essentially, it allows one to assume “for free” that the solution in question already
obeys some quantitative bound which allows one to to prove, with the estimates afforded

by the vector field method, another stronger bound (to avoid circularity). See Appendix

A.1 for an abstract formulation of the bootstrap principle and the proof of Theorem 1.2.3

for a concrete application of the bootstrap principle. The reader should keep in mind that

the analysis of Chapters 2 – 4 will encompass the techniques and methods mentioned in

this prelude as a template.

1.2.1 The model

The Vlasov equation describes how a distribution of particles evolves in time. Newton’s

laws of motion assert that its evolution is determined by its initial position and velocity.

4

Consequently, we define the classical phase space Rd
x

× Rd

v as the space of all possible

configurations. A distribution of particles is then a function

ρ : Rt × Rd

x

× Rd

v

→ R+

which measures the density of the particles at a particular position x, velocity v, and time

t. The Vlasov equation then manifests itself as Newton’s first law of motion under the

simplification that particles don’t interact:

∂tρ(t, x, v) + v · ∂xρ(t, x, v) = 0.

In this chapter we prove the existence of unique global-in-time solutions for the initial

value problem

∂tρ + v · ∂xρ =

ρ(0, x, v) = ρ0(x, v)

(cid:82)
Rd ρ(t, x, v − v

(cid:48)

)ρ(t, x, v + v

(cid:48)

(cid:48)

) dv

(1.2.1)

in the small data regime whenever d ≥ 2 and small data exponential time existence for
d = 1. The nonlinearity is not chosen to be physically meaningful; rather it was cho-

sen for ease of exposition to aid in illustrating the vector field method through the lens

of a concrete example. Despite this, the nonlocal term is sensible in that it averages
particle interactions with different speeds (at the same point in space-time) over all pos-

sible velocities. In this way this model can be thought of as a poor man’s Boltzmann.

Moreover, the method of characteristics shows that a naive quadratic nonlinearity such
∞
as ∂tρ + v · ∂xρ = ρ2 would blow up in finite time for any ρ0 ∈ C
c (R2d). Finally, similar
nonlinearities are recovered after taking an appropriate transform of some well known

equations. For example, it is known that the nonlinear Schr¨odinger equation

i∂tu − ∆xu = |u|σ u

has small data global-in-time existence when d ≥ 3 and σ ∈(cid:16)
(cid:100)|u|σ ∗(cid:98)u.

i∂t(cid:98)u +|ξ|2(cid:98)u =

of (1.2.2) is precisely

(cid:17)

(1.2.2)

1, 4
d−2

. The Fourier analogue

1

d
2

(2π)

5

This shows that the nonlinearity in (1.2.1) is of a similar type as the one found in the

Fourier ODE analogue of the nonlinear Schr¨odinger equation.

1.2.2 Notational conventions

Several notational conventions used in this subsection are now established. For the classi-
cal phase space we introduce the shorthand notation R2d
x,v

v. The Vlasov operator

def= Rd
x

×Rd

will be denoted as

X def= ∂t + v · ∂x.

(1.2.3)

It is convenient to define the bilinear form

F(ρ1, ρ2) def= F(ρ1, ρ2)(t, x, v) =

(cid:90)

for functions ρ1, ρ2 : Rt × R2d
(1.2.1) can be written as

x,v

ρ1(t, x, v − v

(cid:48)

)ρ2(t, x, v + v

(cid:48)

(cid:48)

.

) dv

(1.2.4)

(cid:48)

Rd
v

→ R+. With this at hand, our nonlocal Cauchy problem

Let ρ ∈ C

∞
c (R2d

x,v) and define the norms

Xρ = F(ρ, ρ)
|ρ(x, v)| dxdv
†

ρ|t=0 = ρ0.
(cid:90)
(cid:88)

R2d
x,v

|α|≤k

R2d
x,v

|∂α
x ρ(x, v)| dxdv.

(cid:107)ρ(cid:107)
xL1
L1
v

(cid:107)ρ(cid:107)

W k,1
x L1
v

def=

def=

xL1
x,v) under the respective norms. The space L

The phase-space Lebesgue and Sobolev spaces L1
∞
c (R2d
C
measurable functions on R2d

v, W k,1
∞
x L1

x L1
v will be the completion of
v will be the space of Lebesgue

(cid:107)ρ(cid:107)

L

∞
x L1
v

x,v such that
(cid:107)ρ(x,·)(cid:107)

def= esssup
x∈Rd
x

L1(Rd
v )

= esssup
x∈Rd
x

Rd
v

6

(cid:90)

|ρ(x, v)| dv < ∞.

Finally, we also introduce the shorthand

(Y k,(cid:107)·(cid:107)

Y k ) def=

(cid:32)

W k

x L1

v,(cid:107)·(cid:107)

(cid:33)

W k,1
x L1
v

v. In the sequel, when there is no ambiguity on k, we simply denote (cid:107)·(cid:107)
xL1

and the ball centered at zero of radius R as BY k (R) := {ρ ∈ Y k | (cid:107)ρ(cid:107)
Y 0 = L1
Remark 1.2.1. In this notation, for functions ρ : Rt × R2d
(A.1.2) reads

Y k < R}. Note that
Y k = (cid:107)·(cid:107)Y .
→ R, the Sobolev inequality

x,v

(cid:107)ρ(t,—)(cid:107)

∞
x L1
L
v

(cid:46)k,d (cid:107)ρ(t,—)(cid:107)

Y k

(1.2.5)

for any k ≥ d.

1.2.3 Methodology

An initial value problem is said to be wellposed in the sense of Hadamard if there exists

a unique solution with continuous dependence on the initial data. Wellposedness results

are twofold; local wellposedness (LWP) and global wellposedness (GWP). Typical results

for the former are that for arbitrary initial data, there exists a positive time of existence.

Results for the latter are those in which the time of existence is infinite.

The main approach to prove LWP results for nonlinear evolution equations is to con-

struct approximating solutions that, in a limit, converge to a true solution. One such

method is Picard iteration which generalizes the proof of the Picard-Lindel¨of theorem.

Broadly speaking, one linearizes the problem by solving iteratively the corresponding lin-

ear equation where the nonlinearity uses the previous iterate. For semilinear equations

this can be formulated as a contraction mapping argument because the equation is linear
on the highest derivative terms and their coefficients do not depend on lower derivative

terms. For quasilinear equations this is not the case because the principal term can have
coefficients which depend on the solution itself, and as a consequence, convergence issues

are more delicate, see [H¨or97, Sog08].

7

The mechanism for the GWP results of this dissertation is the so called breakdown cri-

terion. Roughly stated, a solution exists globally if and only if its “size” (as measured by

an appropriate norm) does not diverge to infinity in finite time. The intuition behind the

breakdown criterion is that a singularity can form if the solution becomes too large to iter-

ate the approximating solutions from the LWP result indefinitely. The fundamental tool

which provides control on the solution at later times are the (approximate)-conservation
laws1. They control the solution by an integral that measures the interactions between

the linear and nonlinear dynamics. In order to show that these interactions are integrable,

and hence the solution is global as per the breakdown criterion, we rely wholeheartedly

on the decay estimates which serve to dampen their feedback.

The local and global wellposedness results for the nonlocal Vlasov model (1.2.1) are:

Theorem 1.2.2 (Local wellposedness). For any R > 0 there exists a time T > 0 such that,
for any ρ0 ∈ BY 2d+2
(R) ⊂ Y 2d+2, there exists a unique ρ ∈ C0([0, T ]; Y 2d+2) solving (1.2.1).
In addition, solutions have a Lipschitz dependence on initial data, i.e. there exists a universal

constant C > 0 such that, for any ρ0, ρ

(cid:48)
0

∈ BY 2d+2

(cid:107)ρ− ρ

(cid:48)(cid:107)

C0([0,T ];Y 2d+2)

(R), their respective solutions satisfy
≤ C(cid:107)ρ0 − ρ

(cid:107)
Y 2d+2.

(cid:48)
0

Theorem 1.2.3 (Global existence). Let ρ0 ∈ Y 2d+2 and consider the initial value problem

Xρ = F(ρ, ρ)

ρ(0, x, v) = 
ρ0(x, v).

(1.1
)

Let T 
∗ := sup{T > 0 | ∃ρ ∈ C0([0, T ]; Y 2d+2)∩ C1([0, T ]; Y 2d+1) solving (1.1
)}. Then
(a) if d ≥ 2 then there exists an 
0 > 0 depending on at most 2d + 1 derivatives of ρ0 such that

T 
∗ = ∞ for all 
 < 
0;
1In the wave equation setting, these are the energy estimates.

8

(b) if d = 1 then there exist constants B, 
0 > 0 depending on at most 3 derivatives of ρ0 such

(cid:18)B

(cid:19)




that

for all 
 < 
0.

T 
∗ ≥ sinh

1.2.4 Approximate-conservation laws

We begin by examining the inhomogeneous linear Vlasov equation

∂tρ + v · ∂xρ = F(t, x, v)

ρ(0, x, v) = ρ0(x, v).

(1.2.6)

Duhamel’s principle explicitly gives the solution as

t(cid:90)

ρ(t, x, v) = ρ0(x− tv, v) +

F(s, x + (s− t)v, v) ds.

0

From this one immediately sees that if ρ0 ∈ S (R2d) and F ∈ C
ρ ∈ C

([0, T ]; S (R2d)).

∞

∞

([0, T ]; S (R2d)), then

The vector field method described in the next subsection has its origins rooted in

commutation relations such as

[X, ∂α

x ] = 0,

(1.2.7)

where X is as in (1.2.3) and α is any multi-index. This fact can be used to prove the

([0, T ]; S (R2d)) and suppose that ρ is a solution to

(cid:107)ρ(t,—)(cid:107)

Y s ≤ (cid:107)ρ(0,—)(cid:107)

Y s +

(cid:107)F(τ,—)(cid:107)

Y s dτ.

(1.2.8)

following estimate.
Lemma 1.2.4. Let ρ0 ∈ S (R2d), F ∈ C
(1.2.6). Then

∞

t(cid:90)

for any s ∈ N∪{0}.

0

9

(cid:90)

R2d

(cid:90)
(cid:90)
(cid:90)

R2d

R2d

R2d

=

=

=

X(G(∂α

x ρ(t, x, v))) dxdv

(cid:48)

(cid:48)

G

G

(∂α

x ρ(t, x, v))X∂α

x ρ(t, x, v) dxdv

(∂α

x ρ(t, x, v))∂α

x Xρ(t, x, v) dxdv.

(cid:90)

∂t

R2d

Proof. Let G : R → R be any smooth function. Fix a multi-index α and use Stoke’s theo-
rem, the Schwarz assumption, and (1.2.7) to compute

G(∂α

x ρ(t, x, v)) dxdv = ∂t

G(∂α

x ρ(t, x, v)) dxdv +

(cid:90)
v · ∂x(G(∂α
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

x ρ(t, x, v))) dxdv

R2d

=0

R2d

(cid:90)

G(∂α

G(∂α

x ρ(t, x, v)) dxdv =

x ρ(0, x, v)) dxdv +

Integrate this equality from 0 to t to find

(cid:90)
t(cid:90)
Approximate x (cid:55)→ |x| by the smooth function G : x (cid:55)→(cid:112)
(cid:90)
(cid:90)
t(cid:90)
(cid:90)
t(cid:90)

|∂α
x ρ(0, x, v)| dxdv +

|∂α
x ρ(t, x, v)| dxdv =

(cid:90)
(cid:90)

R2d

R2d

R2d

0

R2d

≤

|∂α
x ρ(0, x, v)| dxdv +

(cid:90)

(cid:48)

G

(∂α

x ρ)∂α

x Xρ(τ, x, v) dxdvdτ.

0

R2d


 +|x|2 as 
 (cid:38) 0 to find

sgn(∂α

x ρ)∂α

x Xρ(τ, x, v) dxdvdτ

|∂α
x Xρ(τ, x, v)| dxdvdτ.

R2d

0

R2d

Summing up this inequality over all multi-indices of length s yields the claim.
Remark 1.2.5. If F ≡ 0 in (1.2.6), then (1.2.8) is the usual conservation of mass result

(cid:107)ρ(t,—)(cid:107)

Y s = (cid:107)ρ(0,—)(cid:107)

Y s.

1.2.5 Wellposedness

In this final section we provide proofs for the wellposedness results of Theorems 1.2.2

and (1.2.3). The proof of LWP will be established via a Picard approximation argument.

10

We then establish the breakdown criterion for our non-local Vlasov model and exploit it

by using the vector field method to prove global existence.

Proof of Theorem 1.2.2. Assume the initial data ρ0 is compactly supported in phase-space;
the full statement of Theorem 1.2.2 can be recovered by a density argument because
c (R2d) is dense in Y 2d+2 (which will be denoted as Y in this proof). Let ρ(−1) ≡ 0 and
∞
define {ρ(j)}∞

j=0 inductively as solutions to

C

(1.2.9)

∞

ρ(j)(0, x, v) = ρ0(x, v).
∞
We begin by showing that each ρ(j) ∈ C
c (R2d)) with an induction argument.
Note that compact support of ρ0 implies the existence of constants Rx, Rv > 0 such that
ρ0(x, v) = 0 whenever |x| > Rx or |v| > Rv. The first iterate ρ(0)(t, x, v) solves the ho-
mogeneous Vlasov equation Xρ(0) = 0 and so the explicit solution is given to be ρ(0) =
ρ0(x− tv, v) by the method of characteristics. An immediate consequence is

([0, T ]; C

Xρ(j) = F(ρ(j−1), ρ(j−1)),

ρ(0)(t, x, v) ∈ C

∞

([0, T ]; C

∞
c (R2d)),

t(cid:90)

and specifically, ρ(0)(t, x, v) = 0 whenever |v| > Rv. Assume the induction hypothesis
∞
ρ(j−1) ∈ C
c (R2d)) with ρ(j−1)(t, x, v) = 0 whenever |v| > Rv. The initial value
problem (1.2.9) is solved by Duhamel’s formula

([0, T ]; C

∞

ρ(j)(t, x, v) = ρ0(x− vt, v) +

F(ρ(j−1), ρ(j−1))(s, x + (s− t)v, v) ds.

It is clear that ρ(j) is compactly supported in the spatial variables from the induction

0

hypothesis. However, it is not clear that this is the case for the momentum variables
(cid:48). The
because ρ(j) depends on the non-local term F which is an integral over all of Rd
v
induction hypothesis states that ρ(j−1)(s, x + (s − t)v, v − v
(cid:48)| > Rv and
(cid:48)| > Rv. The Duhamel expression is then
ρ(j−1)(s, x + (s − t)v, v + v

) = 0 whenever |v + v

) = 0 whenever |v − v

(cid:48)

(cid:48)

11

equivalent to

ρ(j)(t, x, v) = ρ0(x− vt, v) +

t(cid:90)

{|v−v

0

(cid:90)
(cid:48)|≤Rv}∩{|v+v

(cid:48)|≤Rv}

ρ(j−1)(s, x + (s− t)v, v − v

(cid:48)

)

× ρ(j−1)(s, x + (s− t)v, v + v

(cid:48)

(cid:48)

ds.

) dv

The intersection {|v − v
cludes the proof of the induction.

(cid:48)| ≤ Rv} ∩ {|v + v

(cid:48)| ≤ Rv} is empty whenever |v| > Rv, and this con-

def= (cid:107)ρ0(cid:107)Y and the mass at time t as
Denote the initial mass by A0
(cid:107)∂α
x ρ(j)(t,—)(cid:107)
xL1
L1
v

(cid:88)

(cid:88)

Aj(t) def=

(cid:107)∂α
x ρ(j)(t,—)(cid:107)
Y 0

=

|α|≤2d+2

|α|≤2d+2

= (cid:107)ρ(j)(t,—)(cid:107)Y .

We claim that there exists a time T > 0, depending only on A0 and d, such that

Aj(t) ≤ 2A0

∀j and ∀t ∈ [0, T ].

The proof of (1.2.10) follows from induction. Firstly, for any of the iterates,

Aj(0) = A0.

(1.2.10)

(1.2.11)

Since ρ(0) solves the homogeneous Vlasov equation Xρ(0) = 0 with initial condition ρ0,
conservation of mass implies

A0(t) = (cid:107)ρ(0)(t,—)(cid:107)Y = (cid:107)ρ(0)(0,—)(cid:107)Y = A0(0) = A0.

Now assume that there exists a constant T > 0 depending only on the size of the initial
data and d such that (1.2.10) holds for ρ(j−1). We will derive estimates for (cid:107)∂α
x ρ(j)(t,—)(cid:107)
Y 0
for |α| ≤ 2d + 2 in order to control Aj(t) = (cid:107)ρ(j)(t,—)(cid:107)Y . Denote the volume form on R2d

x,v

12

as dV and compute with the almost conservation law (1.2.4)

(cid:107)∂α
x ρ(j)(t,—)(cid:107)

Y 0 ≤ (cid:107)∂α

x ρ(j)(0,—)(cid:107)

Y 0 +

= (cid:107)∂α

x ρ(j)(0,—)(cid:107)

Y 0 +

≤ (cid:107)∂α

x ρ(j)(0,—)(cid:107)

Y 0 +

Y 0 ds

(cid:107)∂α
x Xρ(j)(s,—)(cid:107)
†
†

(cid:12)(cid:12)(cid:12)(cid:12)∂α
x F(ρ(j−1), ρ(j−1))(s)
(cid:88)

R2d
x,v

(cid:18)|∂

F

 dvds
(cid:12)(cid:12)(cid:12)(cid:12)
x ρ(j−1)|(cid:19)

β

γ

x ρ(j−1)|,|∂

|γ|+|β|≤|α|

0

R2d
x,v

0

t(cid:90)
t(cid:90)
t(cid:90)

0




ds.

(s) dv

(1.2.12)

For the sake of clarity, we examine the term in the sum above where all of the derivatives
act on ρ(j−1)(t, x, v− v
†
By Fubini-Tonelli we can interchange the order of integration

) and consider, for a fixed v ∈ Rd

v, the change of variables v + v

†

= w.

(cid:48)

(cid:48)

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12)∂α

x ρ(j−1)(s, x,2v − w)

(cid:16)|∂α
x ρ(j−1)|,|ρ(j−1)|(cid:17)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ρ(j−1)(s, x, w)
(cid:12)(cid:12)(cid:12)(cid:12) dwdV
|ρ(j−1)(s, x, w)| dxdw.

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L

dx

∞(Rd
w)

x ρ(j−1)(s,—)(cid:107)
(cid:107)∂α
xL1
L1
v
= (cid:107)ρ(j−1)(s,—)(cid:107)
∞
x L1
v

L

x ρ(j−1)(s,—)(cid:107)
(cid:107)∂α

Y 0.

13

By H¨older’s inequality this can be controlled by

F

R2d
x,v

(cid:90)

Rd
x

≤

R2d
x,v

Rd
w

(s, x, v) dv =

†

=

R2d
x,w

(cid:90)



Rd
v

x ρ(j−1)(s, x,2v − w)| dv
|∂α

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)L

∞
w

dx

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

(cid:90)

Rd
v

x ρ(j−1)(s, x,2v −·)| dv
|∂α
(cid:90)

(cid:90)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

≤ (cid:107)ρ(j−1)(s,—)(cid:107)

Rd
x

Rd
v
∞
x L1
v

L

≤ (cid:107)ρ(j−1)(s,—)(cid:107)

L

∞
x L1
w

x ρ(j−1)(s, x,2v −·)| dv
|∂α

(cid:107)ρ(j−1)(s, x,·)(cid:107)

L1(Rd
w)

Note that we relabeled w and v in the first factor of the third inequality. The Sobolev

inequality (1.2.5) allows us to estimate the factor without any derivatives and so the in-

ductive hypothesis gives us

(cid:16)|∂α
x ρ(j−1)|,|ρ(j−1)|(cid:17)

†

F

R2d
x,v

(s, x, v) dv ≤ (cid:107)ρ(j−1)(s,—)(cid:107)

L

x ρ(j−1)(s,—)(cid:107)
(cid:107)∂α
Y 0

∞
x L1
v

≤ (cid:107)ρ(j−1)(s,—)(cid:107)Y (cid:107)∂α
x ρ(j−1)(s,—)(cid:107)
≤ 2A0(cid:107)∂α

x ρ(j−1)(s,—)(cid:107)
Y 0
Y 0.

(1.2.13)

Of course, this is only one term in the binomial expansion in (1.2.12) which comes from

x acting on the product ρ(j−1)(s, x, v − v
∂α

(cid:48)

)ρ(j−1)(s, x, v + v

(cid:48)

). In general, the terms will be a

product

γ

x ρ(j−1)(s, x, v − v

(cid:48)

x ρ(j−1)(s, x, v + v

β
)∂

(cid:48)

)

∂

with |γ| +|β| ≤ |α|. We note that at most one factor in the sum will be differentiated more
than |α|/2 times. If it is the ρ(s, x, v − v
) term, then the same analysis that led to (1.2.13)
yields†

(cid:48)

(cid:18)|∂

x ρ(j−1)|(cid:19)

β

γ

x ρ(j−1)|,|∂

(s, x, v) dv ≤ (cid:107)∂

β

x ρ(j−1)(s,—)(cid:107)

(cid:107)∂

γ

x ρ(j−1)(s,—)(cid:107)
Y 0

∞
x L1
v

L

F

R2d
x,v

≤ (cid:107)ρ(j−1)(s,—)(cid:107)Y (cid:107)∂γ ρ(j−1)(s,—)(cid:107)
Y 0
≤ 2A0(cid:107)∂

x ρ(j−1)(s,—)(cid:107)
Y 0

γ

(1.2.13’)

If instead the β derivatives satisfy |β| ≥ |α|/2, applying the change of variables v − v
yields the same result

(cid:48)

= w

(cid:18)|∂

†

F

R2d
x,v

x ρ(j−1)|(cid:19)

β

γ

x ρ(j−1)|,|∂

(s, x, v) dv ≤ (cid:107)∂

γ

x ρ(j−1)(s,—)(cid:107)

L

(cid:107)∂

β

x ρ(j−1)(s,—)(cid:107)
Y 0

∞
x L1
v

≤ (cid:107)ρ(j−1)(s,—)(cid:107)Y (cid:107)∂βρ(j−1)(s,—)(cid:107)
≤ 2A0(cid:107)∂

x ρ(j−1)(s,—)(cid:107)

β

Y 0.

Y 0

(1.2.13”)

14

We combine these results and apply them to the right hand side of (1.2.12) to find

(cid:107)∂α
x ρ(j)(t,—)(cid:107)

Y 0 ≤ (cid:107)∂α

x ρ(j)(0,—)(cid:107)

Y 0 + CA0

x ρ(j−1)(s,—)(cid:107)
(cid:107)∂

β

Y 0 ds.

t(cid:90)

(cid:88)

|β|≤|α|

0

Here C is a constant depending only on d that comes from the binomial expansion. Esti-

mate (1.2.11), the induction hypothesis, and summing up over over the multi-indices of

length less than or equal to 2d + 2 yields

Aj(t) ≤ Aj(0) + CA0
t(cid:90)

≤ A0 + CA0

0

= A0 + CA2
0t.

t(cid:90)

0

Aj−1(s) ds

2A0 ds

Choose T = 1/(CA0) to find that Aj(t) ≤ 2A0 for all t ∈ [0, T ]. This concludes the induction
and so (1.2.10) holds for all j.

Let T = 1/CA0 be the time constructed for estimate (1.2.10) for the initial data ρ0 ∈ Y
(cid:48)
with norm (cid:107)ρ0(cid:107)Y = A0. We now show that equation (1.2.1) has a unique solution on [0, T
]
for some T

< T to be determined later. Denote the balls

(cid:48)

B(R) = {ρ ∈ C0([0, T

(cid:48)

];Y) | (cid:107)ρ(cid:107)

C0([0,T

for any R > 0. Define the map Gρ0 by

Gρ0 : ρ(t, x, v) (cid:55)→ ρ0(x− tv, v) +

(cid:48)];Y) < R}
t(cid:90)

and

Y
B

(R) = {ρ ∈ Y | (cid:107)ρ(cid:107)Y < R}

F(ρ, ρ)(s, x + (s− t)v, v) ds.

(1.2.14)

One explicitly checks that

0

X(Gρ0(ρ)) = F(ρ, ρ)

Gρ0(ρ)(0, x, v) = ρ0(x, v)

and so a solution to equation (1.2.1) would be given by a fixed point Gρ0(ρ) = ρ. Estimate
(1.2.10) and a density argument show that Gρ0 is actually a map Gρ0 : B(2A0) → B(2A0).

15

Banach’s Fixed Point Theorem will show existence of a unique ρ such that Gρ0(ρ) = ρ
(cid:48) ∈ B(2A0) be arbitrary functions.
provided that Gρ0 is a contraction mapping. Let ρ, ρ
Existence and uniqueness of (1.2.1) will then follow from
≤ γ(cid:107)ρ− ρ

(1.2.15)

(cid:107)Gρ0(ρ)− Gρ0(ρ

(cid:48)(cid:107)
C0([0,T

)(cid:107)
C0([0,T

(cid:48)];Y)

(cid:48)];Y)

(cid:48)

for some 0 < γ < 1. We will prove (1.2.15) and Lipschitz dependence on initial data

simultaneously.

Pick two arbitrary ρ0, ρ

Y

(cid:48)
0

∈ B

X(Gρ0(ρ)− Gρ

(ρ)− Gρ

(cid:48)
(Gρ
0

(cid:48)
0

(A0). The function Gρ0(ρ)− Gρ

(cid:48)

(ρ

) solves

(cid:48)
0

(cid:48)
0
(ρ

(cid:48)

(cid:48)
(cid:48)
)) = F(ρ, ρ)− F(ρ
)
(ρ
(cid:48)
(cid:48)
))(0, x, v) = ρ0(x, v)− ρ
0(x, v).

, ρ

The conservation law (1.2.4) shows

(cid:107)(Gρ0(ρ)− Gρ

(cid:48)
0

(cid:48)

(ρ

))(t,—)(cid:107)Y ≤ (cid:107)ρ0 − ρ

(cid:48)
0

(cid:107)Y +

t(cid:90)

0

(cid:107)(Fi(ρ)− Fi(ρ

(cid:48)

))(s,—)(cid:107)Y ds.

(1.2.16)

For a fixed multi-index α the L1

xL1

v (= Y 0) norm of the difference ∂α

x F(ρ, ρ)− ∂α

x F(ρ

(cid:48)

(cid:48)

, ρ

) is

(cid:18)

∂

F

γ
x ρ, ∂

β

x ρ− ∂
β
x ρ

(cid:48)(cid:19)

(t) + F

(cid:18)

x ρ− ∂

γ
∂

γ
x ρ

(cid:48)

β
x ρ

, ∂

(cid:48)(cid:19)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dV .

(t)

equivalent to

(cid:12)(cid:12)(cid:12)∂α

†

R2d
x,v

x (F(ρ, ρ)− F(ρ

))(t)

(cid:12)(cid:12)(cid:12) dv
(cid:88)

(cid:48)

(cid:48)

, ρ

†

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

=

R2d
x,v

|γ|+|β|≤|α|

†
†

R2d
x,v

(cid:88)
(cid:88)

|γ|+|β|≤|α|

I2(t) def=

F

|γ|+|β|≤|α|

R2d
x,v

With this we reduce our attention to the two integrals
x ρ|,|∂

I1(t) def=

F

β

x ρ− ∂

β
x ρ

(t) dV

(cid:48)|(cid:19)
(cid:48)|(cid:19)

γ

x ρ|,|∂

β
x ρ

(t) dV .

γ

(cid:18)|∂
(cid:18)|∂γ ρ− ∂

16

(cid:48)

β

x ρ − ∂
β
x ρ

We note that when |γ| ≥ |α|/2, the Sobolev inequality bounds the L
(cid:48)
ference terms ∂
Y norms, and vice-versa when |β| ≥ |α|/2. Keeping the difference terms ρ − ρ
tegral norms and bounding the monomial terms of ρ, ρ
G : B(2A0) → B(2A0) shows

∞
x L1
v norm of the dif-
terms in I2(t) by their
in the in-
by their Y norms, and using

in I1(t) by their Y 0 norms and the ∂

β
x ρ

(cid:48)

(cid:48)

I1(t) + I2(t) ≤ C

(cid:88)

|β|≤|α|

(cid:107)(∂
(cid:88)

(cid:48)

β

x ρ− ∂
(cid:107)(∂

β
x ρ
x ρ− ∂

)(t,—)(cid:107)
L1
xL1
v
)(t,—)(cid:107)

β
x ρ

β

(cid:48)

Y 0.

(2A0 + 2A0)

= 4CA0

|β|≤|α|

Here C is a constant depending only on d that comes from the binomial expansion. We

apply this estimate to (1.2.16) to find

(cid:107)(Gρ0(ρ)− Gρ

(cid:48)
0

(cid:48)

(ρ

))(t,—)(cid:107)Y ≤ (cid:107)ρ0 − ρ

(cid:48)
0

(cid:107)Y +

≤ (cid:107)ρ0 − ρ

(cid:48)
0

(cid:107)Y +

≤ (cid:107)ρ0 − ρ

(cid:48)
0

(cid:107)Y +

0

t(cid:90)
t(cid:90)
t(cid:90)

0

(cid:107)(F(ρ, ρ)− F(ρ
(cid:88)

4CA0

|β|≤|α|,
|α|≤2d+2
(cid:48)
4CA0(cid:107)(ρ− ρ

(cid:48)

(cid:48)

))(s,—)(cid:107)Y ds

, ρ

(cid:107)(∂

β

x ρ− ∂
β
x ρ

(cid:48)

)(s,—)(cid:107)

Y 0 ds

)(s,—)(cid:107)Y ds.

(1.2.17)

Note that if ρ0 = ρ

(cid:48)
0 the first term in inequality (1.2.17) vanishes. Then

(cid:107)(Gρ0(ρ)− Gρ0(ρ

(cid:48)

))(t,—)(cid:107)Y ≤ 4CA0

(cid:107)(ρ− ρ

(cid:48)

)(s,—)(cid:107)Y ds
t(cid:90)

C0([0,T

(cid:48)];Y)

C0([0,T

(cid:48)];Y)t.

1 ds

0

0

t(cid:90)

0

≤ 4CA0(cid:107)ρ− ρ

(cid:48)(cid:107)

≤ 4CA0(cid:107)ρ− ρ

(cid:48)(cid:107)

(cid:48)

< 1

17

Choose the time of existence to be controlled by T
bounded by 1
2

(cid:48)(cid:107)
C0([0,T ];Y). This does not depend on t ∈ [0, T

(cid:107)ρ − ρ

8CA0

(cid:48)

. Then the right hand side is

] so we can take the

supremum to find

(cid:107)Gρ0(ρ)− Gρ0(ρ

(cid:48)

)(cid:107)
C0([0,T

(cid:48)];Y )

≤ 1
2

(cid:107)ρ− ρ

(cid:48)(cid:107)
C0([0,T ];Y).

This concludes the proof of (1.2.15) and so the fixed point Gρ0(ρ) = ρ is the unique solu-
tion of (1.2.1).

Now suppose that ρ0 (cid:44) ρ

(cid:48)
0 with ρ0, ρ

(cid:48)
0

Y

∈ B

be the two solutions of (1.2.1)

with these respective initial conditions. Applying Gronwall’s inequality to (1.2.17) shows

(cid:107)(ρ− ρ

(cid:48)

)(s,·)(cid:107)Y ≤ (cid:107)ρ− ρ

(cid:48)
0

4CA0 ds



(cid:48)

(A0). Let ρ, ρ

(cid:107)Y exp

 t(cid:90)
(cid:18) 4CA0
(cid:19)
(cid:107)Y exp(4CA0t)
(cid:107)Y exp
(cid:48)
(cid:107)Y .
0

8CA0

0

(cid:48)
= (cid:107)ρ− ρ
0
(cid:48)
≤ (cid:107)ρ− ρ
0
≤ C(cid:107)ρ− ρ

Taking the supremum over [0, T
(cid:107)ρ− ρ

] shows
(cid:48)(cid:107)
C0([0,T

(cid:48)

(cid:48)];Y)

≤ C(cid:107)ρ0 − ρ

(cid:48)
0

(cid:107)Y .

This concludes the proof of Lipschitz dependence of initial data and the theorem.
Remark 1.2.6. We have shown that, given a ρ0 ∈ Y 2d+2, there exists a T > 0 depending
on the size of ρ0 such that (1.2.1) has unique solution ρ ∈ C0([0, T ]; Y 2d+2) with ρ0 as its
initial value. That is, ρ satisfies

∂tρ + v · ∂xρ = F(ρ, ρ).

The function ρ is actually in C0([0, T ]; Y 2d+2)∩ C1([0, T ]; Y 2d+1) because

∂tρ = −v · ∂xρ + F(ρ, ρ)

and the right hand side is in C0([0, T ]; Y 2d+1).

With local wellposedness established, we now introduce a key technical tool used for

proving global existence: the breakdown criterion.

18

Theorem 1.2.7 (Breakdown criterion). Define

T∗ := sup{T > 0 | ∃ρ ∈ C0([0, T ]; Y 2d+2) solving (1.2.1)}.

(1.2.18)

Then either T∗ = ∞ or (cid:88)
(1.2.19)
Proof. The proof follows by a contradiction argument. If T∗ = ∞ then there is nothing to
prove so suppose that T∗ < ∞. Assume that (1.2.19) does not hold. Then we claim that

(cid:107)∂α
x ρ(t, x,·)(cid:107)
L1
v(Rd)

([0, T∗)× Rd
x).

|α|≤d+1

∞
(cid:60) L

(cid:88)

A0

def= sup
t∈[0,T∗)

|α|≤d+1

< ∞

∞
x L1
v

(1.2.20)

L

implies

A(t) def= (cid:107)ρ(t,—)(cid:107)

(1.2.21)
for all t ∈ [0, T∗). Fix a multi-index α with length |α| ≤ 2d + 2. The conservation of mass
estimate (1.2.8) gives

|α|≤2d+2

Y 2d+2 =

xL1
L1
v

x ρ(t,—)(cid:107)
(cid:107)∂α

< ∞

(cid:107)∂α
x ρ(t,—)(cid:107)
(cid:88)

t(cid:90)

The term in the integral is explicitly

Y 0 +

(cid:107)∂α
x Xρ(s,—)(cid:107)

Y 0 ds.

(1.2.22)

(cid:107)∂α
x ρ(t,—)(cid:107)
†

(cid:12)(cid:12)(cid:12)∂α

x ρ(0,—)(cid:107)
†

Y 0 ≤ (cid:107)∂α
(cid:12)(cid:12)(cid:12) dv ≤

x F(ρ, ρ)(s, x, v)

R2d
x,v

R2d
x,v

0

(cid:88)

|γ|+|β|≤|α|

(cid:18)|∂

F

x ρ|(cid:19)

β

γ

x ρ|,|∂

dV .

Then either |γ| ≥ |α|/2 or |β| ≥ |α|/2 in each term of the sum. In the former case, the change
†
of variables v + v

= w and the same analysis as in (1.2.13’) shows
x ρ(s,—)(cid:107)
xL1
L1
v

x ρ(s, x, v − v

(cid:107)∂
x ρ(s,—)(cid:107)

β
x ρ(s, x, v + v

dV ≤ (cid:107)∂

∞
x L1
v

.

)| dv

)||∂

(cid:90)

|∂

γ

γ

L

β

(cid:48)

(cid:48)

(cid:48)

(cid:48)

R2d
x,v

(cid:48)

Rd
v

†
In the latter case the change of variables v − v
)| dv

x ρ(s, x, v − v

β
x ρ(s, x, v + v

)||∂

(cid:90)

|∂

γ

(cid:48)

(cid:48)

(cid:48)

(cid:48)

R2d
x,v

(cid:48)

Rd
v

19

= w yields
dV ≤ (cid:107)∂

β

x ρ(s,—)(cid:107)
L1
xL1
v

(cid:107)∂

γ

x ρ(s,—)(cid:107)

L

∞
x L1
v

.

norm will be differentiated at most d + 1 times
In both instances, the factor in the L
because |α| ≤ 2d + 2. The assumption (1.2.20) allows us to bound these factors by A0.
Applying this estimate to (1.2.22), summing over all multi-indices, and using Gronwall’s

∞

inequality yields(cid:88)
(cid:107)∂α
x ρ(t,—)(cid:107)

Y 0 ≤ (cid:88)

|α|≤2d+2

t(cid:90)

0

(cid:88)

|β|≤|α|

(cid:107)∂

β

x ρ(s,—)(cid:107)

Y 0 ds

Y 0 + A0

|α|≤2d+2
A(t) ≤ A(0) + A0

A(t) ≤ A(0)exp

(cid:107)∂α
x ρ(0,—)(cid:107)
t(cid:90)
 t(cid:90)

A(s) ds

A0 ds



0

0

= A(0)exp(tA0) ≤ A(0)eT∗A0
< ∞.

The last inequality follows from the assumption T∗ < ∞ and the initial data assumption
ρ0 ∈ Y 2d+2. This concludes the proof of the claim (1.2.21). Now let {tn} be an increasing
sequence such that tn (cid:37) T∗ as n → ∞. The claim shows that ρ(tn,—) can be taken as an
initial condition for the Cauchy problem and so Theorem 1.2.2 shows that there exists a

small T > 0 such that a solution exists on [tn, tn+T ]. The uniqueness of the solution shows
that ρ can be extended past T∗, contradicting the definition of supremum.

1.2.6 Vector field method in action

It remains to show global existence for equation (1.2.1). In order to exploit the breakdown

criterion, we need to show

(cid:88)

|α|≤d+1

sup
t∈[0,T∗)

(cid:107)∂α
x ρ(t,—)(cid:107)

L

∞
x L1
v

< ∞,

(1.2.23)

20

where ρ : [0, T∗) × R2
inequality states

x,v

→ R+ is a solution to (1.2.1) and T∗ is as in (1.2.18). The Sobolev
(cid:88)

(cid:107)∂α
x ρ(t,—)(cid:107)

∞
x L1
L
v

≤ Cd(cid:107)ρ(t,—)(cid:107)

Y 2d+1

for all t ∈ [0, T∗). This shows that getting a bound

|α|≤d+1

(cid:107)ρ(t,—)(cid:107)

Y 2d+1 ≤ C

(1.2.24)

which is uniform in time will prove (1.2.23) and, as a consequence of the breakdown
criterion of Theorem 1.2.7, show that T∗ = ∞.

This approach of constructing a global solution is akin to the proof of local wellposed-

ness in Theorem 1.2.2. Estimate (1.2.24) which is uniform in time is analogous to the

uniform estimate (1.2.10); the proof of which was an induction argument. In the global

existence scenario the induction argument manifests itself as a bootstrapping argument

where we will show that the desired bound holds on a non-empty open and closed subset
of [0, T∗) (and hence on all of [0, T∗) itself). Appendix A.1 shows that, in fact, the bootstrap
mechanism can be thought of as a continuous induction argument.

A fundamental tool in closing the induction for estimate (1.2.10) was the the commu-

− ∂α

x ] = X∂α
x

tation relation [X, ∂α

x X = 0. This identity immediately revealed the Vlasov-
type equation solved by the derivatives ∂α
x ρ, to which the approximate-conservation of
mass result (1.2.4) can apply. Indeed, differentiating the equation that the Picard iterates

solved produced an equation for their derivatives:

X∂α

x ρ(j) = ∂α

x Xρ(j) = ∂α

x F(ρ(j−1), ρ(j−1)) =

(cid:88)

(cid:18)

F

γ

x ρ(j−1), ∂

β

x ρ(j−1)

∂

(cid:19)

.

|γ|+|β|=|α|

Careful analysis of the right hand side involving the conservation estimate (1.2.8), the

Sobolev inequality (A.1.2), and the induction hypothesis led to (1.2.10).

The vector field method is based on finding a collection of temporally weighted vector
fields {W} that preserve the linear part of (1.2.1); they too satisfy the commutation rela-
tion [X, W α] = XW α − W αX = 0. In direct analogy with the LWP scenario, this produces an

21

equation for the W -derivatives of ρ:

XW αρ = W αXρ = W αF(ρ, ρ) =

(cid:88)

|γ|+|β|=|α|

F

(cid:16)

W γ ρ, W βρ

(cid:17)

.

This corresponds to Point I of the general strategy for the vector field method introduced

in the Prologue of this chapter.

A second key step in the vector field method is to use the temporal weight of {W}
Sobolev inequality (A.1.2) to obtain an estimate that reflects the dis-

∞
and the L2—L

persive decay that the density satisfies, see Proposition 1.2.10 and Remark 1.2.8. Our

approach for proving this estimate is adapted from Smulevici et al [Smu16, FJS17]. To

control the terms F(W γ ρ, W βρ) that appear in the conservation law, the decay estimate is

supplemented with the aforementioned bootstrap assumption, where we assume that the

estimate (1.2.24) holds. To avoid circularity, we use the temporal decay to show that the

right hand side of estimate (1.2.24) can be replaced with C/2. This is in direct analogy with
the LWP scenario, where the inductive hypothesis of Aj−1 was used to show that it also
holds for Aj. This analysis is a nonlinear modification of Points II and III introduced in
the Prologue.

Remark 1.2.8. In fact, the pointwise decay in time will be proven for the spatial density

(cid:82)

Rd
v

ρ(t, x) =

ρ(t, x, v) dv. That this is available for the integral over velocity arises from the

intuition that dispersion occurs because the physical extent of the particles spreads out

while the total mass is conserved. With the same mass divided among a greater volume,

the spatial density (which is what ρ measures) must decay.

1.2.6.1 The weighted vectorfields

Galilean relativity is the physical theory that asserts that the the laws of motion pre-

scribed by Newtonian mechanics are the same in all inertial frames. Newtonian interac-

tion of particles reduces to, under the simplification that individual particles don’t inter-

22

act, Newton’s first law of motion: the Vlasov equation

∂tρ(t, x, v) + v · ∂xρ(t, x, v) = Xρ = 0.

The Galilean boost transformation of ρ(t, x, v) is, for any w ∈ Rd, the change of inertial
frame defined by

ρw(t, x, v) := ρ(t, x + tw, v + w).

(1.2.25)

This action is continuously parametrized by w and acts as the identity when w = 0. An ex-

plicit computation shows that if ρ solves the linear Vlasov equation then ρw does indeed
solve it as well:

X(ρw) = 0.

Assuming the family ρw is differentiable in w, the linearity of the Vlasov operator X
implies that dρw

also solves the Vlasov equation:

(cid:48)

dw at any w = w

(cid:32)dρw

dw

(cid:33)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)w=w

(cid:48)

X

= 0.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)w=ei

Taking w = ei as the basis element in Rd we find dρw
dw

= t∂xi ρ+∂vi ρ. We hence define

Wi := t∂xi + ∂vi

(1.2.26)

for i = 1, . . . , d, which are the infinitesimal generators of the Galilean boosts (1.2.25). This

construction of the Wi shows that they satisfy the commutation relation

[X, Wi] = 0, ∀i = 1, . . . , d.

(1.2.27)

αd
α1
Let α = (α1, . . . , αd) be any multi-index and denote W α = W
d . Then (1.2.27) and
1
an induction argument show [X, W α] = 0 for all multi-indices. These computations, and

··· W

the t-weight in each Wi, imply that they are good candidates for getting pointwise time
decay for the Vlasov equation. Denote

(cid:90)

ρ(t, x) :=

ρ(t, x, v) dv

(1.2.28)

Rd
v

23

as the spatial density of ρ. The following lemma is the first step in proving our desired

result.
Lemma 1.2.9. Let ρ ∈ C

x,v) and suppose t > 0. Then

∞
c (Rt×2d
|ρ(t, x)| ≤ 1
td

(cid:88)

(cid:90)

|α|≤d

R2d
y,v

|W αρ(t, y, v)| dydv.

(1.2.29)

Proof. For x = (x1, . . . , xd) ∈ Rd, define the infinite rectangle
Rx := {y ∈ Rd | yi ≤ xi ∀i = 1, . . . d}.

The compact support of ρ implies

ρ(t, x) =

=

Rx

The compact support also tells us(cid:90)

(cid:90)
(cid:90)

Rx

Rd
v

∂x1 ··· ∂xd ρ(t, y) dy
(cid:90)
∂x1 ··· ∂xd ρ(t, y, v) dydv.

Rd
v

∂vi ρ(t, x, v) dv = 0

for all i = 1, . . . , d. We then arrive to

|ρ(t, x)| =

(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:90)
(cid:90)

Rd
v

Rx
≤ 1
td

R2d
y,v

∂x1 ··· ∂xd ρ(t, y, v) dydv

1
td
|W1··· Wdρ(t, y, v)| dydv ≤ 1
td

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

W1··· Wdρ(t, y, v) dydv
(cid:90)

|W αρ(t, y, v)| dydv.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:90)
(cid:90)
(cid:88)

Rd
v

Rx

|α|≤d

R2d
y,v

This is the desired inequality so the proof is complete.

We note that the proof of Lemma 1.2.9 provided a sharper estimate than (1.2.29); we
had bounded |ρ(t, x)| with the same decay but each Wi was used only once. We also note

24

that the estimate is not useful for t = 0 because the right hand side blows up as t (cid:38) 0.
We now show how to overcome this hurdle. Let 0 ≤ t ≤ 1 and define the Japanese bracket
(cid:104)t(cid:105) =

2 so this immediately implies that (cid:104)t(cid:105)−d ≥ 2

−d/2. Then

√

1 + t2. Then (cid:104)t(cid:105) ≤ √
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:90)
(cid:90)
≤ (cid:88)

|ρ(t, x)| =

Rd
v

Rx

∂x1 ··· ∂xd ρ(t, y, v) dydv
(cid:90)
(cid:88)

|∂α
x ρ(t, y, v)| dydv =
(cid:90)

R2d
y,v

|∂α
x ρ(t, y, v)| dydv.

2d/2
2d/2

|α|≤d

R2d
y,v

|α|≤d
≤ C(cid:104)t(cid:105)d

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤

(cid:90)
|∂x1 ··· ∂xd ρ(t, y, v)| dydv
(cid:90)
(cid:88)

|∂α
x ρ(t, y, v)| dydv

R2d
y,v

|α|≤d

R2d
y,v

(1.2.30)

This estimate fixes the weakness of Lemma 1.2.9. Consider the collection {∂xi , Wi} and
denote Γ as an arbitrary element of this collection. Define Γ α in the natural way. Then we

have proved the following.
Proposition 1.2.10 (Pointwise decay for Vlasov). Let ρ ∈ C
there exists a constant C depending only on d such that

∞
c (Rt × R2d

x,v) and t ≥ 0. Then

|Γ αρ(t, y, v)| dydv.

(1.2.31)

|ρ(t, x)| ≤ C(cid:104)t(cid:105)d

(cid:88)

(cid:90)

|α|≤d

R2d
y,v

Remark 1.2.11. We note that the proof for (1.2.31) does not need the full power of ρ ∈
x,v). Our theorem will have ρ ∈ C0([0, T ]; Y s) ∩ C1([0, T ]; Y s−1) for some s so
∞
c (Rt × R2d
C
taking spatial derivatives, at least in the weak sense, is valid so long as s is large enough.

However, we do not have any regularity in v from our LWP theorem. The upshot is that

we do not need it; we only need that ρ is well defined on integral curves of Wi so that the
derivatives Wiρ makes sense.

25

1.2.7 Global existence

Proof of Theorem 1.2.3. We present the proof of (a) first. Assume the spatial dimension
satisfies d ≥ 2. Theorems 1.2.2 and 1.2.7 show that T 
∗ > 0 for every 
 > 0. We need to
show that T 
∗ = ∞ for small enough initial data. Assume that T 
∗ < ∞ for all 
 > 0. Then
Theorem 1.2.7 implies (cid:88)

∞

([0, T 
∗ )× Rd
x).

(cid:60) L

We will arrive to a contradiction by showing that there exists a small 
0 > 0 such that

|α|≤d+1

x ρ(t, x,·)(cid:107)
(cid:107)∂α
L1
v(Rd)
(cid:88)

sup

(cid:107)∂α
x ρ(t, x,·)(cid:107)
L1

v(Rd) < ∞

(1.2.32)

|α|≤d+1
for all 
 < 
0. Define the weighted energy as

(t,x)∈[0,T 
∗ )×Rd
x

(cid:88)

A(t) :=

|α|≤2d+1

(cid:107)Γ αρ(t,—)(cid:107)
L1
xL1
v

.

We will prove (1.2.32) by showing that

A(t) ≤ A
,

0 ≤ t ≤ T 
∗

(1.2.33)

|α|≤d+1

for small enough 
 > 0 and some A > 0 that is independent of time. First we show why
(1.2.33) implies (1.2.32). Fix a multi-index α of length |α| ≤ d + 1. Then the pointwise
estimate (1.2.31) shows(cid:88)
(cid:107)∂α
x ρ(t, x,·)(cid:107)
L1
v(Rd)

≤ (cid:88)
|α|≤d+1
≤ C(cid:104)t(cid:105)d
≤ C(cid:104)t(cid:105)d
Taking the supremum over all (t, x) ∈ [0, T 
∗ )×Rd
x proves (1.2.32) and the theorem. Hence,
it suffices to show that there exists an 
0 > 0 such that (1.2.33) holds for all 
 < 
0. This is
done by bootstrapping.

(cid:107)Γ αρ(t, x,·)(cid:107)
(cid:88)
(cid:88)

|β|≤d
|α|≤d+1
A(t) ≤ CA
(cid:104)t(cid:105)d

(cid:107)Γ βΓ αρ(t,—)(cid:107)
L1
xL1
v

< ∞.

L1
v(Rd)

(1.2.34)

26

We note that the initial energy A(0) depends only on the initial data. Hence we can

find a large constant A depending on 2d + 1 derivatives of ρ0 (but is independent of 
)
such that A(0) ≤ A
. Define the set

E
 := {T ∈ [0, T 
∗ ) | A(t) ≤ A
 ∀0 ≤ t ≤ T}.

(1.2.35)
This set is not empty because 0 ∈ E
 by construction. Moreover, A(t) ∈ C([0, T 
∗ )) so E
 is
also closed. By the continuity principle, (1.2.33) will be proved if we show that E
 is open
because it would imply E
 = [0, T 
∗ ).

Let t0 ∈ E
. Continuity of A(t) shows that there exists a T

(cid:48)

> t0 such that

(cid:48)

|A(T

)− A(t0)| ≤ A
,

and so this immediately implies

A(t) ≤ 2A
,

0 ≤ t ≤ T

(cid:48)

.

(1.2.36)
We show that A(t) ≤ 2A
 actually implies the sharper bound A(t) ≤ A
 for small enough

. This would imply that E
 is open, as we hoped. Let α be a fixed multi-index of length
|α| ≤ 2d + 1. The commutation relation XΓ α = Γ αX and the proof for (1.2.8) immediately
imply

t(cid:90)

Y 0.

(1.2.37)

The integral term is precisely

(cid:107)Γ αρ(t,—)(cid:107)

(cid:12)(cid:12)(cid:12)Γ αF(ρ, ρ)(s, x, v)

†

R2d
x,v

Y 0 ≤ (cid:107)Γ αρ(0,—)(cid:107)
†
(cid:12)(cid:12)(cid:12) dv =

Y 0 +

0

(cid:88)

(cid:107)Γ αXρ(s,—)(cid:107)
(cid:17)
(cid:16)

Γ γ ρ, Γ βρ

F

|γ|+|β|≤|α|

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

R2d
x,v

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dv.

(s, x, v)

Then either |γ| ≥ |α|/2 or |β| ≥ |α|/2. In the former case, we do the change of variables
= w, use the same analysis as in (1.2.13), apply the pointwise decay estimate (1.2.31),
v+v

(cid:48)

27

†

F

R2d
x,v

†

F

R2d
x,v

and the bootstrapping assumption to find

(cid:16)|Γ γ ρ|,|Γ βρ|(cid:17)

(s, x, v) dv ≤ (cid:107)Γ γ ρ(s,—)(cid:107)
L1
xL1
v

∞
x L1
v

L

(cid:107)Γ βρ(s,—)(cid:107)
(cid:88)

|δ|≤d+|β|

(cid:107)Γ δρ(s,—)(cid:107)
L1
xL1
v

≤ C(cid:104)s(cid:105)d
≤ 2CA

(cid:104)s(cid:105)d

(cid:107)Γ γ ρ(s,—)(cid:107)
L1
xL1
v
(cid:107)Γ γ ρ(s,—)(cid:107)
(cid:48)

Y 0.

In the latter case the change of variables v − v

= w and a similar argument yields

(cid:16)|Γ γ ρ|,|Γ βρ|(cid:17)

(s, x, v) dv ≤ (cid:107)Γ βρ(s,—)(cid:107)
xL1
L1
v

∞
x L1
v

L

(cid:107)Γ γ ρ(s,—)(cid:107)
(cid:88)

|δ|≤d+|γ|

(cid:107)Γ δρ(s,—)(cid:107)
xL1
L1
v

≤ C(cid:104)s(cid:105)d
≤ 2CA

(cid:104)s(cid:105)d

(cid:107)Γ βρ(s,—)(cid:107)
xL1
L1
v
(cid:107)Γ βρ(s,—)(cid:107)

Y 0.

These estimates show that

Y 0 ≤ (cid:88)

|β|≤|α|

(cid:107)Γ αXρ(s,—)(cid:107)

2CA

(cid:104)s(cid:105)d

(cid:107)Γ βρ(s,—)(cid:107)

Y 0

for all fixed indeces. We put this bound into (1.2.37) and sum over all multiindices α to

find (cid:88)

|α|≤2d+1

(cid:107)Γ αρ(t,—)(cid:107)

(cid:88)

|α|≤2d+1

t(cid:90)

0

(cid:107)Γ αXρ(s,—)(cid:107)

Y 0 ds

2CA

(cid:104)s(cid:105)d

(cid:107)Γ βρ(s,—)(cid:107)
xL1
L1
v

ds

Y 0 ≤ (cid:88)

|α|≤2d+1
≤ A(0) +

≤ A(0) +

t(cid:90)

0

(cid:107)Γ αρ(0,—)(cid:107)
(cid:88)
(cid:88)

Y 0 +

t(cid:90)

|α|≤2d+1

|β|≤|α|

0

2CA

(cid:104)s(cid:105)d

A(s) ds.

28

We can finally apply Gronwall’s inequality to see

ds


 .

2CA

(cid:104)s(cid:105)d

2CA

(cid:104)s(cid:105)d

ds

 t(cid:90)
 ∞(cid:90)

0

0

A(t) ≤ A(0)exp

≤ A
 exp

(1.2.38)

Choose 
0 > 0 small enough such that the exponential term is bounded by 1, which can
be done because (cid:104)s(cid:105)−d is integrable for d ≥ 2. Then (1.2.38) implies A(t) ≤ A
 for all 
 < 
0
and the proof of (a) is complete.

We now focus our attention on (b) of Theorem 1.2.3. Assume that d = 1 and that ρ is a

solution to (1.1
) on [0, T ]. The proof of Theorem 1.2.7 shows that T can be extended by
a positive value so long as

(cid:88)

sup
t∈[0,T )

|α|≤d+1

v(R2d) < ∞.
∞
x L1

L

x ρ(t,—)(cid:107)
(cid:107)∂α
(cid:18)B

T 
 = sinh

(cid:19)




We prove the desired lower bound for T 
∗ by showing that T can be extended until

for some B and 
 to be determined later. As in the proof of (1.2.34), this will be accom-

plished by showing that

A(t) ≤ A
,

∀0 ≤ t ≤ T 


(1.2.39)

where A (which depends on three derivatives of the initial data, but is independent of 
)
is as in A(0) ≤ A
. Define the set

E
 := {T ∈ [0, T 
) | A(t) ≤ A
 ∀0 ≤ t ≤ T}.

This set is not empty because 0 ∈ E
 by construction. Moreover, A(t) ∈ C([0, T 
)) so E
 is
also closed. Let t0 ∈ E
. Continuity of A(t) shows that there exists a T

> t0 such that

(cid:48)

A(t) ≤ 2A
,

0 ≤ t ≤ T

(cid:48)

.

(1.2.40)

29

This is the bootstrapping assumption. As in the proof of (a), we show that (1.2.40) implies
A(t) ≤ A
 for 0 ≤ t ≤ T
[0, T 
) by the continuity principle.

. This would imply the openness of E
 and consequently E
 =

(cid:48)

Choose B > 0 small enough so that exp(2CAB) < 1 and 
0 > 0 small enough so that
(cid:48) ≤ T 
 for all 
 < 
0. The same computations that led to (1.2.38) then imply

T

 .

2CA
(cid:104)s(cid:105) ds

(1.2.41)

We now use the explicit form of T 
 to evaluate the integral

A(t) ≤ A(0)exp

t(cid:90)

0

2CA
(cid:104)s(cid:105) ds ≤ 2CA


 t(cid:90)

0

 t(cid:90)
2CA
(cid:104)s(cid:105) ds
T 
(cid:90)

0

 ≤ A
 exp
T 
(cid:90)

0

1(cid:104)s(cid:105) ds = 2CA

−1(t)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)t=T 


t=0

0

= 2CA
 sinh

= 2CA
 sinh

ds

1√
1 + s2
−1

(cid:18)

(cid:18)B

(cid:19)(cid:19)




sinh

= 2CAB.

We conclude from these computations and our choices of B, 
0 that

A(t) ≤ A
 exp(2CAB) < A


(cid:48)

for all 0 ≤ t ≤ T
and the theorem.

. This concludes the improvement of the bootstrapping assumption, (b),

1.3 Main results of the current work

We now give a very brief summary of how the results of dissertation tie into a rapidly

advancing research program of wave-type equations. In Subsections 1.3.2 and 1.3.3, we

give a slightly more detailed overview of the results; we refer the reader to Chapters 3

and 4 for further details. The reader should keep in mind that the key a priori estimates

needed for the analysis are proved using the vector field method.

30

1.3.1 Overview

The current work uses the commuting vector field method to analyze the long-time be-
havior of solutions to nonlinear hyperbolic partial differential equations (PDEs). Our

approach is to take a non-trivial symmetric “background” solution, add a symmetry-

breaking perturbation, and study its dynamical stability on the whole space-time. Here

“stability” of the background is quantified by deriving uniform decay estimates for the

perturbations, showing that the inherent symmetries survive in the long-time asymp-

totics.

The classical analytic framework for wellposedness of general equations views the

solutions to the Cauchy initial value problem as perturbations of a trivial constant back-

ground solution [H¨or97,Tao06,Sog08]. The recent breakthrough works of Christodoulou

[Chr07, Chr09] have led to new analytical techniques that can account for non-trivial

symmetric background solutions. For example, in [Chr09], he takes a spherically sym-

metric solution to the Einstein field equations which forms a black hole (which he proved

exists in [Chr91]), and adds to it a symmetry-breaking perturbation. He is able to get quan-

titative control of the perturbation by exploiting special structures (or null conditions in

the literature) of the equations which cancel harmful terms that would compromise the

dynamics. With this control, he is able to show that the lower dimensional behavior is

stable for subsequent times.

In a similar vein, the stability properties of some affine totally geodesic maps from

Minkowski space to a spaceform are studied in Chapter 3. Viewing our background map

as a one dimensional ODE solution of the wave map equation, we perturb the homogene-

ity to prove:

Theorem 1.3.1 (Very rough version of Theorem 1.3.5). Totally geodesic maps from Minkowski
space R1,3 to a spaceform (M, g) are globally stable solutions to the wave map equation.

Using the spaceform structures of the target, we prove that the equations of motion for

31

the perturbation decouple into a nonlinear system of wave-Klein–Gordon equations sat-

isfying a certain weak null condition. Using the technical weighted Sobolev interpolation

estimates proved in Chapter 2, we prove global existence for this system and derive uni-

form decay properties of the perturbation, showing that the homogeneous ODE picture

is stable.

Quantitative control of perturbations by exploiting null conditions is also the strat-

egy taken by Christodoulou in his monograph [Chr07]. There he proves stability of a

background solution of the compressible Euler equations that exhibits shock singulari-

ties akin to the (1+1)-dimensional Burgers’ equation. By the 1970s [Lax64,Lax73,Joh74],
this Burgers-type shock singularity was known to occur for planewave2 solutions to equa-

tions satisfying Lax’s resonant genuinely nonlinear condition. The shock mechanism has

recently been shown [SHLW16] to be a stable phenomenon in the above sense for such

equations. In contrast, global-in-time planewave solutions have been constructed for the

relativistic membrane equation [Lin04, Won17a]. This was possible because the mem-

brane equation enjoys extremely strong null conditions which cancel those terms that

would exhibit the resonant condition of Lax, precluding singular behavior. One can sim-

ilarly ask whether the global results of this lower dimensional phenomena are stable.

The main result of Chapter 4, adapted from original joint work [AW19a], answers this
question in the affirmative. We show:

Theorem 1.3.2 (Very rough version of Theorem 1.3.6). Simple planewave solutions to the

membrane equation are globally stable under symmetry-breaking perturbations.

Remark 1.3.3 (The intimate role played by null conditions). Klainerman [Kla86] and
Christodoulou [Chr86] independently discovered a sufficient algebraic condition on the
nonlinearities of wave equations on R1+3 that guarantees global-in-time existence for
sufficiently small data. A physical interpretation of the null condition is that it prevents

nonlinear interactions between wave packets traveling along the same null geodesic. That
2Planewave solutions are those whose dynamics are only nontrivial in one direction.

32

this has an effect on the large-time behavior makes sense in view of the fact that wave

packets that are localized around the null geodesic can interact for an arbitrary amount
of time, in contrast to those which are traveling in different directions; this latter behavior

is guaranteed by the classical null condition of Klainerman and Christodoulou [Kla86,Chr86].

The equations studied in this dissertation (wave maps, membrane equation) both sat-

isfy the classical null condition. However, since we are considering initial perturba-

tions of large background solutions, we can no longer appeal to the small-data results

of [Kla86, Chr86].

Instead, crucial to our analysis is the existence of “vestigial” null

conditions” that survive the perturbations, see the discussions surrounding (1.3.4) and

(1.3.9).

Remark 1.3.4 (Physical Relevance). The equations studied in this dissertation describe

the dynamics of physical models (i.e. membranes [Neu90, Jer11], nonlinear sigma mod-

els [GML60,CBM96], etc). Historically, mathematicians sought to provide rigorous math-

ematical analysis of these physical models by restricting their attention to symmetrically

reduced solutions. Although symmetric regimes are of mathematical interest in them-

selves, the extent of their applications is limited in that physical phenomena is never truly

symmetric. As is evident by the monumental works describing the stability of Minkowski

space [CK93, LR05, BZ09], shock formation in compressible fluids [Chr07, CM14, LS18],
and formation of black holes [Chr09, AL14], there is an extensive gap in difficulty in

passing from the case of symmetric solutions to those without symmetry. Hence, this

dissertation serve to add to this rapidly advancing and physically relevant sub-field of

PDEs.

1.3.2 Totally geodesic wave maps
A map φ : N → M between two pseudo-Riemannian manifolds (N , h) and (M, g) is said
to be totally geodesic if it maps geodesics in N to geodesics in M. This is characterized by
the total vanishing of the fundamental form of φ, namely the tensor ∇∗

dφ. Here, ∇∗

is the

33

pull-back of the Levi-Civita connection ∇M onto N . An immediate consequence of this is
that totally geodesic maps are critical points of the energy functional

(cid:90)

S[φ] =

(cid:104)dφ,dφ(cid:105)T

∗

T M dvolh
because its Euler-Lagrange equations (ELE) are precisely trh∇∗
dφ = 0. Setting the domain
to be (R1,d, m) and writing the ELE in local coordinates on the target manifold M, we see

N

∗

M⊗φ

that totally geodesic maps automatically satisfy the wave maps equation

(cid:3)mφi + Γ k

ij(φ)(cid:104)φi, φj(cid:105)m = 0.

(1.3.1)

The wave maps equation simultaneously generalize the geodesic and linear wave equa-
tions, as can be seen directly from (1.3.1) and setting d = 0 or M = R, respectively.

In the Riemannian setting, Vilms [Vil70] characterized totally geodesic maps between

Riemannian manifolds as a composition of a Riemannian submersion followed by a Rie-

mannian immersion, both totally geodesic:

M.

N

ΦS

Φ

B

ΦI

With this as a motivational starting point, our work serves to expand on the literature of

totally geodesic maps in the Lorentzian regime by analyzing the dynamical stability of

mappings that factor as

R1+d

ϕS

ϕI

R

M;

(1.3.2)

where, denoting by e the standard Euclidean metric on R, the mapping ϕS is a semi-
Riemannian submersion to either (R, e) or (R,−e), and ϕI is a Riemannian immersion
from (R, e) to a spaceform (M, g). In particular, this factorization implies the background

solution is automatically a totally geodesic wave map that has infinite total energy. The
semi-Riemannian submersion ϕS can be classified as spacelike or timelike3 depending on
3Note that by definition, a semi-Riemannian submersion cannot be null. We always

equip the real line R, as the domain of ϕI, with +e.

34

whether its codomain R is considered as being equipped with e or −e. A rough version of
our theorem can be stated as
Theorem 1.3.5 (Rough version of Theorems 3.6.1 and 3.7.1). Fix d ≥ 3. A totally geodesic
map satisfying the factorization (1.3.2) is globally nonlinearly stable as a solution to the initial

value problem for the wave maps equation under compactly supported smooth perturbations,

provided that either

TL: ϕS is timelike and (M, g) is a negatively-curved spaceform, or

SL: ϕS is spacelike and (M, g) is a positively-curved spaceform.

We derive the equations of motion for the perturbation in a tubular neighborhood
R × N of the geodesic ϕI(R) ⊂ M (here R parametrizes the geodesic and N the normal
(n − 1)-directions). The main geometric contribution of Chapter 3 is to show that the
equations for the perturbation u = (u1, (cid:126)u ) ∈ R × N decouple into a system of wave and

Klein–Gordon equations:(cid:3)u1 = F1u·(cid:104)du,dϕS(cid:105)m + O(|u|,|∂u|3),

(cid:3)(cid:126)u − (cid:126)M (cid:126)u = (cid:126)Fu·(cid:104)du,dϕS(cid:105)m + O(|u|,|∂u|3).

(1.3.3)

Here F1, (cid:126)F are functions of the curvature of (M, g) restricted to the geodesic ϕI. The (cid:126)M
are the masses of (cid:126)u, and as a consequence of the spaceform assumption on M, we prove
that (cid:126)M = κ(cid:104)dϕS,dϕS(cid:105)m where κ is the sectional curvature of M. Hence, the assumptions
on ϕS in Theorem 1.3.5 are there to at minimum guarantee linear stability, i.e. make the
Klein–Gordon terms (cid:126)u have positive masses.

The computations leading to (1.3.3) hinge on a careful Taylor expansion of the Christof-
fel symbols Γ in the wave maps equation (1.3.1) about the geodesic ϕI(R). This is where
we utilize the spaceform assumption on M; it forces certain cancellations of these Taylor
coefficients which reveal hidden null-structures of the equations. More concretely, we

35

prove that the nonlinearities of (1.3.3) can be written schematically as

(cid:126)u ·(cid:104)u,dϕS(cid:105)m + (cid:126)u · O(|u|2 +|∂u|2).

(1.3.4)

From this structure we see a complete vanishing of resonant wave–wave interactions that

could lead to finite-time blow up. Indeed, as we are unable to use the Morawetz vector
field as a multiplier, the available decay rate for the undifferentiated wave u1 in dimen-
−1/2; the exposed null structures serve to completely remove deadly terms of the
sion 3 is t
form (u1)2 or (u1)3. Secondly, notice that every term has as a factor a Klein–Gordon so-

lution (cid:126)u. This is crucially important because the linear decay rate for the Klein–Gordon
−3/2, which serves to dampen the nonlinear feedback
and allow for a global solution. We note, however, that this is the extent of the improved

equation is the integrable power t

decay which we can extract from (1.3.4). This is because the background ϕS, as a conse-
quence of being a semi-Riemannian submersion, cannot be null. Its interaction with the
perturbation (cid:104)du,dϕS(cid:105)m is then a generic derivative ∂u and not a tangential derivative for
which there would be an improved decay rate.

1.3.3 Membranes

The starting point of our discussion is the equation

(cid:112)
mµν∂νφ
1 + m(∇φ,∇φ)

∂
∂xµ



 = 0

(1.3.5)

on R1,d, the (1 + d) dimensional Minkowski space equipped with the metric m which in
standard coordinates is given by the diagonal matrix diag(−1,1,1,··· ,1). In the equation
we used the notation m(∇φ,∇ψ) def= mµν∂µφ∂νψ. This equation is variously known as
the membrane equation, the timelike minimal/maximal surface equation, or the Lorentzian

vanishing mean curvature flow. This is due to the interpretation that the graph of φ in
R1,d × R (cid:27) R1,d+1 is an embedded timelike hypersurface with zero mean curvature.

Solutions to (4.1.1) model extended test objects (world sheets), in the sense that the

case where d = 0 reduces to the geodesic equation which models the motion of a test parti-

36

cle. (The membrane equation can also be formulated with codimension greater than one;

see [AAI06, Mil08].) The membranes can also interact with external forces which man-

ifests as a prescription of the mean curvature; see [AC79, Hop13, Kib76, VS94] for some

discussion of the physics surrounding such objects, and see [Jer11, Neu90] for rigorous

justifications that membranes represent extended particles.

The precise version of our main theorem is Theorem 4.5.8; there we state the result as

a small-data global existence result for the corresponding perturbation equations, after a

nonlinear change of independent variables that corresponds to a gauge choice. Here we

state a slightly less precise version in terms of the original variables.

Theorem 1.3.6 (Rough version of Theorem 4.5.8). Fix the dimension d = 3. Let Υ denote a

smooth simple-plane-symmetric solution to (4.1.1) with finite extent in its direction of travel.
Fix a bounded set Ω ⊂ R3. There exists some 
0 > 0 depending on the background Υ and
∞
the domain Ω, such that for any (ψ0, ψ1) ∈ (H5(R3) ∩ C
0 (Ω)) with
(cid:107)(ψ0, ψ1)(cid:107) < 
0, the initial value problem to (4.1.1) with initial data

∞
0 (Ω)) × (H4(R3) ∩ C

φ(0, x) = Υ (0, x) + ψ0(x),

∂tφ(0, x) = ∂tΥ (0, x) + ψ1(x)

has a global solution that converges in C2(R3) to Υ as t → ±∞.

Here Υ is to be interpreted as a traveling “pulse”; it is a compactly supported function
that is both constant in two of the spatial variables and one whose differential dΥ is

null with respect to the dynamic metric. This has the physical interpretation that Υ

propagates along only one of the characteristic directions of the nonlinear wave equation.
We remark that our results extend to d ≥ 3, with adjustments made to the regularity
assumptions on the initial data.

The equation of motion for φ, upon dropping higher order terms momentarily, is

effectively

(cid:48)(cid:48)

(cid:3)mφ + φΥ

(∂t + ∂x1)2φ + Υ

(cid:48)(cid:48)

(∂tφ + ∂x1φ)2 = 0.

(1.3.6)

37

Here (cid:3)m = −∂2
t + ∆x, and the background pulse is assumed to be traveling in the +x1
direction, so has compact support in the (t − x1) variable. This is derived by a convenient
choice of gauge where the perturbation is written as a graph in the normal bundle of
Υ , interpreted as a submanifold of R1,d+1. This gauge is also used in [DKSW16], where

the authors studied the stability problem for the static catenoid solutions to the mem-

brane equation. Global existence for equations satisfying a verison of the null condition

is known in spatial dimensions two and three [Kla80,Kla82,Kla84,Chr86,Ali01a,Ali01b].
as a coefficient of the resonant terms means that one can’t

However, the presence of Υ

(cid:48)(cid:48)

directly apply the classical null condition arguments because it is not Lorentz invariant.
More sinister still is the fact that these coefficients have a growing weight when differenti-

ated by the Lorentz boosts (which are the weighted commutator fields mentioned in the

previous section adapted to the geometry of this problem):
= (xi − δ1it)Υ

LiΥ

(cid:48)(cid:48)

= (t∂xi + xi∂t)Υ

(cid:48)(cid:48)

(cid:48)(cid:48)(cid:48)

.

(1.3.7)

This growing weight has the physical interpretation of a transfer of energy from the “in-

finite energy” background Υ to the perturbation.

This growing weight requires us to use a modified bootstrap argument where the en-

ergies of order 2 (controlling 3 derivatives) and higher are allowed to grow polynomially.

This is in stark contrast to generic quadratically resonant settings where only the top or-

der energies are allowed to grow; the more derivatives are used, the more energies remain

bounded.

We begin the analysis of (4.1.2) by dropping the quasilinearity and studying the semi-

linear model problem

(cid:48)(cid:48)

(cid:3)mφ + Υ

(∂tφ + ∂x1φ)2 = 0.

(1.3.8)

This simplified problem encapsulates most of the difficulties present in the energy esti-

mates and hence sheds a considerable amount of insight on how to handle the full quasi-

linear problem. That we are able to close the bootstrap in spite of the growing weights is

38

(cid:48)(cid:48)

due to the fact that Υ

has compact support in the (t − x1) variable. Since the resonant
interacting terms (∂tφ + ∂x1φ) represent waves traveling in directions transverse to the
level sets of t − x1, the resonant interaction only takes place for a finite amount of time.

When handling the quasilinear problem (4.1.2) the second term

(cid:48)(cid:48)

φΥ

(∂t + ∂x1)2φ

(1.3.9)

would naively lose a full derivative (and, due to the growth hierarchy, also lose the asso-

ciated decay) and consequently the bootstrap would not close. What allows the argument

to go through is a “vestigial” null condition in the equation: note that when i = 1 equation

(1.3.7) and the compact support imply that the weight is not growing after commuting
with the boost L1. From this we show that (∂t + ∂x1)2φ decays faster than a generic tan-
gential second derivative, and so we can close the auxiliary bootstrap assumptions by
examining the system of equations satisfied by φ and L1φ.

39

CHAPTER 2

THE LINEAR WAVE AND KLEIN–GORDON EQUATIONS

In this chapter we apply the vector field method introduced in the pervious chapter to

obtain decay estimates for the linear wave and Klein–Gordon equations. Since the well-

posedness theory for these equations is standard [H¨or97, Sog08], we focus exclusively on

obtaining the decay estimates through the vector field method approach. As we men-
∞
tioned in the pervious chapter, L1–L

dispersive decay is typically derived from oscil-

latory integral control of the explicit Fourier representations of the fundamental solu-

tions [Tao06]. The physical space nature of the vector field method is then better suited

for the quasilinear regime of Chapter 4, where the principal symbol of the equation de-

pends on the solution itself.

This chapter begins with Section 2.1, where we first record Morrey-type L2–L

Sobolev
embedding estimates adapted to hyperboloidal foliations of Minkowski space1. We also

∞

derive interpolated GNS-type Lp–Lq Sobolev embedding estimates, also adapted to the

hyperboloidal foliations. These estimates are valid for any scalar function of Schwarz

class (and by density arguments for less regular functions), and are not confined to solu-

tions of PDEs. The latter estimates are useful because they allow one to save a derivative
∞
over the former ones. A poignant example occurs on R2. Using only the L

type Sobolev

estimates we can bound

(cid:107)φ2(cid:107)

L2(R2)

≤ (cid:107)φ(cid:107)

L

∞(R2)

(cid:107)φ(cid:107)

L2(R2)

(cid:46) (cid:107)φ(cid:107)

H2(R2)

(cid:107)φ(cid:107)

L2(R2).

(Scaling would have given us the first factor of φ in H1, but as we know the end-point
is false.) Using Lp type Sobolev inequalities instead we can

Sobolev embedding in L

∞

1For motivation on why a hyperboloidal foliation approach is taken, see the introduc-

tory discussion of Section 2.2 and Remark 2.2.6

40

appeal to Ladyzhenskaya’s inequality to get
(cid:46) (cid:107)φ(cid:107)

(cid:107)φ2(cid:107)

L2(R2)

H1(R2)

(cid:107)φ(cid:107)

L2(R2)

for a gain of one derivative.

The main results of this chapter are found in Section 2.2. There, the Morrey- and

GNS-type estimates are used in conjunction with the vector field method to derive a priori

estimates for solutions to the linear wave and Klein–Gordon equations. More specifically,

the former are pointwise decay estimates (see Proposition 2.2.7) whereas the latter are a

family of integrated decay estimates (see Subsections 2.2.2 and 2.2.3).

We point out that the Morrey-type estimates (and their associated pointwise decay

estimates for the linear wave and Klein–Gordon equations) are not original work. They

originated in Le Floch and Ma’s work [LM14] (see also Wong [Won17b]), which adapted
∞
Klainerman’s foundational vector field method [Kla85b] to derive L2–L

estimates for

the wave equation adapted to hyperboloidal foliations of Minkowski space. Hence, we re-

strict our analysis to rigorously deriving the GNS-type estimates in Section 2.1 and their

associated integrated decay for solutions of the linear wave and Klein–Gordon equations

in Subsections 2.2.2 and 2.2.3. This is original work based on [AW19b], and whose re-

sults can be viewed as the counterpart to the Morrey theory extended to weighted Lp

based Sobolev spaces. We chose to include the Morrey-type results in this chapter be-

cause they play a crucial role in the arguments used in Chapters 3 and 4. Moreover, since

the recurring theme of the present work is to showcase the vector field method, we high-

light and overview the role it plays in deriving the a priori estimates of Section 2.2. For

the convenience of the reader, complete proofs of the technical details based on Wong’s

geometric formulation [Won17b] are provided in Appendix A.2.

2.1 Sobolev embeddings adapted to hyperboloidal foliations

Keeping in mind the expectation that these integrals will be viewed as being adapted

to a hyperboloidal foliation, we will set our notation accordingly. By Στ we refer to the

41

hyperboloid in R1+d given by

Στ

We can parametrize it by Rd via the map
(x1, . . . , xd) (cid:55)→ (t =

For convenience throughout we will denote by

def= {t2 −|x|2 = τ2, t > 0}.
(cid:113)
τ2 +|x|2, x1, . . . , xd) ∈ R1+d.
(cid:113)
τ2 +|x|2,

x ∈ Rd.

wτ(x) def=

(2.1.1)

(2.1.2)

(2.1.3)

We note that the value of wτ, when thinking of Στ as embedded in R1+d, of course agrees
with the value of the t coordinate; we use the notation wτ as mental aid to work intrinsi-
cally on Στ whenever appropriate.

The Minkowski metric on R1+d induces a Riemannian metric on Στ, which is given

by the matrix-valued function

hij = δij − xixj
wτ(x)2

(2.1.4)

relative to the parametrization above. This being a rank-one perturbation of the Eu-

clidean metric, the corresponding volume form can be easily computed to be

dvolτ =

τ
wτ

dx1 ∧···∧ dxd.

(2.1.5)

The Minkowski space R1+d admits as Killing vector fields the Lorentzian boosts, given

as

Li def= xi∂t + t∂xi .

(2.1.6)

These vector fields are tangent to the hypersurfaces Στ for every τ > 0 and span the tan-
gent space at every point. In the parametrization above they can be identified with

Li (cid:27) wτ∂xi .

(2.1.7)

We remark that

Liwτ = xi, Lixi = wτ.

42

In particular, we have that for any string of derivatives

(cid:12)(cid:12)(cid:12)Li1 ··· LiK wτ

(cid:12)(cid:12)(cid:12) ≤ wτ.

(2.1.8)
If α is an m-tuple with elements drawn from {1,2,3} (namely that α = (α1, . . . , αm) with
αi ∈ {1,2,3}) we denote

Lαu def= LαmLαm−1 ··· Lα1u.

(2.1.9)

By |α| we refer to its length, namely m.

Almost all of the analysis of this work uses the following weighted Lebesgue and

Sobolev spaces:

• For p ∈ [1,∞) and α ∈ R, by Lp

α we refer to the weighted Lebesgue norm



(cid:90)

Στ

1/p

(cid:107)u(cid:107)Lp

α

=

|u|p dvolτ

wα
τ

.

(2.1.10)

• For p ∈ [1,∞), α ∈ R, and k ∈ N, by ˚W k,p
(cid:88)
Sobolev norm

α we refer to the weighted homogeneous

(cid:107)u(cid:107)

=

˚Wk,p
α

|β|=k

(cid:107)Lβu(cid:107)Lp

α

.

(2.1.11)

The corresponding inhomogeneous version W k,p
α is
(cid:107)u(cid:107)

k(cid:88)

=

(cid:107)u(cid:107)Wk,p

α

˚Wj,p
α

j=0

.

(2.1.12)

∞

estimates along the hyperboloids Στ are

The main L2–L
Theorem 2.1.1. ( [Won17b, Theorem 2.18]). Let (cid:96) ∈ R be fixed. Then the following uniform
estimate holds for functions u defined on the set {t > |x|} ⊂ R1+d (with the implicit constant
depending only on d and (cid:96)):

∞(Στ )

(cid:46) (cid:107)u(cid:107)W(cid:98)d/2(cid:99)+1,2

(cid:96)

.

(2.1.13)

(cid:13)(cid:13)(cid:13)(cid:13)w

τ1/2

(d+l−1)/2
τ

· u

(cid:13)(cid:13)(cid:13)(cid:13)L

43

Proof. See appendix A.2.2.

With regards to the general approach to the vector field method described in the Pro-

logue of Chapter 1, estimate (2.1.13) sets up the “pointwise estimates from higher deriva-

tive integral norms” described in Point III.

2.1.1 The basic global GNS inequalities

The Nirenberg argument [Nir59] is built upon the fundamental theorem of calculus.
Given a point x ∈ Rd, we will write

(cid:48)
i(s) def= (x1, x2, . . . , xi−1, s, xi+1, . . . , xd)

x

as the point where the ith coordinate of x is replaced by the real parameter s. Then

the fundamental theorem of calculus states that, for any smooth, compactly supported

|u(x)| ≤ xi(cid:90)

−∞

function u,

This implies

∞(cid:90)

−∞

(cid:48)
i(s))| ds ≤
|∂iu(x

(cid:90)
d−1 ≤ d(cid:89)

i=1

R

|u(x)| d

1
wτ ◦ x

(cid:48)
i(s)

|Liu(x

(cid:48)
i(s))| ds.

 1

d−1

(cid:48)
|Liu(x
i(s))|
(cid:48)
wτ ◦ x
i(s)

ds

(2.1.14)

.

(2.1.15)

Now, integrating the left hand side and applying H¨older’s inequality (exactly as in [Nir59])

this implies (noting that the volume form is weighted according to (2.1.5))
d−1
1

(cid:90)

d−1 dvolτ ≤ d(cid:89)

d−1
1

τ

wτ(x)|u(x)| d

|Liu(x)| dvolτ

.

(GNS1)

Στ

i=1

Στ

(cid:90)





The extra factor of τ comes from the dvol that appears different number of times on the

two sides. Taking advantage of (2.1.8) which shows that we have really an exponential-
type weight, (GNS1) implies the following arbitrarily-weighted counterpart. For any α ∈

44

R,

(cid:90)

d−1
1

τ

Στ

1+α· d
d−1
w
τ

d−1 dvolτ ≤ d(cid:89)

|u(x)| d

i=1



Στ

(cid:90)

|Liu| +|α|wα

τ

|u| dvolτ

wα
τ

d−1
1



.

(GNAWS1)

(This last inequality follows by replacing u (cid:55)→ wα

So (GNAWS1) asserts the continuous embedding W 1,1

τ u in (GNS1).)

α (cid:44)→ Ld/(d−1)

αd/(d−1)+1.

Remark 2.1.2. To foreshadow our discussion, notice that the standard ∂t-energy of the
linear wave equation (see (2.2.10) and Lemma 2.2.5) controls

τ

−1(cid:107)u(cid:107)2

˚W1,2−1

+ τ(cid:107)∂tu(cid:107)2L2−1

.

On the other hand, the ∂t-energy of the linear Klein–Gordon equation controls

τ

−1(cid:107)u(cid:107)2

˚W1,2−1

+ τ(cid:107)∂tu(cid:107)2L2−1

+ τ

−1(cid:107)u(cid:107)2L2

1

(note the different weight on the final term).

Replacing u by uq, coupled with an application of H¨older’s inequality, gives the stan-

α . Let 1 ≤ p < d, we have
dard extensions of (GNS1) and (GNAWS1) to W 1,p
(cid:46) (cid:107)u(cid:107)
˚W1,p
1−p
(cid:46) (cid:107)u(cid:107)W1,p

τ1/d(cid:107)u(cid:107)Ldp/(d−p)

τ1/d(cid:107)u(cid:107)Ldp/(d−p)

(GNAWSp)
Iterating (GNAWSp) above, we also have as a corollary that, given k ∈ N and p ∈ [1,∞)
such that kp < d, for any β ∈ R,

1+αdp/(d−p)

1−p+αp

(GNSp)

1

,

.

τk/d(cid:107)u(cid:107)Lq

(cid:46) (cid:107)u(cid:107)Wk,p

1−q+q(β+k)

(GNAWSpk)
where q = dp/(d−kp) is the usual Sobolev conjugate of p. We note that the case β +k = 1 is
essentially a re-formulation of the standard Gagliardo-Nirenberg-Sobolev inequality on
Rd.

1−p+pβ

,

45

Remark 2.1.3. Notice that formally setting p = 2, k = d/2, and β = 0, one sees that

(GNAWSpk) has the correct scaling for an inequality of the type
(cid:107)u(cid:107)Wd/2,2
−1

τ1/2(cid:107)wd/2−1

∞ “ (cid:46) ”

u(cid:107)L

τ

.

This inequality, as we know, is not true, due to the failure of the end-point Sobolev in-

∞
equality into L

. On the other hand, the (Morrey-type) global Sobolev inequality as

stated in Theorem 2.1.1 can be restated in the following form

τ1/2(cid:107)wd/2−1

τ

u(cid:107)L

∞ (cid:46) (cid:107)u(cid:107)W(cid:98)d/2(cid:99)+1,2

−1

.

(2.1.16)

2.1.2 Interpolating inequalities: non-borderline case

The inequalities (GNSp) and (GNAWSp) represent the endpoint Sobolev embeddings,
when p < d, in our setting. In this Subsection we prove Gagliardo-Nirenberg type in-

terpolation inequalities. For simplicity we will focus on the case of one derivative: that

is, we examine embeddings of the form
W 1,p
α ∩Lq

β (cid:44)→ Lr

γ

with q ≤ r ≤ dp/(d− p). The case of higher derivatives, based on (GNAWSpk), is analogous
d−p as the Sobolev conjugate of p.
and left to the reader. For convenience we denote p
Proposition 2.1.4. Given q ≤ r ≤ p

∗ def= dp
, and let θ ∈ [0,1] satisfy

∗

1− θ
∗
p
Then the following inequalities hold for any α, β ∈ R:

θ
q

1
r

=

+

.

τ(1−θ)/d(cid:107)u(cid:107)Lr

1+θβr

τ(1−θ)/d(cid:107)u(cid:107)Lr

1+(θβ+(1−θ)α)·r

(cid:46)

(cid:46)

(cid:107)u(cid:107)Lq
(cid:107)u(cid:107)Lq

θ ·
θ ·

(cid:107)u(cid:107)
(cid:107)u(cid:107)W1,p

˚W1,p
1−p

1−θ
1−θ

,

1−p+αp

1+βq

1+βq

46

(GNSpqr)

.

(GNAWSpqr)

Proof. The inequalities hold by applying the following elementary interpolation inequal-
ity of the weighted Lp

α spaces: for all θ ∈ [0,1],
≤ (cid:107)u(cid:107)θLq

(cid:107)u(cid:107)Lr

βθ+(1−θ)α

·(cid:107)u(cid:107)1−θLp

αp/r

,

(2.1.17)

βq/r

whenever

1
r

=

θ
q

+

1− θ
p

.

2.1.3 Interpolating inequalities: borderline case

In the previous Subsection we treated the interpolation inequalities when p < d. In this

Subsection we treat the interpolation inequalities when p = d. Specifically, we examine

embeddings of the form

W 1,d
α ∩Lq

β (cid:44)→ Lr

γ

where now 1 ≤ q ≤ r < ∞. In view of our applications, the case p = d = 2 will be of specific
∗
interest. We occasionally abbreviate the Sobolev conjugate 1
Proposition 2.1.5. Let q ≤ r < ∞, and β ∈ R. Then

= d/(d − 1).

(cid:16)
τ1/d(cid:17) r−q

r (cid:107)u(cid:107)Lr

(cid:46)

1+θβr

(cid:107)u(cid:107)Lq

1+βq

q/r ·

(cid:107)u(cid:107)

(r−q)/r

˚W1,d
(1−d)(1+βθr)

,

(GNSdqr)

where θ ∈ (0,1] is the solution to

Proof. Replacing u (cid:55)→ u1+r/1

∗

(cid:32)(cid:90)

τ1/d

∗
wτ|u|r+1

dvol

1
r

=

θ
q

+

1− θ
r + 1∗ .

(cid:33)1/1

in (GNS1) implies
|u|r/1

(cid:90)

(cid:46)

∗

d(cid:88)
(cid:32)(cid:90)

i=1

(cid:46)

∗|Liu| dvol
(cid:33)1/1
|u|r dvol

1+θβr
τ

w

47

∗

·(cid:107)u(cid:107)

˚W1,d
(1−d)(1+βθr)

by H¨older’s inequality. Here we used that

(1+θβr)/1
1 = w
τ

∗

∗
(−1−θβr)/1
τ

.

· w

This in particular implies

∗
τ1/(d−1)(cid:107)u(cid:107)r+1
Lr+1∗

1

(cid:46) (cid:107)u(cid:107)rLr

1+θβr

We next interpolate using (2.1.17) to find

∗
˚W1,d
(1−d)(1+θβr)

(cid:107)u(cid:107)1
(cid:32)

θ ·

(cid:33)1−θ

.

(cid:107)u(cid:107)Lr+1∗

1

(cid:107)u(cid:107)Lq

1+βq

(cid:107)u(cid:107)Lr

1+θβr

≤

.

(2.1.18)

Plugging (2.1.18) in, cancelling the extra factors on both sides, we get the desired inequal-

ity after noting that θ is given by
∗

θ =

1

q

r(1∗ + r − q)

1− θ =

,

∗
(r − q)(r + 1
r(1∗ + r − q)

)

.

We note that when β = 0, the triple of weights

(1 + θβr,1 + βq,(1− d)(1 + βθr)) = (1,1,1− d).

Replacing u (cid:55)→ wα
(cid:16)
r (cid:107)u(cid:107)Lr

τ1/d(cid:17) r−q

τ u we further have as a corollary

(cid:18)(cid:107)u(cid:107)Lq

(cid:46)

(cid:19)q/r(cid:18)(cid:107)u(cid:107)W1,d

1+θβr+αr

1+βq+αq

(1−d)(1+βθr)+αd

(cid:19)(r−q)/r

.

(GNAWSdqr)

2.2 A priori estimates via the vector field method

2.2.1 Energy formalism and pointwise estimates
Our goal is to derive pointwise and integrated decay for a scalar function φ : R1+d → R
which is a solution of

(2.2.1)

−∂2

t φ + ∆xφ− M2φ = 0,

48

where M ∈ R represents the particle mass. Note that the case where M ≡ 0 reduces (2.2.1)
to the linear wave equation. We will prescribe initial data at on the slice t = 0:

φ(0, x) = f

∂tφ(0, x) = g

(2.2.2)

Based on the strategy introduced in the beginning of Chapter 1 (see also the discussion
leading to (1.2.26)), we look for a collection of vector fields {Z} which preserve the flow
of (2.2.1). That is, if φ solves (2.2.1), so does Zφ. It is straightforward to check that all of

the vector fields that satisfy this requirement are linear combinations of

• space-time translations: ∂t, ∂x1, . . . , ∂xd ;
xj − xj∂xi ;

• spatial rotations: Ωij = xi∂

• Lorentz boosts: Li = t∂xi + xi∂t.

These vector fields are precisely those that preserve the underlying geometry of Minkowski
space (R1+d, m) in the sense that they are Killing vector fields of m: LZm = 0 whenever Z
is any of the vector fields mentioned above and L denotes Lie differentiation. It is in this
way that one can argue that the vector field method identifies the geometry of Minkowski

space with the geometry of the equation (2.2.1).

Analogizing to the Vlasov model problem of Section 1.2, the natural candidates to

provide temporal decay for solutions of (2.2.1) are the boosts Li for i = 1, . . . , d because of

their t-weight. We note, however, that the crucial step in the proof of Lemma 1.2.9 subtly

used that Wi were tangent to constant t-hypersurfaces. This is not the case for the Lorentz
boosts. Instead, we are led to consider Sobolev inequalities on hypersurfaces to which Li

are tangent, namely, the hyperboloids Στ of the previous section. This is a fundamental
reason as to why we use the hyperboloidal foliations in our analysis. We also emphasize

that we have identified the vector fields mentioned in Point I in the Prologue of Chapter

1.

49

The vector field method for the wave and Klein–Gordon equation (2.2.1) has its roots

in the general idea of “multiply the equation by ∂tφ and integrate by parts”. Indeed, by
doing just that, one derives the well known conservation of energy identity
|g|2 +|∇xf |2 + M2|f |2 dx.

|∂tφ|2 +|∇xφ|2 + M2|φ|2 dx =

(2.2.3)

(cid:90)

(cid:90)

{t}×Rd

Rd

The following presentation of the vector field method formalizes and generalizes this

idea in a way that can be used to derive a family of a priori estimates, of which (2.2.3) is

a member.

Let ϕ be any scalar function (in practice ϕ will be a solution of the equation (2.2.1)).

We define the energy-momentum tensor associated to ϕ to be the following symmetric type

(cid:1)-tensorfield:

(cid:0)0

2

Qαβ[ϕ] def= ∂αϕ∂βϕ − 1
2

mαβ(m

−1)µν∂µϕ∂νϕ − 1

2

mαβM2ϕ2.

(2.2.4)

Given ϕ and a “multiplier” vector field X, we define the corresponding X-energy current

to be the vector field

(X)J α[ϕ] def= (m

−1)αβQβγ[ϕ]Xγ .

(2.2.5)

The dominant energy condition is the following well-known result: Q[ϕ](X, Y ) is a

positive definite quadratic form in ∂ϕ and ϕ whenever X and Y are both future-directed
(i.e. X0, Y 0 > 0) and timelike. In the case that X is timelike and Y is null, then Q[ϕ](X, Y )

is a positive semi-definite quadratic form. These properties are what allow one to con-

struct coercive energies and fluxes for wave equation solutions. For example, we note

that

Q[ϕ](∂t, ∂t) =

|∂tϕ|2 +

1
2

1
2

|∇xϕ|2 + M2|ϕ|2.

This is precisely the energy density that appears in the standard conservation of energy

identity (2.2.3). The following lemma makes this precise when X = ∂t and Y is causal (i.e.
null or timelike).

50

Lemma 2.2.1 (Dominant energy condition). Let Y be a future oriented causal vector field,
that is, m(Y , Y ) ≤ 0 and Y 0 > 0. Then

(2.2.6)

Proof. We note that Y being causal implies

(cid:17) ≥ 0.

m

(cid:16)(∂t)J [φ], Y
(Y 0)2 ≥ d(cid:88)

(Y i)2.

i=1

(cid:16)(∂t)J [ϕ], Y

Then we compute by completing the square
(∂t)J [ϕ]αY β = mαβ(m

(cid:17)

m

−1)αµQ[ϕ]µν(∂t)νY β = Q[ϕ]µν(∂t)νY µ

m(∂t, Y )m

−1(dϕ,dϕ)− 1
2

m(∂t, Y )M2ϕ2

Y i∂tϕ∂iϕ +

Y 0(∂iϕ)2 +

Y 0M2ϕ2

= mαβ
= ∂tϕY ϕ − 1
2
1
2

Y 0(∂tϕ)2 +

=

Y 0(∂tϕ)2 +

Y 0(∂tϕ)2 +

=

1
2
≥ 1
2

d(cid:88)
=
≥ 0.

i=1

√

i=1

d(cid:88)
d(cid:88)
d(cid:88)

i=1

i=1

√
√

Y 0√
2
Y 0√
2

∂iϕ +

∂iϕ +

2

Y 0√
2

∂iϕ +

Y i√
2Y 0

∂tϕ

+

1
2

Y 0M2ϕ2

1
2
Y i√
2Y 0
Y i√
2Y 0

1
2

2 − (Y i)2
2 − Y 0

2Y 0 (∂tϕ)2 +
1
2

(∂tϕ)2 +

2

∂tϕ

∂tϕ

Y 0M2ϕ2

1
2

Y 0M2ϕ2

Motivated by the dominant energy condition, for any spacelike hypersurface ˜Σ with fu-
ture oriented unit normal ˜N (which is timelike by definition), and for any future oriented
timelike multiplied vectorfield X, we define the X-energy of ϕ along ˜Σ to be

.

(2.2.7)


(cid:90)

˜Σ

1/2

Q[ϕ](X, ˜N ) dvol ˜Σ

51

For example, for a solution φ of (2.2.1), the ∂t-energy of φ along {t}× Rd, namely

(cid:90)

Q[φ](∂t, ∂t) d,

{t}×Rd

is the square of the standard conserved energy of (2.2.3).

A straightforward computation yields the following identity, which will form the

starting point for our a priori estimates for the wave and Klein–Gordon equations:

(cid:16)(X)J [ϕ]
(cid:17)

div

= ((cid:3)m − M2)ϕXϕ +

Qαβ[ϕ]LXmαβ.

1
2

(2.2.8)

We pause at this juncture to make a few remarks. Firstly, the vector field X was called
a “multiplier” because the term ((cid:3)m − M2ϕ)Xϕ on the right hand side of (2.2.8) is being
multiplied by Xϕ. It is in this way that we are generalizing the previous notion of “mul-

tiplying the equation by ∂tφ” when deriving (2.2.3). Secondly, we note that the second
2Qαβ[ϕ]LXmαβ completely vanishes whenever X is a Killing field of the Minkowski
term 1

metric m. From the discussions in the previous paragraph, for the rest of this work we

will always use the multiplier vector field X to be the Killing field ∂t. The a priori esti-
over a
mates will be a consequence of applying the divergence theorem to div

spacetime domain sandwiched between spacelike hypersurfaces (so that the correspond-

ing normals are timelike in order to obtain coercivity as per the dominant energy condi-

tion). This generalizes the notion of “integrating by parts” when deriving the standard

conservation of energy identity (2.2.3). It also makes precise the discussion of conserva-
tion laws of Point II in the prologue of Chapter 1. Indeed, if supp{φ|t=0, ∂tφ|t=0} (cid:98) B(0,1),
the proof of Proposition 2.2.2 shows that the energy is conserved because estimate (2.2.9)

(cid:16)(X)J [ϕ]
(cid:17)

is achieved with equality.
Proposition 2.2.2. Suppose φ solves (cid:3)φ− M2φ = 0. Then

(cid:90)

Στ

Q[φ](∂t, ∂τ) dvolτ ≤

Q[φ](∂t, ∂t) dx.

(2.2.9)

{0}×Rd

(cid:90)

52

Remark 2.2.3. This proof is standard and is adapted from notes taken in Willie Wong’s

Introduction to Dispersive Equations course.

Proof. Define the regions

Dτ def= {(t, x) ∈ R1+d | t > 0, t2 ≤ τ2 +|x|2},

Dτ

µ

def= Dτ ∩{(t, x) ∈ R1+d | |x| < µ− t}.

Then we analyze the boundary of Dτ

µ as

0 =

Dτ
µ

(cid:90)

(cid:90)

The identity (2.2.8) and the divergence theorem imply

((cid:3)φ− M2φ2)∂tφ dvol =

div

Dτ
µ

dvol =

Στ,µ+Cτ,µ+Rµ

ι(∂t)J [φ]

dvol .

Along Στ,µ, the volume form2 can be factored as dvol = dτ∧dvolτ (see (2.1.4) and (2.1.5)),
so

(cid:90)

ι(∂t)J [φ]

Στ,µ

On Rµ, we factor by dvol = dt ∧ dx and use the change in orientation to see

Στ,µ

Στ,µ

m(∂τ,(∂t)J [φ]) dvolτ =

Q[φ](∂τ, ∂t) dvolτ .

Rµ

Cτ,µ

Στ,µ

def= {(0, x) ∈ R1+d | |x| < µ};
def= {(t, x) ∈ R1+d |t2 −|x|2 ≤ τ2, t = µ−|x|};
def= {(t, x) ∈ R1+d | t2 −|x|2 = τ2, t < µ−|x|}.
(cid:90)

(cid:90)

(cid:16)(∂t)J [φ]

(cid:17)

(cid:90)

dτ ∧ dvolτ =
(cid:90)

ι(∂t)J [φ]

Rµ

(cid:90)

Q(∂t, ∂t) dx.

t=0, |x|<µ

dt ∧ dx = −
(cid:90)

ι(∂t)J [φ]

dvol ≥ 0.

Finally, we notice that the dominated energy condition (2.2.6) implies

Cτ,µ

2Note that the Riemannian metric h induced on Στ by the Minkowski metric agrees

with the one on Στ,µ

53

(cid:90)

Hence we conclude

Q(∂τ, ∂t) dvolτ ≤
Letting µ → ∞ concludes the proof.

Στ,µ

t=0, |x|<µ

(cid:90)

Q(∂t, ∂t) dx ≤

(cid:90)

{0}×Rd

Q(∂t, ∂t) dx.

Motivated by the discussion immediately following Lemma 2.2.1, given a solution φ

of (2.2.1), we define the ∂t-energy of φ along the hyperbola Στ to be

(cid:90)



Στ

Eτ[φ] def=

1/2

Q[φ](∂t, ∂τ) dvolτ

.

(2.2.10)

In this notation estimate (2.2.9) naturally reads as

Eτ[φ] ≤ (cid:107)f (cid:107)

H1(Rd) +(cid:107)g(cid:107)

L2(Rd).

Remark 2.2.4. The flexibility of the dominant energy condition allows one to define
i=1 txi∂xi . It
is straight forward to check that K is a future oriented causal vector field. Define the

other kinds of energies, see (2.2.7). For example, let K def= (t2 + |x|2)∂t + 2(cid:80)d
(cid:1)-tensor
(cid:0)0

1

˜Q[ϕ](K,·) def= Q[ϕ](K,·) +

and its associated the modified current

d{trLK m} ϕ2.

(2.2.11)

{trLK m}d(φ)2 − d − 1

8(d + 1)

d − 1
8(d + 1)
(K)(cid:101)J [ϕ] def= ˜Q[ϕ](K,·)#.
(cid:16)(K)(cid:101)J [ϕ]
(cid:17)

= (cid:3)φ (Kϕ + (d − 1)tϕ) .

Then a tedious but straightforward calculation implies the following divergence identity

Hence, for solutions to (2.2.1) with M = 0, the proof of Proposition 2.2.2 can be repli-

for the modified current:

div

cated3 to show that (cid:90)

(cid:90)

˜Q[φ](∂t, ∂τ) dvolτ ≤

Στ

Q[φ](∂t, ∂t) dx,

{0}×Rd

3For compactly supported initial data, say.

54

where the integral on the left hand side of the above inequality is defined as the K-energy
of φ along ˜Σ.

Because our analysis will use the weighted Lebesgue and Sobolev spaces (see defini-

tions (2.1.10) – (2.1.12)), we set up the following useful lemma.
Lemma 2.2.5. Suppose d ≥ 3. Then the following estimates holds with a universal implicit
constant, with φt

def= ∂tφ:
Eτ[φ] ≈ τ
Eτ[Lαφ] ≈ τ

(cid:88)

|α|≤k

−1/2(cid:107)φ(cid:107)
˚W1,2−1
−1/2(cid:107)φ(cid:107)Wk+1,2
−1

+ τ1/2(cid:107)φt(cid:107)L2−1
+ τ1/2(cid:107)φt(cid:107)Wk,2−1

+ τ

−1/2(cid:107)Mφ(cid:107)L2

1

−1/2(cid:107)Mφ(cid:107)Wk,2

+ τ

1

,

(2.2.12)

.

(2.2.13)

Proof. The proof can be found in Appendix A.2.2

Remark 2.2.6. A feature of (2.2.12) is its anisotropy. Consider momentarily the wave

equation case of M = 0. The classical energy estimates of wave equations control inte-
grals of |∂tφ|2 + |∇φ|2 where all components appear on equal footing. Here, however,
the transversal (to Στ) derivative ∂tφ has a different weight compared to the tangential
derivatives Liφ. Noting that by its definition, ∂t has unit-sized coefficients with expressed
relative to the standard coordinates of Minkowski space. The coefficients for Li (within
the light cone {t > |x|}) have size ≈ t. Therefore an isotropic analogue would be expected to
contain integrals of 1
t2 (Liφ)2 along with integrals of ∂tφ. This indicates that an isotropic

analogue would contain, instead of the integral given in (2.2.12), the integral

τ1/2(cid:107)φ(cid:107)

˚W1,2−3

+ τ1/2(cid:107)φt(cid:107)L2−1

.

In other words, the integral for Liφ in the energy (2.2.12) has a better wτ weight than
would be expected from an isotropic energy, such as that controlled by the standard en-

ergy estimates.

This improvement reflects the fact that the energy estimate described in this sec-

tion captures the peeling properties of linear waves within the energy integral itself. It

55

is well-known that derivatives tangential to an out-going light-cone decay faster along the

light-cone, than derivatives transverse to the light-cone. As asymptotically hyperboloids

approximate light-cones, we expect the same peeling property to survive. Indeed, the

energy inequality (2.2.9) shows that we can capture this in the integral sense.

Proposition 2.2.7. ( [Won17b, Proposition 3.2, Remark 5.8]). Let φ solve (2.2.1). Then the

following estimate holds (with the implicit constant depending only on d)

(cid:13)(cid:13)(cid:13)(cid:13)wd/2

τ φ

(cid:13)(cid:13)(cid:13)(cid:13)L

M

(cid:13)(cid:13)(cid:13)(cid:13)w

∞(Στ )

+ τ

(d−2)/2
τ

∂tφ

∞(Στ )

+

(cid:13)(cid:13)(cid:13)(cid:13)L

(cid:13)(cid:13)(cid:13)(cid:13)w

d(cid:88)

i=1

(d−2)/2
τ

Liφ

(cid:13)(cid:13)(cid:13)(cid:13)L
(cid:88)
∞(Στ )

(cid:46)

|α|≤(cid:98)d/2(cid:99)+1

Eτ[Lαφ].

(2.2.14)

∞
Proof. The proof follows from the global Sobolev pointwise L2–L
estimate provided by
Theorem 2.1.1 and Lemma 2.2.5 by setting (cid:96) = −1 for ∂tφ and Liφ and by setting (cid:96) = 1
for φ.

Remark 2.2.8. We note that the Klein–Gordon mass term Mφ has improved decay over the
wave derivatives ∂tφ and Liφ. This is a well-known fact [Kla85a] and is a consequence
of the positive wτ-weight in the mass term Mφ of the energy density (see (2.2.12) and
(A.2.17)). Note that Proposition 2.2.7 does not provided pointwise decay for φ itself in

the wave equation case of M = 0. In this scenario, one can still get decay for φ itself by
appealing to Hardy’s inequality (see Lemma A.2.5) when d ≥ 3.
Proposition 2.2.9. Suppose d ≥ 3. Let φ solve the linear wave equation, that is, (2.2.1) with
M = 0. Then the following estimate holds (with the implicit constant depending only on d)

(cid:107)w

(d−2)/2
τ

φ(cid:107)L

∞(Στ ) (cid:46)

Eτ[Lαφ].

(2.2.15)

(cid:88)

|α|≤(cid:98)d/2(cid:99)

Proof. The estimate follows from the global Sobolev pointwise estimate provided by The-
orem 2.1.1, Lemma 2.2.5 with (cid:96) = −1, by Hardy’s inequality (see Lemma A.2.5) to control
the term without any derivatives, and by the coercivity afforded by Lemma 2.2.5.

56

Remark 2.2.10. Note that estimate (2.2.15) is not sharp. By estimating the fundamental
solution of the linear wave equation, we expect φ to decay like t(d−1)/2 = w
sharp decay rate can be obtained by hyperboloidal techniques by using the K-energy as-

(d−1)/2
τ

. The

sociated to φ (see Remark 2.2.4 and [Won17b, Proposition 5.6]). We omit this computation

because we do not use the sharp decay rate for φ in Chapter 3 nor in 4 when M = 0.

2.2.2 Wave equation, d = 2,3,4
In this section we apply our results to obtain Lr∗ bounds by L2∗ integrals that occur as part
of the conserved energy for the linear wave equation. As we will see there is often more

than one way to obtain interpolated estimates, depending on the number of derivatives

one is willing to sacrifice. Rather than attempt to be exhaustive in this section, we will

opt for concreteness and list several possible estimates for dimensions d = 2,3,4, where

the choices are more limited. As we will see, the most delicate case is when d = 2 because

Hardy is not available. For clarity we save those estimates until the end of this subsection.
Throughout we will let u be a smooth function on R1+d, and ut will denote its time
derivative. Motivated by Lemma 2.2.5, we will denote by Ek the kth order energy quantity

−1/2(cid:107)u(cid:107)Wk+1,2
−1
Proposition 2.2.11 (d = 3). When r ∈ [2,6],

Ek(τ) = τ

+ τ1/2(cid:107)ut(cid:107)Wk,2−1 (Στ )

.

(Στ )

(cid:32)

−1/r

τ

(cid:107)u(cid:107)

˚Wk+1,r
r/2−2(Στ )

τ

−1/r(cid:107)u(cid:107)Lr
+ τ(cid:107)ut(cid:107)

r/2−2(Στ ) (cid:46) E0(τ),
(cid:46) (Ek(τ))

˚Wk,r
r/2−2(Στ )

(cid:33)

6−r
2r

· (Ek+1(τ))

3r−6
2r

.

(2.2.16)

(2.2.17)

When r > 6,

(cid:32)
(cid:107)u(cid:107)

˚Wk+1,r
r/2−2(Στ )

−1/r

τ

τ

−1/r(cid:107)u(cid:107)Lr
+ τ(cid:107)ut(cid:107)

r/2−2(Στ ) (cid:46) (E0(τ))
(cid:46) (Ek+1(τ))

˚Wk,r
r/2−2(Στ )

r+6
2r

r−6
· (E1(τ))
2r ,
· (Ek+2(τ))

r+6
2r

r−6
2r .

(2.2.18)

(2.2.19)

(cid:33)

57

(cid:32)
(cid:107)u(cid:107)

For higher derivatives, the latter of the above estimate in r > 6 can be replaced by
· (Ek+2)

+ τ(cid:107)ut(cid:107)

(cid:46) (Ek(τ))

r−2
2r

−1/r

τ

r−2
2r

˚Wk+1,r
r/2−2(Στ )

˚Wk,r
r/2−2(Στ )

(2.2.20)

Proof. Estimate (2.2.16) follows by applying (GNSpqr) with d = 3, q = 2. Indeed, we see

(cid:33)

4

2r · (Ek+1(τ))
(cid:32)
(cid:33)1−θ
(cid:107)u(cid:107)

(cid:33)θ ·

˚W1,2−1

,

(cid:32)
(cid:107)u(cid:107)L2−1

τ(1−θ)/3(cid:107)u(cid:107)Lr
where θ ∈ [0,1] is the solution to
1− θ
6

θ
2

1
r

=

+

(cid:46)

1−θr

=⇒

6− r
2r

,

θ =

1− θ =

3r − 6
2r

.

Rearranging using Hardy on the first factor and the definition of the energy we see that
(2.2.16) follows. Similarly, if α is a k-tuple with elements drawn from {1,2,3} and v is any
function we have

τ(1−θ)/3(cid:107)Lαv(cid:107)Lr

(cid:46)

1−θr

(cid:32)
(cid:107)Lαv(cid:107)L2−1

(cid:33)θ ·

(cid:32)
(cid:107)Lαv(cid:107)

˚W1,2−1

(cid:33)1−θ

,

with the same θ as before. Replacing v (cid:55)→ Liu or ut, and since we can estimate (cid:107)LαLiu(cid:107)L2−1

by the kth order energy without invoking Hardy, (2.2.17) follows using the definition of

their energies with the respective weights.

For larger r, we first appeal to (GNAWSdqr) with d = 3, q = 6, and

1 + βq + αq = 1

−2(1 + βθr) + 3α = −1
2

θr =

9

3

2 + r − 6

which is solved by

This implies

−α = β = − 3

2 + r − 6

3 + 2r

.

r−6
3r (cid:107)u(cid:107)Lr

r/2−2

τ

(cid:46) (cid:107)u(cid:107)6/rL6

1

·(cid:107)u(cid:107)(r−6)/r
W1,3−1/2

.

58

Applying (2.2.16) and (2.2.17) to the two terms on the right we get

(cid:46)(cid:16)
(cid:46)(cid:16)

(cid:17)6/r ·(cid:16)
τ1/3E0(τ)1/2E1(τ)1/2(cid:17)(r−6)/r
τ1/3Ek(τ)1/2Ek+1(τ)1/2(cid:17)(r−6)/r
(cid:17)6/r ·(cid:16)

.

r−6
3r (cid:107)u(cid:107)Lr

τ

r/2−2

τ1/6E0(τ)

and

r−6
3r (cid:107)u(cid:107)

˚Wk,r
r/2−2

τ

τ1/6Ek(τ)

Rearranging this gives (2.2.18) and (2.2.19)

To find the other estimate for r > 6 we appeal to the borderline (GNAWSdqr) inequality

slightly differently. Using d = 3 and q = 2 now, with
1 + βq + αq = −1,

(1− d)(1 + θβr) + αd = −1/2,
∗
1/r = θ/q + (1− θ)/(r + 1
),

we can solve to find

θ =

6

r(2r − 1)

,

(−3)(2r − 1)
2(3 + 2r)

,

β =

2r − 9
2(3 + 2r)

.

α =

Let α be a k-tuple with elements drawn from {1,2,3} and v be any function. Then the
inequality reads(cid:16)

(cid:33) r−2

(cid:32)

r

τ1/3(cid:17) r−2

(cid:33)2/r ·

(cid:32)
(cid:107)Lαv(cid:107)W1,3−1/2

r (cid:107)Lαv(cid:107)Lr

(cid:46)

r/2−2

(cid:107)Lαv(cid:107)L2−1

.

Estimating the second factor using (2.2.17) with r = 3 and the choice v = Liu or ut, we can
then rearrange to obtain (2.2.20).
Proposition 2.2.12 (d = 4). When r ∈ [2,4],

(cid:32)
(cid:107)u(cid:107)

˚Wk+1,r
r−3

−1/r

τ

τ

−1/r(cid:107)u(cid:107)Lr
+ τ(cid:107)ut(cid:107)

˚Wk,r
r−3(Στ )

r−3(Στ ) (cid:46) E0(τ),
(cid:46) (Ek(τ))

4−r
r

2r−4
r

.

(Ek+1(τ))

(2.2.21)

(2.2.22)

(cid:33)

59

When r > 4,

(cid:32)
(cid:107)u(cid:107)

˚Wk+1,r
r−3 (Στ )

−1/r

τ

or

(cid:33)

τ

−1/r(cid:107)u(cid:107)Lr
+ τ(cid:107)ut(cid:107)

r−3(Στ ) (cid:46) (E0(τ))2/r · (E1(τ))
(cid:46) (Ek(τ))2/r (Ek+2(τ))
˚Wk,r
r−3(Στ )

r−2
r
r−2
r

(2.2.23)

(2.2.24)

,

(cid:32)
(cid:107)u(cid:107)

˚Wk+1,r
r−3 (Στ )

−1/r

τ

τ

−1/r(cid:107)u(cid:107)Lr
+ τ(cid:107)ut(cid:107)

˚Wk,r
r−3(Στ )

(cid:33)

r−3(Στ ) (cid:46) (E0(τ))4/r (E1(τ))

r−4
r

,

(cid:46) (Ek+1(τ))4/r (Ek+2(τ))

r−4
r

.

(2.2.25)

(2.2.26)

Proof. The proofs of (2.2.21) and (2.2.22) are the same as (2.2.16) and (2.2.17) except that

now d = 4 and θ solves

1
r

=

θ
2

+

1− θ
4

=⇒

4− r
r

,

θ =

1− θ =

2r − 4
r

.

To find estimate for r > 4 we appeal to the borderline (GNAWSdqr) inequality. We will

first be applying the inequality with

d = 4,

q = 2,

1 + βq + αq = −1,

(1− d)(1 + θβr) + αd = 1,
∗
1/r = θ/q + (1− θ)/(r + 1
).

These equations are solved by

θ =

8

r(3r − 2)

,

(−2)(3r − 2)

4 + 3r

β =

,

α =

3r − 8
4 + 3r

,

and so the weight 1 + θβr + αr = r − 3.

Let α be a k-tuple with elements drawn from {1,2,3,4} and v be any function. Then

(cid:16)

τ1/4(cid:17) r−2

r (cid:107)Lαv(cid:107)Lr
r−3

(cid:46)

(cid:32)
(cid:107)Lαv(cid:107)L2−1

(cid:33)2/r ·

(cid:32)
(cid:107)Lαv(cid:107)W1,4

(cid:33) r−2

r

.

1

60

This inequality holds for r > 2, so in particular for r > 4. If k = 0 and v = u, then the

first factor can be estimated by the energy after invoking Hardy. The second factor can

∗
by treated with (GNSp) because 2
= (cid:107)u(cid:107)L4

(cid:107)u(cid:107)W1,4

= 4:
+(cid:107)u(cid:107)

1

1

(cid:46) τ

−1/4((cid:107)u(cid:107)

˚W1,2−1

+(cid:107)u(cid:107)

˚W2,2−1

).

˚W1,4
1

This gives (2.2.23) after applying the definition of the energy. Again, note that if k is

arbitrary and v = Liu, then we do not have to invoke Hardy to estimate the first factor by
−1/rE2/r
the energy τ1/rE2/r
k .
The second factor in the case of v (cid:55)→ (Liu, ut) can again be treated with (GNSp). Rearrang-
ing the inequalities and using the coercivity of their energies with the respective weights

k . On the other hand, if v = ut, the first factor is bounded by τ

gives (2.2.24).

Alternatively, we can also solve with

d = 4

q = 4

1 + βq + αq = 1

(1− d)(1 + θβr) + αd = 1
∗
1/r = θ/q + (1− θ)/(r + 1
).

Let α be a k-tuple now and compute again with (GNAWSdqr) and (GNSp)

(cid:16)
τ1/4(cid:17) r−4

r (cid:107)Lαv(cid:107)Lr

(cid:32)
(cid:107)Lαv(cid:107)L4

(cid:33)4/r ·

(cid:32)
(cid:107)Lαv(cid:107)W1,4

(cid:33) r−4

r

.

(cid:46)

1

1+θβr+αr

1

The prior equations are solved by

θ =

16

r(3r − 8)

,

−β = α =

3r − 8
4 + 3r

and so the weight 1+θβr +αr = r−3. We control each factor with (GNSp) in the two cases
k = 0, v = u and arbitrary k and v = (Liu, ut) as above. This finishes the proof of (2.2.25)
and (2.2.26).

61

When d = 2, Hardy’s inequality is generally unavailable for the wave equation energy.

So the kth order energy should only be

k+1(cid:88)

j=1

−1/2

Ek(τ) = τ

(cid:107)u(cid:107)

˚Wj,2−1 (Στ )

+ τ1/2(cid:107)ut(cid:107)Wk,2−1 (Στ )

.

So we cannot in general control (cid:107)u(cid:107)Lr∗ ; but we can control the first derivatives of u in Lr∗

with suitable weights.
Proposition 2.2.13. When r ∈ [2,∞),
+ τ(cid:107)ut(cid:107)

(cid:32)
(cid:107)u(cid:107)

−1/r

τ

˚Wk+1,r
−1

(Στ )

˚Wk,r−1 (Στ )

(cid:33)

(cid:46) (Ek(τ))2/r (Ek+1(τ))

r−2
r

,

(2.2.27)

Proof. We appeal to the borderline (GNAWSdqr) inequality with

d = 2,

q = 2,

1 + βq + αq = −1,

(1− d)(1 + θβr) + αd = −1,
1/r = θ/2 + (1− θ)/(r + 2).

These equations are solved by

θ =

4
r2 ,

−r
2 + r

,

β =

−2
2 + r

,

α =

and so the weight 1 + θβr + αr = −1. Let α be a k-tuple and let v be an arbitrary function.
Then we compute(cid:16)
Replacing v (cid:55)→ Liu or ut and using the coercivity of their energies with the respective
weights concludes the proof.

(cid:32)
(cid:107)Lαv(cid:107)W1,2−1

(cid:32)
(cid:107)Lαv(cid:107)L2−1

τ1/2(cid:17) r−2

r (cid:107)Lαv(cid:107)Lr−1

(cid:33)2/r ·

(cid:33) r−2

r

(cid:46)

.

62

2.2.3 Klein–Gordon equation, d = 2,3,4

The Klein–Gordon energies control additionally a differently weighted L2 term. More-

over, as we will see below, it is useful to distinguish between the energies of u and ut (the
latter of which also solves the Klein–Gordon equation). We write the kth order energy as

Ek[v](τ) = τ

−1/2(cid:107)v(cid:107)Wk+1,2
−1

(Στ )

+ τ1/2(cid:107)vt(cid:107)Wk,2−1 (Στ )

+ τ

−1/2(cid:107)v(cid:107)Wk,2

1 (Στ )

,

where v can play the roll of u or ut. Here we’ve assumed M = 1 for simplicity. Moreover,

we assume that τ ≥ 1, so that (cid:107)u(cid:107)L2−1

≤ (cid:107)u(cid:107)L2

.

1

Proposition 2.2.14 (d = 2). When r > 2, we have

τ

τ

−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)

τ

τ

˚Wk,r

1 (Στ )

˚Wk,r
r−1(Στ )

(Στ )

˚Wk+1,r
−1
˚Wk+1,r
r−3 (Στ )

(cid:46) Ek[u](τ),
(cid:46) (Ek[u](τ))2/r · (Ek+1[u])
(cid:46) (Ek[u](τ))2/r · (Ek+1[u])
(cid:46) (Ek[u](τ))2/r · (Ek+2[u](τ))

r−2
r
r−2
r

,

,
r−2
r

.

For the time derivatives the following estimates hold:

τ

τ1−3/r(cid:107)ut(cid:107)
−1/r(cid:107)ut(cid:107)
τ1−1/r(cid:107)ut(cid:107)
τ1/r(cid:107)ut(cid:107)

˚Wk,r

1 (Στ )

˚Wk,r
r−1(Στ )
˚Wk,r−1 (Στ )
˚Wk,r
r−3(Στ )

r−2
(cid:46) (Ek[ut](τ))2/r · (Ek+1[u](τ))
r
,
r−2
(cid:46) (Ek[ut](τ))2/r · (Ek+1[ut](τ))
r
r−2
(cid:46) (Ek[u](τ))2/r · (Ek+1[u](τ))
r
,
r−2
(cid:46) (Ek[u](τ))2/r · (Ek+1[ut](τ))
r

.

,

(2.2.28)

(2.2.29)

(2.2.30)

(2.2.31)

(2.2.32)

(2.2.33)

(2.2.34)

(2.2.35)

Proof. Throughout this proof α will be a k-tuple and v will be an arbitrary function. We

63

solve (GNAWSdqr) for

d = 2

q = 2

1 + βq + αq = µ

(1− d)(1 + θβr) + αd = ν
1/r = θ/2 + (1− θ)/(r + 2),
where µ, ν can take the values ±1. Denoting the weight

ρ(µ, ν) def= 1 + θβr + αr,

the borderline inequality yields

(cid:16)
τ1/2(cid:17) r−2

r (cid:107)Lαv(cid:107)Lr

ρ(µ,ν)

(cid:32)
(cid:107)u(cid:107)L2

µ

(cid:33)2/r ·(cid:18)(cid:107)u(cid:107)W1,2

ν

(cid:19) r−2

r

.

(cid:46)

(2.2.36)

One explicitly computes the weights as
ρ(1,1) = r − 1,

ρ(1,−1) = 1,

ρ(−1,−1) = −1,

ρ(−1,1) = r − 3.

Replacing v (cid:55)→ (u, ut) in (2.2.36) and using the definition of the energies with their re-
spective weights with µ = 1, ν = −1 proves (2.2.28), (2.2.32). When µ = ν = 1, this proves
(2.2.29), and (2.2.33). On the other hand, replacing v (cid:55)→ (Liu, ut) in (2.2.36) and using the
definition of the energies with their respective weights with µ = ν = −1 shows (2.2.30),
(2.2.34). Finally, using µ = −1, ν = 1 proves (2.2.31), and (2.2.35).

Remark 2.2.15. We note that (2.2.30) and (2.2.34) are identical to the estimates (2.2.27)

derived for the wave equation. Indeed, the Klein–Gordon and wave t-energies both con-

k+1(cid:88)
(cid:107)u(cid:107)
˚Wj,2−1
−1/2(cid:107)u(cid:107)L2

j=1

1

64

trol

−1/2

τ

+ τ1/2(cid:107)ut(cid:107)W1,2−1

.

The takeaway is that the mass term τ

allows for estimates with different weights.

Remark 2.2.16. One can summarize the proof of Proposition 2.2.14 by saying that its
estimates correspond to the four endpoint cases of µ, ν = ±1 when applying (GNAWSdqr).
Of course, various interpolations of these hold. One can interpolate, for example, equa-
tion (2.2.28) with (2.2.30) to see, for any θ ∈ [0,1],

−1/r(cid:107)u(cid:107)

τ

˚Wk+1,r
1−2θ

(cid:46) (Ek[u](τ))2θ/r · (Ek+1[u](τ))1−2θ/r .

For the sake of brevity and clarity, we leave these straightforward computations to the

reader.
Proposition 2.2.17 (d = 3). When r ∈ [2,6],

(2.2.37)

(2.2.38)

(2.2.39)

(2.2.40)

(2.2.41)

(2.2.42)

(2.2.43)

(2.2.44)

,

−1/r(cid:107)u(cid:107)

τ

˚Wk,r

1 (Στ )

τ

−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)

τ

τ

˚Wk,r
3r/2−2(Στ )
˚Wk+1,r
r/2−2(Στ )
˚Wk+1,r
2r−5 (Στ )

(cid:46) (Ek[u])

(cid:46) Ek[u](τ),
6−r
2r
6−r
2r
6−r
2r

(cid:46) (Ek[u])

(cid:46) (Ek[u])

· (Ek+1[u])
· (Ek+1[u])
· (Ek+2[u])

3r−6
2r
3r−6
2r
3r−6
2r

,

,

.

˚Wk,r

1 (Στ )

(cid:46) (Ek[ut](τ))

For the time derivatives, the following estimates hold:
6−r
2r
6−r
2r
6−r
2r
6−r
2r

τ3/2−4/r(cid:107)ut(cid:107)
−1/r(cid:107)ut(cid:107)
τ1−1/r(cid:107)ut(cid:107)
−1/2+2/r(cid:107)ut(cid:107)

˚Wk,r
3r/2−2(Στ )
˚Wk,r
r/2−2(Στ )
˚Wk,r
2r−5(Στ )

(cid:46) (Ek[ut](τ))

(cid:46) (Ek[u](τ))

(cid:46) (Ek[u](τ))

τ

τ

· (Ek+1[u](τ))
· (Ek+1[ut](τ))
· (Ek+1[u](τ))
· (Ek+1[ut](τ))

3r−6
2r
,
3r−6
2r
3r−6
2r
,
3r−6
2r

.

65

For the time derivatives, we have:

−1/r(cid:107)u(cid:107)

τ

˚Wk,r
r−1(Στ )

τ

τ

−1/r(cid:107)u(cid:107)

τ

τ

τ

r/2(Στ )

˚Wk,r
r−1(Στ )

˚Wk,r
3r/2−2(Στ )
−1/r(cid:107)u(cid:107)
˚Wk,r
−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)

˚Wk+1,r
3r/2−4(Στ )
˚Wk+1,r
r/2−2(Στ )
˚Wk+1,r
r−3 (Στ )

˚Wk+1,r
r−3 (Στ )

τ

τ

τ

r/2(Στ )

˚Wk,r
r−1(Στ )

τ1/2−2/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)
˚Wk,r
3r/2−2(Στ )
τ1−3/r(cid:107)ut(cid:107)
˚Wk,r
τ1/2−2/r(cid:107)ut(cid:107)
τ1/2(cid:107)ut(cid:107)
τ1/r(cid:107)ut(cid:107)
˚Wk,r
3r/2−4(Στ )
τ1−1/r(cid:107)ut(cid:107)
˚Wk,r
r/2−2(Στ )
τ1/2(cid:107)ut(cid:107)
˚Wk,r
r−3(Στ )

˚Wk,r
r−1(Στ )
˚Wk,r
r−3(Στ )

When r > 6, the following estimates hold:

r−2
r

(cid:46) (Ek[u](τ))2/r · (Ek+1[u])

(cid:46) (Ek[u](τ))2/r ·(cid:16)

Ek+1[u](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2

r

,

(2.2.45)

,

(2.2.46)

r−2
2r ,

r+2
2r

(cid:46) (Ek[u](τ))

· (Ek+1[u](τ))
r−2
· (Ek+2[u])
(cid:46) (Ek[u](τ))
2r ,
r−2
(cid:46) (Ek[u](τ))2/r · (Ek+2[u](τ))
r

r+2
2r

,

r

r

,

,

.

r

(cid:46) (Ek[u](τ))2/r ·(cid:16)
(cid:46) (Ek[u](τ))2/r ·(cid:16)
(cid:46) (Ek[u](τ))2/r ·(cid:16)
(cid:46) (Ek[ut](τ))2/r ·(cid:16)
(cid:46) (Ek[ut](τ))2/r ·(cid:16)
(cid:46) (Ek[ut](τ))2/r ·(cid:16)
(cid:46) (Ek[ut](τ))2/r ·(cid:16)
(cid:46) (Ek[u](τ))2/r ·(cid:16)
(cid:46) (Ek[u](τ))2/r ·(cid:16)
(cid:46) (Ek[u](τ))2/r ·(cid:16)
(cid:46) (Ek[u](τ))2/r ·(cid:16)

Ek+2[u](τ)1/2 · Ek+3[u](τ)1/2(cid:17) r−2
Ek+1[u](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2
Ek+1[u](τ)1/2 · Ek+3[u](τ)1/2(cid:17) r−2
Ek+1[ut](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2
Ek+1[ut](τ)1/2 · Ek+2[ut](τ)1/2(cid:17) r−2
Ek+1[u](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2
Ek+1[u](τ)1/2 · Ek+2[ut](τ)1/2(cid:17) r−2
Ek+1[ut](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2
Ek+1[ut](τ)1/2 · Ek+2[ut](τ)1/2(cid:17) r−2
Ek+1[u](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2
Ek+1[ut](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2

r

r

r

r

r

r

r

r

,

,

,

.

,

,

,

(2.2.47)

(2.2.48)

(2.2.49)

(2.2.50)

(2.2.51)

(2.2.52)

(2.2.53)

,

(2.2.54)

(2.2.55)

(2.2.56)

(2.2.57)

(2.2.58)

(2.2.59)

(2.2.60)

66

Proof. Throughout this proof α will be a k-tuple and v will be an arbitrary function. For
r ∈ [2,6], we can solve (GNAWSpqr) with

d = 3

q = 2

p = 2

1 + βq = µ
1− p + αp = ν

1/r = θ/q + (1− θ)/p

∗

,

where µ, ν can again take the values ±1. Denoting the weight
ρpqr(µ, ν) def= 1 + (θβ + (1− θ)· α)r

the interpolation inequality yields

τ1/2−1/r(cid:107)Lαv(cid:107)Lr

ρpqr (µ,ν)

(cid:33) 6−r

2r

(cid:32)
(cid:107)Lαv(cid:107)L2

µ

(cid:46)

·(cid:18)(cid:107)Lαv(cid:107)W1,2

ν

(cid:19) 3r−6

2r

.

(2.2.61)

One explicitly computes the weights as
ρpqr(1,1) = 3r/2− 2,

ρpqr(1,−1) = 1,

ρpqr(−1,−1) = r/2− 2,

ρpqr(−1,1) = 2r − 5.

We note that we are unable to simply replace v (cid:55)→ u in (2.2.61) and use the definition of
the energies with their respective weights with µ = 1, ν = −1 because the second factor in
(2.2.61) is the inhomogeneous Sobolev norm. To remedy this, we again use the extra mass

term in the energy:

(cid:107)Lαu(cid:107)W1,2−1

+(cid:107)Lαu(cid:107)
+(cid:107)Lαu(cid:107)

˚W1,2−1
˚W1,2−1

= (cid:107)Lαu(cid:107)L2−1
≤ (cid:107)Lαu(cid:107)L2
1
(cid:46) τ1/2Ek[u].

67

Now we can replace v (cid:55)→ (u, ut) in (2.2.61) to prove (2.2.37), (2.2.41) (note that this prob-
lem did not occur for v = ut). When µ = ν = 1, v (cid:55)→ (u, ut) in (2.2.61) also proves (2.2.38),
and (2.2.42).

On the other hand, replacing v (cid:55)→ (Liu, ut) in (2.2.61) and using the definition of the
energies with their respective weights with µ = ν = −1 shows (2.2.39), (2.2.43). Finally,
using µ = −1, ν = 1 proves (2.2.40), and (2.2.44).

For the estimates when r > 6, we appeal to the borderline (GNAWSdqr) inequality with

d = 3,

q = 2,

1 + βq + αq = σ ,

(1− d)(1 + θβr) + αd = ρpq3(µ, ν),

∗
1/r = θ/q + (1− θ)/(r + 1

),

where σ can take the values ±1 and ρpq3(µ, ν) is as above. This inequality is valid for r > 2
so in particular r > 6. Denoting the weight

ρd=3(σ , µ, ν) def= 1 + θβr + αr,

the borderline inequality yields

(cid:16)

τ1/3(cid:17) r−2

r (cid:107)Lαv(cid:107)Lr

(cid:18)(cid:107)Lαv(cid:107)L2

σ

(cid:19)2/r ·

(cid:46)

(cid:107)Lαv(cid:107)W1,3

ρpq3(µ,ν)

 r−2

r

.

(2.2.62)

ρd=3(σ ,µ,ν)

One explicitly computes the weights
ρd=3(1,1,−1) = r − 1,
ρd=3(1,1,1) = 3r/2− 2,
ρd=3(1,−1,−1) = r/2,
ρd=3(1,−1,1) = r − 1,

ρd=3(−1,1,−1) = r − 3,
ρd=3(−1,1,1) = 3r/2− 4,
ρd=3(−1,−1,−1) = r/2− 2,
ρd=3(−1,−1,1) = r − 3.

68

Note that even though

ρd=3(1,1,−1) = ρd=3(1,−1,1),
ρd=3(−1,1,−1) = ρd=3(−1,−1,1),

that µ, ν are different implies that we have different estimates. We note that replacing
v (cid:55)→ u, Liu whenever σ = 1,−1 (respectively) is not enough to prove the estimates because
the second factor in (2.2.62) is

(cid:107)Lαv(cid:107)W1,3

ρpq3(µ,ν)

= (cid:107)Lαv(cid:107)L3

ρpq3(µ,ν)

+(cid:107)Lαv(cid:107)

˚W1,3

ρpq3(µ,ν)

.

Consequently, special care must be taken to analyze the two different derivative terms
because the left hand sides in (2.2.37) - (2.2.40) are all with respect to the homogeneous
spaces ˚W∗,r∗

.

Fix v = u. When µ = 1,−1 we can estimate
−1/3(cid:107)Lαu(cid:107)W1,3

τ

(cid:46) Ek+1[u]

1

by using Ek ≤ Ek+1. Arguing in the same way, when µ = ν = 1 one finds

−1/3(cid:107)Lαu(cid:107)W1,3

τ

(cid:46) Ek+1[u]1/2 · Ek+2[u]1/2.

For µ = ν = −1, on the other hand, we estimate

τ

−1/3(cid:107)Lαu(cid:107)W1,3−1/2

−1/3

≤ τ

+(cid:107)Lαu(cid:107)

˚W1,2−1/2

The first term was controlled again using (2.2.38). Finally, when µ = −1, ν = −1 we see

5/2

(cid:32)
(cid:107)Lαu(cid:107)L3
(cid:32)
(cid:107)Lαu(cid:107)L3

−1/3

5/2

(cid:33)

(cid:46) Ek[u]1/2 · Ek+1[u]1/2.
(cid:33)

−1/3(cid:107)Lαu(cid:107)W1,3

1

τ

+(cid:107)Lαu(cid:107)

1

˚W1,3
1

= τ
(cid:46) Ek[u]1/2 · Ek+1[u]1/2 + Ek[u]1/2 · Ek+2[u]1/2
(cid:46) Ek[u]1/2 · Ek+2[u]1/2.

Using these estimates in (2.2.62) with σ = 1 proves (2.2.45) - (2.2.48) after appealing to

the definition of the energy with the respective weights.

69

Fix now v = Liu. Then, arguing as above with Ek ≤ Ek+1 for arbitrary k to control
, equation (2.2.62) with σ = −1 and the estimates (2.2.37) - (2.2.40)

(cid:107)LαLiu(cid:107)W1,3
with the respective choices of µ, ν = ±1 prove (2.2.49) - (2.2.52).

ρpq3(µ,ν)

The time derivative estimates are more straight forward, the σ = ±1 cases are treated

separately but similarly. The first factor in (2.2.62) is treated by

(cid:107)Lαut(cid:107)L2

1

≤ τ1/2Ek[ut],

(cid:107)Lαut(cid:107)L2−1

≤ τ

−1/2Ek[u].

Simply replacing v (cid:55)→ ut in (2.2.62) and using (2.2.41) - (2.2.44) to control the second
factor (cid:107)Lαut(cid:107)W1,3
with the respective choices of µ, ν = ±1 proves (2.2.53) - (2.2.60)

after appealing to the energies with the respective weights.

ρpq3(µ,ν)

, see (2.2.62). As we saw previously in the wave case, specif-

Remark 2.2.18. For the estimates when r > 6 in the previous proof we made the choice
of interpolating L2∗ with W 1,3∗
ically the proof of (2.2.19), we can also obtain estimates interpolating L6∗ with W 1,3∗
stead. For brevity we leave out these cases and various other interpolations.
Proposition 2.2.19 (d = 4). When r ∈ [2,4],

in-

τ

τ

−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)
−1/r(cid:107)u(cid:107)

τ

τ

˚Wk,r

1 (Στ )

˚Wk,r
2r−3(Στ )

˚Wk+1,r
r−3 (Στ )
˚Wk+1,r
3r−7 (Στ )

(cid:46) (Ek[u])

(cid:46) Ek[u](τ),
4−r
r
4−r
r
4−r
r

(cid:46) (Ek[u])

(cid:46) (Ek[u])

· (Ek+1[u])
· (Ek+1[u])
· (Ek+2[u])

2r−4
r
2r−4
r
2r−4
r

,

,

.

(2.2.63)

(2.2.64)

(2.2.65)

(2.2.66)

70

For the time derivatives, the following estimates hold:

τ

τ2−5/r(cid:107)ut(cid:107)
−1/r(cid:107)ut(cid:107)
τ1−1/r(cid:107)ut(cid:107)
−1+3/r(cid:107)ut(cid:107)

˚Wk,r

1 (Στ )

˚Wk,r
2r−3(Στ )
˚Wk,r
r−3(Στ )

(cid:46) (Ek[ut](τ))

(cid:46) (Ek[ut](τ))

(cid:46) (Ek[u](τ))

(cid:46) (Ek[u](τ))

4−r
r
4−r
r
4−r
r
4−r
r

· (Ek+1[u](τ))
· (Ek+1[ut](τ))
· (Ek+1[u](τ))
· (Ek+1[ut](τ))

2r−4
r
,
2r−4
r
2r−4
r
,
2r−4
r

.

τ

˚Wk,r
3r−7(Στ )
When r > 4, the following estimates hold:

(2.2.67)

,

(2.2.68)

(2.2.69)

(2.2.70)

(2.2.71)

(2.2.72)

(2.2.73)

(2.2.74)

,

,

,

.

Ek+1[u](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2

r

,
r−2
2r ,

r+2
2r

(Ek[u](τ))

r−2
r
· (Ek+1[u](τ))

(Ek[u](τ))2/r · (Ek+1[u])
(Ek[u](τ))2/r ·(cid:16)
(Ek[u](τ))2/r · (Ek+2[u](τ))
(Ek[u](τ))2/r ·(cid:16)
(Ek[u](τ))2/r ·(cid:16)
(Ek[u](τ))2/r ·(cid:16)

· (Ek+2[u])

(Ek[u](τ))

r+2
2r

r−2
2r ,
r−2
r

,

Ek+1[u](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2
Ek+2[u](τ)1/2 · Ek+3[u](τ)1/2(cid:17) r−2
Ek+1[u](τ)1/2 · Ek+3[u](τ)1/2(cid:17) r−2

r

r

r

τ

−1/r(cid:107)u(cid:107)

˚Wk,r
r−1(Στ )

τ

−1/r(cid:107)u(cid:107)

˚Wk,r
2r−3(Στ )

τ

τ

−1/r(cid:107)u(cid:107)

˚Wk+1,r
r−3 (Στ )

−1/r(cid:107)u(cid:107)

˚Wk+1,r
2r−5 (Στ )

(cid:46)

(cid:46)

(cid:46)

(cid:46)

71

For the time derivatives, we have:

(Ek[ut](τ))2/r ·(cid:16)
(Ek[ut](τ))2/r ·(cid:16)
(Ek[ut](τ))2/r ·(cid:16)
(Ek[ut](τ))2/r ·(cid:16)
(Ek[u](τ))2/r ·(cid:16)
(Ek[u](τ))2/r ·(cid:16)
(Ek[u](τ))2/r ·(cid:16)
(Ek[u](τ))2/r ·(cid:16)

r

,

r

r

,

Ek+1[ut](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2
Ek+1[u](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2
Ek+1[ut](τ)1/2 · Ek+2[ut](τ)1/2(cid:17) r−2
Ek+1[u](τ)1/2 · Ek+2[ut](τ)1/2(cid:17) r−2
Ek+1[ut](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2
Ek+1[u](τ)1/2 · Ek+2[u](τ)1/2(cid:17) r−2
Ek+1[ut](τ)1/2 · Ek+2[ut](τ)1/2(cid:17) r−2
Ek+1[u](τ)1/2 · Ek+2[ut](τ)1/2(cid:17) r−2

r

r

r

,

r

τ1−3/r(cid:107)ut(cid:107)

˚Wk,r
r−1(Στ )

τ

−1/r(cid:107)ut(cid:107)

˚Wk,r
2r−3(Στ )

τ1−1/r(cid:107)ut(cid:107)

˚Wk,r
r−3(Στ )

τ1/r(cid:107)ut(cid:107)

˚Wk,r
2r−5(Στ )

(cid:46)

(cid:46)

(cid:46)

(cid:46)

(2.2.75)

,

(2.2.76)

(2.2.77)

(2.2.78)

r

,

,

,

.

Proof. The proofs of these estimates are treated in the same way as the proof of Propo-
sition 2.2.17, so we merely highlight the differences. For estimates (2.2.63) - (2.2.70) we

solve (GNAWSpqr) with

d = 4

q = 2

p = 2

1 + βq = µ
1− p + αp = ν

1/r = θ/q + (1− θ)/p

∗

,

where µ, ν can again take the values ±1. Denoting the weight
ρpqr(µ, ν) def= 1 + (θβ + (1− θ)· α)r,

the interpolation inequality yields

τ1/2−1/r(cid:107)Lαv(cid:107)Lr

ρpqr (µ,ν)

(cid:33) 4−r

r

(cid:32)
(cid:107)Lαv(cid:107)L2

µ

(cid:46)

·(cid:18)(cid:107)Lαv(cid:107)W1,2

ν

(cid:19) 2r−4

r

.

72

(2.2.79)

One explicitly computes the weights as
ρpqr(1,1) = 2r − 3,

ρpqr(1,−1) = 1,

ρpqr(−1,−1) = r − 3,

ρpqr(−1,1) = 3r − 7.

Replacing µ, ν = ±1 and v (cid:55)→ (u, ut) or (Liu, ut) in (2.2.79) then proves (2.2.63) - (2.2.70)
by following the same analysis as in the proof of Proposition 2.2.17.

For the estimates when r > 4, we appeal to the borderline (GNAWSdqr) inequality with

d = 4,

q = 2,

1 + βq + αq = σ ,

(1− d)(1 + θβr) + αd = ρpq4(µ, ν),

∗
1/r = θ/q + (1− θ)/(r + 1

),

where σ can take the values ±1 and ρpq4(µ, ν) is as above. Denoting the weight

ρd=4(σ , µ, ν) def= 1 + θβr + αr,

the borderline inequality yields

(cid:16)

τ1/4(cid:17) r−2

r (cid:107)Lαv(cid:107)Lr

(cid:18)(cid:107)Lαv(cid:107)L2

σ

(cid:19)2/r ·

(cid:46)

(cid:107)Lαv(cid:107)W1,4

ρpq4(µ,ν)

 r−2

r

.

(2.2.80)

ρd=4(σ ,µ,ν)

This inequality is valid for r > 2 so in particular r > 4. One explicitly computes the

weights

ρd=4(1,1,−1) = r − 1,
ρd=4(1,1,1) = 2r − 3,
ρd=4(1,−1,−1) = r − 1,
ρd=4(1,−1,1) = 2r − 3,

ρd=4(−1,1,−1) = r − 3,
ρd=4(−1,1,1) = 2r − 5,
ρd=4(−1,−1,−1) = r − 3,
ρd=4(−1,−1,1) = 2r − 5.

Replacing σ , µ, ν = ±1 and v (cid:55)→ (u, ut) or (Liu, ut) in (2.2.79) then proves (2.2.71) - (2.2.78)
by following the same analysis as in the proof of Proposition 2.2.17.

73

Remark 2.2.20. We note that even though the estimates for r > 4 in Proposition 2.2.19

had almost the same proofs as the ones for r > 6 in Proposition 2.2.17, there is a no-
table difference between the two: there are only four distinct weights for ρd=4(±1,±1,±1)
while there are six distinct weights for ρd=3(±1,±1,±1). The reason for this is that we
controlled the second factor of (2.2.61) using the non-borderline estimates derived from
(GNAWSpqr) with 3 ∈ [2,6]. On the other hand, the second factor of (2.2.79) was esti-
mated with the end point 4 ∈ [2,4].

74

CHAPTER 3

TOTALLY GEODESIC WAVE MAPS

3.1 Introduction

Geodesics are a central object of study in both Riemannian and Lorentzian geometry.

In the former, they are the curves representing the shortest paths between nearby points.

In the latter, timelike geodesics describe the motion of a free falling test particle. It is

therefore not surprising that functions between manifolds that preserve geodesics have
received extensive attention by mathematicians. More precisely, a map f : N → M be-
tween pseudo-Riemannian manifolds is said to be totally geodesic if it maps geodesics to

geodesics.

Particular attention has been payed to totally geodesic maps in the elliptic setting
because they are automatically harmonic maps. A map φ : N → M between Riemannian
manifolds is said to be harmonic if it is a critical point of the energy functional

(cid:90)

S[φ] def=

1
2

(cid:104)dφ,dφ(cid:105)

∗

N⊗φ

−1T M dvolh .

T

(3.1.1)

In local coordinates on the target (M, g), the Euler-Lagrange equations (ELE) take the

N

form

jk(φ)(cid:104)dφj,dφk(cid:105)h = 0.
jk(φ) are the Christoffel sym-
Here ∆h is the Laplace-Beltrami operator on (N , h) and Γ i
bols of M evaluated along the image of φ. Harmonic maps simultaneously generalize

∆hφi + Γ i

(3.1.2)

geodesics and harmonic functions, as can be seen directly from the equation (3.1.2) and
setting N = R or M = R, respectively. The history of totally geodesic maps and harmonic

maps has been intertwined since the foundational paper of Eells and Sampson [ES64].

There the authors gave restrictions on the curvatures of M and N that imply the exis-

75

tence of harmonic maps, as well as sufficient conditions for harmonic maps to be totally

geodesic.

Our work on totally geodesic maps in this chapter serves to expand the literature

to the Lorentzian regime by setting (N , h) = (R1+3, m).

In this case (3.1.2) becomes a

hyperbolic equation, and analyzing the solution φ amounts to an initial value problem.

In this setting we say a solution to the ELE is a wave map if it solves

(cid:3)mφi + Γ i

jk(φ)(cid:104)dφj,dφk(cid:105)m = 0.

(3.1.3)

The theory of wave maps has a rich history, for a general review see [SS98, Kri07]. Our
results apply to the physically relevant cases where the target Mn = Sn or Hn. The former

case models the nonlinear sigma model in plasma physics [GML60], while the latter has

applications in general relativity [CBM96].

The starting point of our discussion is motivated by the Riemannian setting. A well

known result of Vilms shows that “if N is complete, then every totally geodesic map
φ : N → M factors as

N

ΦS

ΦI

B

M;

(3.1.4)

with ΦS a Riemannian submersion and ΦI a Riemannian immersion, both being totally
geodesic” [Vil70]. This chapter is concerned with the global stability of certain infinite
energy totally geodesic maps from Minkowski space R1+d with d ≥ 3 into a spaceform
(Mn, g). We consider as our background solutions those mappings that factor as

R1+d

ϕS

ϕI

R

M;

(3.1.5)

where, denoting by e the standard Euclidean metric on R, the mapping ϕS is a semi-
Riemannian submersion1 to either (R, e) or (R,−e), and ϕI is a Riemannian immersion
from (R, e) to (M, g). The semi-Riemannian submersion ϕS can be classified as spacelike
1A semi-Riemannian submersion ϕ : N → M is necessarily an isometry on the hori-

zontal space normal to fibres. See [O’N83, P. 212] for a precise definition.

76

or timelike2 depending on whether its codomain R is considered as being equipped with
e or −e. Theorem 3.1.1 is a rough version of our main results. See Theorem 3.6.1 for the
precise statement of TL and Theorem 3.7.1 for the precise statement of SL.
Theorem 3.1.1 (Rough version). Fix d ≥ 3. A totally geodesic map satisfying the factorization
(3.1.5) is globally nonlinearly stable as a solution to the initial value problem for the wave maps

equation under compactly supported smooth perturbations, provided that either

TL ϕS is timelike and (M, g) is a negatively-curved spaceform, or

SL ϕS is spacelike and (M, g) is a positively-curved spaceform.

We emphasize the following key point: as ϕS : R1+d → R is an orthogonal projection
onto a 1-dimensional subspace, it automatically satisfies the linear wave equation. Further-

more, the total geodesy of the composed map trivially implies that the image of ϕI is a
geodesic in M. This latter point follows from [ES64, Corollary 5A; (21)].

Stability of factored (non-totally geodesic) wave maps of the form

R1+d

ϕW

R

ϕG

M

has been studied by Sideris and Grigoryan [Sid89, Gri10]. In his paper, Sideris was mo-

tivated to study the stability of wave maps localized to a geodesic to overcome singu-

larity issues discovered in [Sha88], where singular solutions for the nonlinear σ-model
R1+3 → S3 were constructed whose range contained a hemisphere. Our problem is re-
lated to [Sid89, Gri10] in that their background is also the composition of a geodesic ϕG
and a solution to the linear wave equation ϕW. Contrastingly, their ϕW is an arbitrary fi-
nite energy solution to the linear wave equation and hence ϕG◦ϕW is not totally geodesic.
This provides yet another motivation for our problem where we assume that ϕS is as-
sumed to be a semi-Riemannian submersion and hence has infinite total energy. This

2Note that by definition, a semi-Riemannian submersion cannot be null. We always

equip the real line R, as the domain of ϕI, with +e.

77

introduces considerable difficulties as the finite energy backgrounds of [Sid89, Gri10] de-

cay at the expected rate of finite-energy waves, whereas ours are non-decaying.

3.2 Explanation of results

In this section we clarify the geometric set-up for Theorem 3.1.1 and expand on the

precise analytical difficulties and conclusions of the result.

In this chapter we adapt the geometric framework of [Sid89, Gri10], where we write
the equations of motion for the perturbation in a tubular neighborhood R × N of the
geodesic ϕI(R) ⊂ M (here R parametrizes the geodesic and N the normal (n−1)-directions).
The main geometric contribution of this chapter is Proposition 3.4.5, which shows that
the equations for the perturbation u = (u1, (cid:126)u ) ∈ R×N decouple into a system of wave and

Klein–Gordon equations:(cid:3)u1 = F1u· m(du,dϕS) + O(|u|3 +|∂u|3),

(cid:3)(cid:126)u − (cid:126)M (cid:126)u = (cid:126)Fu· m(du,dϕS) + O(|u|3 +|∂u|3).

(3.2.1)

Here F1, (cid:126)F are functions of the curvature of (M, g) restricted to the geodesic ϕI. The (cid:126)M are
the masses of (cid:126)u, and as a consequence of the spaceform assumption on M, Proposition
3.4.5 implies (cid:126)M = κm(dϕS,dϕS) where κ is the sectional curvature of M. Hence, the
assumptions on ϕS in Theorem 3.1.1 are there to at minimum guarantee linear stability,
i.e. make the Klein–Gordon terms (cid:126)u have positive masses.

The computations leading to Proposition 3.4.5 and (3.2.1) hinge on a careful Taylor
expansion of the Christoffel symbols Γ about the geodesic ϕI(R). This is where our geo-
metric approach differs from that of [Sid89, Gri10]. In their works, the authors need only

perform a rough quadratic Taylor expansion because they are able to utilize the decay

properties of their background. In our work, we perform a precise cubic expansion to

capture the lowest order nonlinear structures. Our precise control on these Taylor coef-

ficients reveal weak null-structures that prevent resonant interactions that could lead to

finite-time blow up, see (3.4.2) and Lemma 3.4.4. Finally, we remark that the geometry

78

of the target manifold M in [Sid89, Gri10] is arbitrary. Morally, the premise for their sta-
bility result is that their background solution converges to the same point in ϕG(0) ∈ M
(as a consequence of finite energy!) as one moves in any direction on R1+d to infinity.
As our background is not decaying, moving along generic directions on R1+d does not
imply that the image ϕI◦ϕS in M converges to a single point. Our spaceform assumption
is then a natural way to ensure some sort of homogeneity of the geometry of M as one
moves towards infinity on R1+d along the mapping ϕI ◦ ϕS. See Figure 3.1 below.

(a) The image ϕG ◦ ϕW ⊂ M

(b) The image φI ◦ φS ⊂ M

Figure 3.1: The two different background solutions.

Remark 3.2.1. As we will see, for the energy estimates of higher derivatives of u we need

first and second order commutations of the equations (3.2.1) with the Lorentz boosts
Li = t∂xi + xi∂t. Under the spaceform assumption, the functions F and (cid:126)F are constant and
hence vanish when differentiated. In the case that the curvature is not constant, these
coefficients can grow: LiF ≈ t(F
and (3.4.9), each order of Taylor expansions introduce an additional Klein–Gordon factor
(which has a linear decay rate of |(cid:126)u| (cid:46) t
(cid:88)

). Using the weak null structures revealed in (3.4.2)

−d/2):
∂αF(0)((cid:126)u )N + O(|(cid:126)u|N +1).

As d ≥ 3, this decay can overcome the aforementioned growth. And, consequently, we can
easily relax the spaceform assumption to targets (M, g) with the following property: along
ϕI(R), the metric g agrees with a spaceform up to fourth order. See also Remark 3.3.1

(cid:48)

F =

|α|≤N

79

MϕG◦ϕWϕG(0)bMϕI◦ϕSThe main analytic contributions of the present chapter are Theorems 3.6.1 and 3.7.1,

which are precise versions of 3.1.1. They provide an open set (in a suitable Sobolev topol-

ogy) of initial data such that the Cauchy problem for (3.2.1) has a global solution in spatial

dimension d = 3. For our analysis of the equations of motion we use the physical space

vector field method and its related energy estimates.

Remark 3.2.2 (Dimensionality). We restrict the proof of the main theorem and the dis-

cussions below to d = 3 because stability of quadratic wave-Klein–Gordon systems is a
known standard result in dimensions d ≥ 4. This leaves the case of spatial dimension two
open for this problem. Recently Ma has made headway in the two dimensional analy-

sis of wave-Klein–Gordon systems [Ma19]. However, using Ma’s terminology, the result
of [Ma19] does not apply to the “strongly coupled” nonlinearities u·m(du,dϕS) of (3.2.1).

We note that, for d = 3, global existence of coupled wave and Klein–Gordon equations

is known, see the monograph by LeFloch and Ma [LM14].

In this chapter we give a

short proof of their result using a variant (see [Won17b]) of the the hyperboloidal method

developed in [LM14]. We remark that in that same article, Wong proved global existence
for (3.1.3) for all d ≥ 2 where φ is a small perturbation of a constant. Our main analytical
tool is the vector field method adapted to the hyperbolas Στ introduced in Chapter 2.
This allows us to avoid using the purely spatial rotations, and as our most important
analytic contribution, to prove stability of (3.2.1) assuming that the initial data is in H3,

see Remark 3.2.4. To the best of our knowledge, the best prior results in d = 3 using purely

physical space techniques for wave-Klein–Gordon systems was stability with initial data
at the level of H6 [LM14].

Remark 3.2.3. We note that there are technical differences between the systems studied

in [LM14] and (3.2.1). LeFloch and Ma considered a quasilinear system of wave-Klein–
Gordon equations, which introduces additional difficulties. On the other hand, their non-

linearities satisfy the classical null condition of Klainerman [Kla84] which allows them to

80

extract improved decay from all quadratic nonlinearities. We emphasize that our nonlin-

earities do not satisfy the classical null condition, and hence we are not able to extract the

improved decay present in [LM14].

Remark 3.2.4 (Regularity). To guarantee global existence it suffices that the initial per-
turbation is sufficiently small in H3 × H2; this level of smallness is enough to guarantee
C1 convergence. Note that a standard persistence of regularity argument implies that if
initial data is in H4 × H3 with smallness in H3 × H2, this guarantees that the solutions
remain small in H3× H2, converges to 0 in C1, and has bounded C2 norm globally. As we
will see, pushing the regularity down to H3 × H2 requires our bootstrap mechanism to
allow for growth in the top order energies, see Proposition 3.6.6. Roughly speaking, this is
−3/2 is available
because the improved linear decay for Klein–Gordon derivatives |∂(cid:126)u| (cid:46) t
using only the third order energies (at the level of H4). Instead, by sacrificing a decay
−1/2, we can rely on the interpolated Sobolev embeddings of [AW19b] to close
factor of t
the argument at the level of H3. Had we assumed smallness in H4, our arguments could
easily be adapted to prevent this growth, guaranteeing C2 convergence.

What makes our argument run through is that the spaceform curvature restrictions

expose hidden weak null structures that make harmful wave–wave resonant terms from

the quadratic and cubic nonlinearities vanish. More precisely, we will show that the
undifferentiated factor u1 is missing in u· m(du,dϕS) and |u|3 in equations (3.2.1). We also
show that the quadratic nonlinearity for the wave solution u1 is of the form (cid:126)u · ∂(cid:126)u.

dimension 3 is t

There are numerous ramifications of these exposed null conditions. Firstly, as we are
unable to use the Morawetz vector field as a multiplier, the available decay rate for u1 in
−1/2. This means that terms of the form (u1)2 or (u1)3 (which are excluded
by our exposed null structures) could lead to finite-time blow up. Secondly, it is crucial
that only (cid:126)u appear in the quadratic nonlinearity for u1 because its expected decay rate is
−1. This improved decay
|(cid:126)u|+|∂(cid:126)u| (cid:46) t

−3/2, compared to the derivative wave decay |∂u1| (cid:46) t

81

for the nonlinearity of u1 will feed-back into the Klein–Gordon equations when we try to
estimate |(cid:126)u · m(du1,dϕS)|, allowing us to close our estimates.
3.3 Geodesic normal coordinates

In this section we set up the geometric tools and notations needed for the rest of the

sequel. We first consider the case where M is an arbitrary complete Riemannian manifold

and later specialize to the spaceform setting.

We will construct a system of coordinates for a tubular neighborhood of an arbitrary
geodesic, in which the restriction of the Christoffel symbols to the geodesic vanish. For

a comprehensive treatment of the geometry of geodesic normal coordinates, see the book

by Alfred Gray [Gra04]. He analyzes a generalization of geodesic normal coordinates

called Fermi coordinates. They give a local description of a tubular neighborhood about
an embedded submanifold P ⊂ M of arbitrary codimension.

Consider our complete Riemannian manifold (Mn, g) and let γ : R → M be a fixed
⊥} denote the
geodesic parametrized by arc-length. Let V = {(γ(t), v) | t ∈ R, v ∈ Tγ(t)M
normal bundle along the geodesic γ. We write Vγ for the fibres above γ and Vγ(t0) when
we wish to specify the fibre above a specific point γ(t0). We now construct an explicit
local orthonormal frame of a subbundle of V and use it to define the so called geodesic
normal coordinates by the exponential map.

We will parametrize the tubular neighborhood of γ by R×N , where

N def= {(cid:126)x = (x2, . . . , xn) ∈ Rn−1 | |(cid:126)x| < rfoc(γ)}.

Here rfoc(γ) is the focal radius of γ, which is defined to be the maximal radius such that
the normal exponential map, see (3.3.1), is non-critical on the normal disc bundle of γ

of radius rfoc(γ). In the subsequent analysis of the wave map problem, we can guarantee
that rfoc(γ) > 0 because of the spaceform assumption.

82

Remark 3.3.1. The generalization of Remark 3.2.1 to targets (M, g) such that the met-
ric agrees with a spaceform up to fourth order along the geodesic ϕI(R) should also be
accompanied with the following assumptions:

• the focal radius rfoc(γ) is bounded away from zero;

• higher derivatives of the metric in geodesic normal coordinates are bounded away

from infinity.

Denote e1

def= ˙γ(0) and use it to define an orthonormal basis

⊥ def= (e2, . . . , en).

e

⊥

of Vγ(0). For arbitrary x1 ∈ R, let (e1(x1), e
(e1, e
e1(x1) = ˙γ(x1) by definition.

) along γ and note that e

⊥

For (x1, (cid:126)x ) ∈ R×N , define γ

⊥

(x1)) be defined by parallel transporting

⊥

(x1) is an orthonormal frame for V

γ(x1). Also note that

defined by

⊥

γ

(x1; (cid:126)x,0) = γ(x1),

⊥

˙γ

(x1; (cid:126)x,0) =

n(cid:88)

xkek(x1).

(x1; (cid:126)x, s) as the unique geodesic (with path parameter s)

Remark 3.3.2. We identify the original geodesic γ with {(cid:126)x ≡ 0} because we have γ
⊥
γ

(x1;0,0) = γ(x1) for any s ∈ [0,1] by uniqueness of ODEs.

k=2

We can now define the normal exponential map

which is a map

⊥
exp
γ(x1)

⊥

((cid:126)x ) def= γ

(x1, (cid:126)x,1),

exp

⊥
γ(x1)

: V

γ(x1)

→ M.

⊥

(x1;0, s) =

(3.3.1)

This normal exponential map shares many features with the usual one from Riemannian

geometry. For example, the inverse function theorem and the following computation

83

⊥
show that N is non-trivial and that exp
γ(·)(·) is indeed a smooth immersion from R×N
to a a tubular neighborhood, which we denote as T , around γ ⊂ M:

(cid:18)

⊥
exp
γ(x1)

d

(cid:19)
{(cid:126)x=0} ((cid:126)y) =
=

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s=0
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s=0

d
ds
d
ds

= (cid:126)y.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s=0

⊥
exp
γ(x1)

(s(cid:126)y) =

d
ds

γ(x1; (cid:126)y, s)

γ(x1; s(cid:126)y,1)

(3.3.2)

Remark 3.3.3. The “d” in (3.3.2) denoted the differential of the map in the (cid:126)x variables.
⊥
γ(·)(·) is technically a map on domain R × N , in the future we write dx1 as the

Since exp
differential in the x1 variable. It is easy to see that, restricted to {(cid:126)x = 0},

(cid:18)

(cid:19)

dx1

exp

⊥
γ(y1)

0

(e1) = e1(y1).

This allows us to define the geodesic normal coordinates by the preimage of the expo-

nential map

More explicitly, if

−1
γ

exp

T

R×N .

q = expγ(x1)((cid:126)x) ∈ T ,

then q can be written in geodesic normal coordinates by (x1, (cid:126)x ) = (exp

⊥

−1
γ (q).

)

⊥
−1 is the pre-image of the exponential map. Technically, exp
⊥
Remark 3.3.4. Here (exp
γ )
γ
it is not a bona fide diffeomorphism near self intersections of γ. In the case that γ is an
⊥
γ is a true diffeomorphism. As we are in the perturbative
embedded geodesic, then exp
regime for the ensuing analysis of the wave maps equation, our considerations are local

so we will ignore any self intersections.

The following lemma sets up the key geometric tools that we need for our analysis.

Even though the proof is standard, we include it for the sake of completion:

84

Lemma 3.3.5. Let ∂

∂xi , i = 1, . . . , n be the coordinate vector fields defined by (x1, (cid:126)x ). Let y1 ∈ R

be arbitrary. Then the following identities hold

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)

∇

gij

= 0

= 0

∂kgij

∂
xj

∂

xi

= δij

∂
∂xi

= ei(y1)

i ∈ {1, . . . , n}
i, j ∈ {1, . . . , n}
i, j, k ∈ {1, . . . , n}
i, j ∈ {1, . . . , n}
i, j, k ∈ {1, . . . , n}
i, j, k, m ∈ {1, . . . , n}
i, j, k, m, p ∈ {1, . . . , n}
Proof. The proof of (3.3.3) for i ∈ {2, . . . , n} follows by definition and (3.3.2)
(cid:17)

xj , ∂xk(cid:105)g
xj , ∂xk(cid:105)g
(cid:18)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)
(cid:19)

= 0
= ∂m(cid:104)∇
(cid:104)∇

= ∂2
mp

∂2
mpΓ k
ij

∂mΓ k
ij

∂

∂
xi

∂

∂

xi

Γ k
ij

(cid:16)

= d

expγ(y1)

(cid:126)x=0

ei(y1)

= ei(y1).

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)

∂
∂xi

The case of i = 1 follows from the discussion in Remark 3.3.3:

(3.3.3)

(3.3.4)

(3.3.5)

(3.3.6)

(3.3.7)

(3.3.8)

(3.3.9)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)

∂
∂x1

= e1(y1).

Equation (3.3.4) follows immediately because {ei(y1) | i = 1, . . . , n} were defined by parallel
transporting an orthonormal set and because parallel transport is an isometry.

The definition of parallel transport implies
= ∇

∇

x1 ∂x1
∂

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)

∂

x1 ∂

xj

= 0

for j ∈ {2, . . . , d}. Because we are using coordinate vector fields and ∇ is torsion free,

∇
∂xm ∂xk = ∇

∂

∂xm

xk

85

even away from γ. This implies

∇

∂

∂x1

xj

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)

= 0.

Next we note that any X ∈ V
a geodesic by definition, we have that ∇XX = 0. In particular we have that

γ(y1) is tangent to the curve expγ(y1)(sX). Since this curve is

0 = ∇

∂

(∂xi + ∂

xj ) = ∇

xj +∇

∂

∂

∂
xi

∂xi

xj

xi +∂

xj

whenever i, j ∈ {2, . . . , n}. The torsion condition and setting s = 0 proves

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)

= 0.

∇

∂

∂
xj

xi

This concludes the proof of (3.3.6).

Using that the Levi-Civita connection is metric, we see

∂xk gij = ∂xk(cid:104)∂xi , ∂

xj(cid:105)g = (cid:104)∇

xk ∂xi , ∂

xj(cid:105)g +(cid:104)∂xi ,∇

xj(cid:105)g.

xk ∂

Restricting this computation to γ concludes the proof of (3.3.5) using (3.3.6).

For the Christoffel symbols, recall that they are defined by

∇

∂

xi

∂

xj = Γ m

ij ∂xm.

Taking the inner product with ∂xk , using the metric structure (3.3.4), and (3.3.6) proves

0 = (cid:104)∇

∂

xj , ∂xk(cid:105)g

∂

xi

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)γ(y1)

.

= Γ k
ij

Finally, we compute
∂m(cid:104)∇

∂

xi

xj , ∂xk(cid:105)g = ∂m(Γ l

∂

ijglk) = ∂mΓ l

ijglk + Γ l

ij∂mglk.

This and (3.3.4), (3.3.5) show (3.3.8). Similarly, (3.3.9) follows.

86

3.4 Perturbed system and reduction to wave-Klein–Gordon system

We now return to the wave map equation and describe the precise construction for

the perturbation to our totally geodesic background

R1+d

ϕS

ϕI

R

M.

Recall that ϕS is a semi-Riemannian submersion, so in accordance with the discussion
in Section 3.1 regarding [Vil70], we prescribe ϕS to be a linear function (cid:96) : R1+d → R
satisfying m(d(cid:96),d(cid:96)) = ±1. As ϕI is an immersed geodesic in M, we identify it with the
zero cross-section about the normal bundle of ϕI(R) ⊂ M (see Remark 3.3.2):

R

ι

R×N

⊥
exp
ϕI

M;

⊥
where the first map is the inclusion and the second map is the restriction exp
ϕI

Equipping R × N with the pull-back metric, we then look for maps of the form φ def=

(cid:12)(cid:12)(cid:12)(cid:126)x=0.

ι◦ (cid:96) + u which are solutions to the wave maps equation (3.1.3) on3

R1+d

ι◦(cid:96)+u

R×N .

Here addition is taken coordinate wise on R×N . Consequently, the perturbation of our
totally geodesic background takes the form

exp

⊥
ϕI(ι◦ (cid:96) + u) : R1+d

M

which is also a solution to the wave maps equation. Of course, u ≡ 0 corresponds to the
background ϕI ◦ ϕS. As we consider (cid:96) fixed, the equations of motion (3.1.3) for φ reduce
to a Cauchy problem for the perturbation u.

Let (x1, . . . , xn) be the geodesic normal coordinates about ϕI constructed in section 3.3.

In these coordinates u = (u1, u2, . . . , un) = (u1, (cid:126)u ) and hence φ takes the form

φ = ((cid:96) + u1, u2, . . . , un) = ((cid:96) + u1, (cid:126)u ),

87

Figure 3.2: The perturbation as a section of the normal bundle about ϕI.

see Figure 3.2.

The equations of motion (3.1.3) take the form

(cid:16)
(cid:16)

(cid:96) + u1, (cid:126)u

(cid:96) + u1, (cid:126)u

(cid:17)· m(dφj,dφk) = 0,
(cid:17)· m(dφj,dφk) = 0,

i ∈ {2, . . . , n}.

(3.4.1)

(cid:3)u1 + Γ 1
jk
(cid:3)ui + Γ i
jk

We compute

m(dφj,dφk) = m(d(cid:96),d(cid:96))δ

1 + m(d(cid:96),duj)δk
Taylor expanding Γ about the geodesic ϕI ◦ (cid:96) we see

j
1δk

1 + m(d(cid:96),duk)δ

j
1 + m(duj,duk).

jk((cid:96) + u1, (cid:126)u) = Γ i
Γ i

jk((cid:96),(cid:126)0) +

jk((cid:96),(cid:126)0)um + O(|u|2).

∂mΓ i

n(cid:88)

m=1

(cid:124)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:125)
∂1··· ∂1

q times

We pause at this juncture to make some reductions. From (3.3.7) we see that the first term
jk((cid:96),(cid:126)0) = 0 for arbitrary (cid:96), we see that

on the right hand side vanishes. Moreover, since Γ i

Γ i
jk((cid:96),(cid:126)0) = 0

(3.4.2)

for all i, j, k ∈ {1, . . . , n} and any positive integer q.

Before we expand the Christoffel symbols up to third order, we introduce the fol-
if A is an m-tuple with elements drawn from {1, . . . , n} (namely that

lowing notation:
A = (A1, . . . , Am) with Ai ∈ {1, . . . , n}), for a scalar function f we denote

∂Af def= ∂

xA1

··· ∂xAn f .

3As we will see, initial data for u can be chosen small enough so that the perturbation

ι◦ (cid:96) + u = ((cid:96) + u1, (cid:126)u ) ∈ R×N .

88

u1~uϕIBy |A| we refer to its length, namely m. Given a vector x = (x1, . . . , xn) ∈ Rn, we denote

xA def= xA1 ··· xAm.

We also introduce the shorthand

σ def= m(d(cid:96),d(cid:96))

(3.4.3)

to denote the size of d(cid:96) as measured by the Minkowski metric. Expanding out the
Christoffel symbols up to third order in u, we see that the equations can be expressed

as

n(cid:88)

m=2

(cid:3)u1 +

n(cid:88)

m=2

(cid:3)ui +

11((cid:96),(cid:126)0)uA · σ

∂AΓ 1

∂mΓ 1

11((cid:96),(cid:126)0)um · σ =
d(cid:88)

− 2

∂mΓ 1

m=2

− n(cid:88)

m=2

j1((cid:96),(cid:126)0)um · m(duj,d(cid:96))− (cid:88)
(cid:88)
n(cid:88)

|A|=2
A(cid:44)(1,1)
−

|A|=2
A(cid:44)(1,1)

|A|=3
A(cid:44)(1,1,1)

∂mΓ 1

jk((cid:96),(cid:126)0)um · m(duj,duk)− 2

∂AΓ 1

j1((cid:96),(cid:126)0)uA · m(duj,d(cid:96))

11((cid:96),(cid:126)0)uA · σ + h.o.t.,

∂AΓ 1

(3.4.4)

∂mΓ i

11((cid:96),(cid:126)0)um · σ =
d(cid:88)

− 2

∂mΓ i

m=2

− n(cid:88)

m=2

j1((cid:96),(cid:126)0)um · m(duj,d(cid:96))− (cid:88)
(cid:88)
− (cid:88)

|A|=2
A(cid:44)(1,1)

|A|=2
A(cid:44)(1,1)

jk((cid:96),(cid:126)0)um · m(duj,duk)− 2

∂mΓ i

j1((cid:96),(cid:126)0)uA · m(duj,d(cid:96))

∂AΓ i

11((cid:96),(cid:126)0)uA · σ

∂AΓ i

11((cid:96),(cid:126)0)uA · σ + h.o.t.

∂AΓ i

(3.4.5)

|A|=3
A(cid:44)(1,1,1)

89

Remark 3.4.1. To clarify, the sums involving A on the right hand side of (3.4.4) and

(3.4.5) are summing over m-tuples A = (A1, . . . , Am) excluding the vertex Ai = 1 for all i.

That is, for example,(cid:88)

n(cid:88)

α1,α2=1

(α1,α2)(cid:44)(1,1)

∂AΓ 1

11uα def=

|A|=2
A(cid:44)(1,1)

∂xα1 ∂xα2 Γ 1

11uα1 · uα
2 .

That we are able to do this is of course a consequence of (3.4.2).

Remark 3.4.2. As stated previously, repeated latin indices are implicitly summed over
n(cid:88)
{1, . . . , n} unless otherwise stated. For example,
j1um · m(duj,d(cid:96)) def=

j1um · m(duj,d(cid:96)).

n(cid:88)

∂mΓ i

∂mΓ i

m=2

m=2
j=1

Remark 3.4.3. In the equations “h.o.t.” represents higher order terms of the form

h.o.t. (cid:46) CM(|u|4 +|∂u|4)· f (u, ∂u),

where CM denotes some constant depending on the derivatives of the Christoffel symbols
of the target manifold restricted to the geodesic. Here f : Rn(d+2) → R is an arbitrary
smooth function.

We are able to find explicit formulas for the coefficients of the linear terms:
Lemma 3.4.4. Let i, k, m ∈ {1, . . . , n}. Then, restricted to the geodesic ϕI, we have

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ϕI

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ϕI

.

∂mΓ i
k1

= Rm1ki

Proof. Denote the coordinate vector fields ∂xi = Xi. Then compute

∇Xk X1, Xi(cid:105)g +(cid:104)∇Xk X1,∇XmXi(cid:105)
∇X1Xk, Xi(cid:105)g +(cid:104)∇Xk X1,∇XmXi(cid:105)g

∂m(cid:104)∇Xk X1, Xi(cid:105)g = (cid:104)∇Xm
= (cid:104)∇Xm
= (cid:104)R(Xm, X1)Xk, Xi(cid:105)g +(cid:104)∇X1
= (cid:104)R(Xm, X1)Xk, Xi(cid:105)g + ∂1(cid:104)∇XmXk, Xi(cid:105)g

∇XmXk, Xi(cid:105)g +(cid:104)∇Xk X1,∇XmXi(cid:105)g

−(cid:104)∇XmXk,∇X1Xi(cid:105)g +(cid:104)∇Xk X1,∇XmXi(cid:105)g.

90

Restricting to ϕI, equations (3.3.6) and (3.3.8) yield

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ϕI

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ϕI

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ϕI

.

∂mΓ i
k1

= Rm1ki

+ ∂1Γ i
mk

The second term on the right hand side vanishes from the discussion immediately before

the statement of the lemma.

From this lemma we immediately see that ∂mΓ 1
11

|ϕI = Rm111|ϕI = 0 from the anti-
|ϕI =
symmetric property of the Riemann curvature tensor. On the other hand, ∂mΓ i
11
Rm11i|ϕI, which in general does not vanish. We have then proved the following proposi-

tion.

Proposition 3.4.5. The perturbation equation (3.4.1) decouples into the following system of
wave and Klein–Gordon equations for the unknowns (u1, (cid:126)u ):

(cid:3)u1 =

− 2

− n(cid:88)

m=2

m=2

d(cid:88)

Rm1j1((cid:96),(cid:126)0)um · m(duj,d(cid:96))− (cid:88)
(cid:88)
jk((cid:96),(cid:126)0)um · m(duj,duk)− 2
n(cid:88)

|A|=2
A(cid:44)(1,1)
−

|A|=2
A(cid:44)(1,1)

∂mΓ 1

11((cid:96),(cid:126)0)uA · σ

∂AΓ 1

∂AΓ 1

j1((cid:96),(cid:126)0)uA · m(duj,d(cid:96))

11((cid:96),(cid:126)0)uA · σ + h.o.t.,

∂AΓ 1

(3.4.6)

|A|=3
A(cid:44)(1,1,1)

91

n(cid:88)

m=2

(cid:3)ui +

Rm11i((cid:96),(cid:126)0)umσ =

− 2

∂mΓ i

m=2

d(cid:88)

Rm1ji((cid:96),(cid:126)0)um · m(duj,d(cid:96))− (cid:88)
(cid:88)
jk((cid:96),(cid:126)0)um · m(duj,duk)− 2
− (cid:88)

|A|=2
A(cid:44)(1,1)

|A|=2
A(cid:44)(1,1)

− n(cid:88)

m=2

11((cid:96),(cid:126)0)uA · σ

∂AΓ i

j1((cid:96),(cid:126)0)uA · m(duj,d(cid:96))

∂AΓ i

11((cid:96),(cid:126)0)uA · σ + h.o.t.

∂AΓ i

(3.4.7)

|A|=3
A(cid:44)(1,1,1)

3.4.1 Reductions when M is a spaceform

We now suppose that (M, g) is a spaceform with constant sectional curvature κ (cid:44) 0. In

this case the Riemann curvature tensor has the following form:

Rijkl = κ(gikgjl − gilgjk).

(3.4.8)

This curvature restriction has the following immediate consequence:
Lemma 3.4.6. Let m, p ∈ {2, . . . , n} and denote • for any element of {1, . . . , n}. Then, restricted
to the geodesic ϕI,

••1 = ∂2
pmΓ

•
11 = ∂3

1••Γ

•
11 = 0.

∂2
1mΓ

(3.4.9)

Proof. Denoting the coordinate vector fields ∂xi as Xi, we have already seen in Lemma
3.4.4 that

∂m(cid:104)∇Xk X1, Xi(cid:105)g = Rm1ki + ∂1(cid:104)∇XmXk, Xi(cid:105)g −(cid:104)∇XmXk,∇X1Xi(cid:105)g

+(cid:104)∇Xk X1,∇XmXi(cid:105)g

(3.4.10)

92

for any k, i ∈ {1, . . . , n}. Taking ∂1 of both sides shows
(cid:104)∇XmXk, Xi(cid:105)g −(cid:104)∇X1
∂2
1m
∇X1Xi(cid:105)g +(cid:104)∇X1

(cid:104)∇Xk X1, Xi(cid:105)g = ∂1Rm1ki + ∂2
−(cid:104)∇XmXk,∇X1

11

∇XmXk,∇X1Xi(cid:105)g

∇Xk X1,∇XmXi(cid:105)g +(cid:104)∇Xk X1,∇X1

∇XmXi(cid:105)g.

Restricting this identity on the geodesic proves

∂2
1mΓ i

k1 = ∂1Rm1ki + ∂2

11Γ i
mk

using Lemma 3.3.5. The second term vanishes because of (3.4.2). The first term vanishes
1•Γ
using the spaceform restriction (3.4.8) and (3.3.5), proving ∂2

••1 = 0.

∂2
pm

We can instead differentiate (3.4.10) by ∂p and setting k = 1 to deduce
(cid:104)∇X1X1, Xi(cid:105)g = ∂pRm11i + ∂2
∇XmX1,∇X1Xi(cid:105)g
−(cid:104)∇XmX1,∇Xp

(cid:104)∇XmX1, Xi(cid:105)g −(cid:104)∇Xp
∇X1Xi(cid:105)g +(cid:104)∇Xp

∇X1X1,∇XmXi(cid:105)g +(cid:104)∇X1X1,∇Xp

p1

∇XmXi(cid:105)g.

Restricting to the geodesic and using Lemma 3.3.5 similarly proves

∂2
pmΓ i

11 = ∂pRm11i + ∂2

p1Γ i

m1.

Again the curvature term vanishes using the spaceform restriction (3.4.8) and (3.3.5),
1•Γ
while the second term vanishes using the already proved ∂2

••1 = 0 and that regular

partial derivatives commute.

We have show that, for arbitrary (cid:96), ∂2••Γ
1••Γ
∂3

•
11((cid:96),(cid:126)0) = 0, as desired.

•
11((cid:96),(cid:126)0) = 0. Arguing as in (3.4.2), this proves

This lemma has important ramifications. Firstly, it shows that the only quadratic
terms in (3.4.6) and (3.4.7) are of the form (cid:126)u · m(d(cid:126)u,d(cid:96)) and (cid:126)u · m(du1,d(cid:96)). Secondly, it
shows that the undifferentiated wave factor u1 is missing from the nonlinearities. This
null-structure allows our argument to run because the missing resonant terms such as
(u1)2 or (u1)3 could potentially blow up in finite time due to the lack of the availability

of the Morawetz multiplier.

93

Lemmas 3.3.5 and 3.4.6, and Proposition 3.4.5 immediately imply that the perturba-

∂2
mpΓ 1

j1((cid:96),(cid:126)0)umup · m(duj,d(cid:96))
n(cid:88)

∂3
mpqΓ 1

11((cid:96),(cid:126)0)umupuq · σ + h.o.t.,

(3.4.11)

umm(dum,d(cid:96))− 2

n(cid:88)

m,p=2

tion equations simplify to

n(cid:88)

m=2

∂mΓ 1

(cid:3)u1 = −2κ
n(cid:88)

+

m=2

jk((cid:96),(cid:126)0)um · m(duj,duk)−
n(cid:88)

m,p,q=2

n(cid:88)

m=2

+

m,p=2

n(cid:88)

m,p,q=2

(cid:3)ui − κui · σ = 2κui · m(du1,d(cid:96))− 2

∂2
mpΓ i

j1((cid:96),(cid:126)0)umup · m(duj,d(cid:96))

jk((cid:96),(cid:126)0)um · m(duj,duk)−

∂mΓ i

∂3
mpqΓ i

11((cid:96),(cid:126)0)umupuq · σ + h.o.t.

(3.4.12)

3.4.1.1 Negatively curved case
Without loss of generality, in the case of negative sectional curvature we assume κ ≡ −1.
Consequently we demand that the line (cid:96) be timelike (σ < 0) in order to make the masses

of the Klein–Gordon solutions (cid:126)u positive. Without loss of generality, up to a change of
coordinates, (cid:96) ≡ t. Equations of motions (3.4.11) and (3.4.12) then reduce to

∂2
mpΓ 1

j1((cid:96),(cid:126)0)umup · m(duj,d(cid:96))

∂mΓ 1

jk((cid:96),(cid:126)0)um · m(duj,duk)−

∂3
mpqΓ 1

11((cid:96),(cid:126)0)umupuq + h.o.t.,

(3.4.13)

∂2
mpΓ i

j1((cid:96),(cid:126)0)umup · m(duj,d(cid:96))

jk((cid:96),(cid:126)0)um · m(duj,duk)−

∂mΓ i

∂3
mpqΓ i

11((cid:96),(cid:126)0)umupuq + h.o.t.

(3.4.14)

um · um

t

− 2

n(cid:88)

m,p=2

(cid:3)u1 = −2

+

n(cid:88)
n(cid:88)

m=2

m=2

n(cid:88)

m,p=2

− 2

(cid:3)ui − ui = 2ui · u1
n(cid:88)

t

+

m=2

n(cid:88)

m,p,q=2

n(cid:88)

m,p,q=2

94

3.4.1.2 Positively curved case
In the case of positive sectional curvature, we assume κ ≡ +1 and hence, without loss of
generality, we can prescribe (cid:96) ≡ x1. This reduces (3.4.11) and (3.4.12) to

(cid:3)u1 = 2

− 2

∂2
mpΓ 1

j1((cid:96),(cid:126)0)umup · m(duj,d(cid:96))

∂mΓ 1

jk((cid:96),(cid:126)0)um · m(duj,duk)−

∂3
mpqΓ 1

11((cid:96),(cid:126)0)umupuq + h.o.t.,

(3.4.15)

n(cid:88)

m,p=2

n(cid:88)

m,p=2

− 2

n(cid:88)
um · um
n(cid:88)
x1

m=2

+

m=2

n(cid:88)

m=2

+

n(cid:88)

m,p,q=2

n(cid:88)

m,p,q=2

(cid:3)ui − ui = −2ui · u1
x1

∂2
mpΓ i

j1((cid:96),(cid:126)0)umup · m(duj,d(cid:96))

jk((cid:96),(cid:126)0)um · m(duj,duk)−

∂mΓ i

∂3
mpqΓ i

11((cid:96),(cid:126)0)umupuq + h.o.t.

(3.4.16)

Remark 3.4.7. Starting now and for the remainder of the chapter we restrict ourselves
to the most difficult case of spatial dimension d = 3. This is a borderline case in the
−1 barely misses to be integrable. We will
overcome this growth by exploiting the weak null-condition present in (3.4.13) – (3.4.16),

sense that the linear decay rate for waves t

namely that resonant wave–wave nonlinearities are not present. Instead, we see that the

strongest nonlinear interactions are of wave-Klein–Gordon type. Our estimates will close

by exploiting the stronger linear decay rate of t

Using the integrable decay rate of t(1−d)/2 when d ≥ 4 for linear waves, it is straight

−3/2 for the Klein–Gordon equation.

forward to show that our results hold for higher dimensions as well.

3.5 Preliminary L2 analysis

In this short section we introduce some minor changes in the notation of the energies

introduced for the linear wave and Klein–Gordon equation in Chapter 2. We will denote

95

by

EW
τ [u1] def=

EKG

τ

[ui] def=

(cid:90)
(cid:90)

Στ




Στ

τ
τ

−1

−1

3(cid:88)
3(cid:88)

j=1

j=1

|Lju1|2 + τ(∂tu1)2

1/2

,

−1
τ dvolΣτ

 w
 w

|Ljui|2 + τ(∂tui)2

−1
τ + τ

−1|ui|2wτ dvolΣτ

1/2

(3.5.1)

(3.5.2)

as the energies of (u1, (cid:126)u ). We emphasize that the Klein–Gordon energy EKG
[ui] is adapted
to Klein–Gordon solutions with mass 1, this corresponds to our assumption of κ = ±1 in
Subsubsections 3.4.1.1 and 3.4.1.2 (see also Remark 2.2.8). As a consequence of Lemma

τ

2.2.5, these energies are comparable to the following weighted Lebesgue and Sobolev

norms of Chapter 2 (see (2.1.10) – (2.1.12)):

EW
τ [u1] ≈ τ
[ui] ≈ τ
EKG

τ

−1/2(cid:107)u(cid:107)
−1/2(cid:107)ui(cid:107)

,

+ τ1/2(cid:107)∂tu(cid:107)L2−1
+ τ1/2(cid:107)∂tui(cid:107)L2−1

˚W1,2−1
˚W1,2−1

+ τ

−1/2(cid:107)ui(cid:107)L2

1

.

The proof of Proposition 2.2.2 can be adapted to derive the following fundamental energy

estimate

EW
τ1 [u1]2 +

n(cid:88)

i=1

where we wrote

EKG
τ1 [ui]2 (cid:46) EW

τ0 [u1]2 +

n(cid:88)
τ1(cid:90)
(cid:90)

i=1

+

τ0

Στ

EKG
τ0 [ui]2

(cid:3)mu1∂tu1 +(cid:104)(cid:3)m(cid:126)u − (cid:126)u, ∂t (cid:126)u(cid:105) dvolΣτ dτ,
n(cid:88)

(3.5.3)

(cid:104) (cid:126)φ, (cid:126)ψ(cid:105) def=

φiψi.

To see how (3.5.3) can be derived, define
Dτ2
τ1

def= {(t, x) ∈ R1+d | t > 0, τ2

≤ t2 −|x|2 ≤ τ2

1

2

i=2

},

µDτ2

τ1

def= Dτ2
τ1

∩{(t, x) ∈ R1+d | |x| < µ− t}.

96

The boundary of µDτ2

τ1 is the union of

≤ t2 −|x|2 ≤ τ2

2 , t = µ−|x|};

1

def= {(t, x) ∈ R1+d | τ2
def= {(t, x) ∈ R1+d | t2 −|x|2 = τ2
def= {(t, x) ∈ R1+d | t2 −|x|2 = τ2

Cµ

Στ2,µ

Στ1,µ

2 , t < µ−|x|};
1 , t < µ−|x|}.

(cid:82)

Integrating the identity (2.2.8) with M = 0 for u1 and M = 1 for ui over µDτ2
divergence theorem, using the coercivity for

τ1, applying the
Q[φ](∂t, ∂τ)dvolτ afforded by Lemma
2.2.5, throwing out the boundary integrals along Cµ (which is possible because of the
dominant energy condition [see Lemma 2.2.1]), and letting µ → ∞, one arrives to the
desired inequality (3.5.3).

Στ

It is useful to work with the total energy of the coupled wave-Klein–Gordon system.

We then schematically write

EW
τ [u1]2 +

n(cid:88)

i=1

EKG

τ

[ui]2.

(cid:118)(cid:116)

Eτ[u] def=
(cid:90)
τ1(cid:90)

With this the fundamental estimate simplifies to

Eτ1[u]2 −Eτ0[u]2 (cid:46)

(cid:3)mu1∂tu1 +(cid:104)(cid:3)m(cid:126)u − (cid:126)u, ∂t (cid:126)u(cid:105) dvolΣτ dτ.

(3.5.4)

τ0

Στ

In order to apply (3.5.3) to attain higher derivative estimates of u, we commute the system

(3.4.13) – (3.4.14) with the Lorentz boosts. It is useful to introduce a notation for higher

order energies in order to close the bootstrap assumption in a systematic way. We define

Ek(τ) def= τ

−1/2(cid:107)u(cid:107)Wk+1,2
−1

+ τ1/2(cid:107)∂tu(cid:107)Wk,2−1

+ τ

−1/2(cid:107)(cid:126)u(cid:107)Wk,2

1

.

(3.5.5)

We note that the second estimate (2.2.13) of Lemma 2.2.5 implies

Ek(τ) ≈ (cid:88)

|α|≤k

Eτ[Lαu].

For convenience, we record the pointwise and integrated decay estimates of Chapter 2 in

the notation adapted to u:

97

Proposition 3.5.1. For any x ∈ Στ, the following pointwise estimates hold:

|(cid:126)u(x)| (cid:46) wτ(x)
|Liu(x)| + τ|∂tu(x)| (cid:46) wτ(x)

−3/2E2(τ),
−1/2E2(τ).

The following Sobolev estimates hold for any r ∈ [2,6]:

(cid:107)(cid:126)u(cid:107)
+ τ(cid:107)∂tu(cid:107)

˚Wk,r

1 (Στ )

˚Wk,r
r/2−2(Στ )

(cid:107)u(cid:107)

˚Wk+1,r
r/2−2(Στ )

(cid:46) τ1/rEk(τ),

(cid:46) τ1/rEk(τ)

6−r
2r

· Ek+1(τ)

3r−6
2r

.

Proof. The estimates are an immediate consequence of Propositions 2.2.7, 2.2.11, 2.2.17

and the definition of Ek(τ).

3.6 Global stability in the setting of TL

In this section we use the estimates recorded in the former in order to prove global

existence to the following wave-Klein–Gordon system:

(cid:3)mu1 = −2(cid:104)(cid:126)u, ∂t (cid:126)u(cid:105) + ((cid:126)u )3 + ((cid:126)u )2∂tu + (cid:126)u · m(du,du),
(cid:3)mui − ui = 2ui∂tu1 + ((cid:126)u )3 + ((cid:126)u )2∂tu + (cid:126)u · m(du,du),

i = 2, . . . , n

(3.6.1)

where here ((cid:126)u )3, ((cid:126)u)2∂tu, and (cid:126)u · m(du,du) is an abuse of notation representing a linear
combination of terms of the form

umupuq, m, p, q ∈ {2, . . . , n},
m, p ∈ {2, . . . , n},
umup∂tuj,
m ∈ {2, . . . , n},
umm(duj,duk),

j ∈ {1, . . . , n},
j, k ∈ {1, . . . , n}.

(3.6.2)

For our convenience, we will prescribe initial data at t = 2:

u(2, x) = φ0(x),

∂tu(2, x) = ϕ0(x).

98

Even though this system is a simplification of (3.4.13)–(3.4.14), it captures all of the
analytical difficulties and extending the results to the full equations of motion is merely
a matter of bookkeeping. Indeed, as the coefficients of (3.6.2) in the full system are of the
≤3Γ ((cid:96),0), they can be regarded as constant as a consequence of the manifold (M, g)
form ∂
having constant curvature4. Moreover, the higher ordered terms in (3.4.13)–(3.4.14) are

(cid:16)|(cid:126)u|4 +|(cid:126)u|3|∂tu| +|(cid:126)u|2|m(du,du)|(cid:17)

f (u, ∂u),

where f : Rn(d+2) → R is an arbitrary smooth function. The standard argument, using
the energy method, for either the stability problem or the local existence problem for

nonlinear waves, handles the nonlinearities with the general prescription of “putting the
highest order derivative factor in L2 and the remainder in L

estimates we

.” As the L

∞

∞

will be using are the pointwise bounds from Proposition 3.5.1, we see that higher order
nonlinearities lead to more available decay, and hence add no difficulties when improving

the bootstrap assumptions.

Our main theorem asserts that a geodesic wave map affinely parametrized by a time-

like linear free wave is stable under small (in an appropriate Sobolev norm) perturbations,

and that the perturbed solution stays within a small tubular neighborhood of the back-

ground geodesic.

Theorem 3.6.1. For any γ < 1/2, there exists some 
0 (which depends only on γ) such that
whenever φ0,ϕ0 are compactly supported in the ball of radius 1 centered at the origin satisfying

(cid:107)φ0(cid:107)

H3 +(cid:107)ϕ0(cid:107)

H2 < 
0,

4We now give a few more details explaining this fact for ∂mΓ i

jk; higher derivatives of
the Christoffel symbols follow similarly. Similar analysis used to prove Lemma 3.4.4 and
the geodesy of ϕI show that L
jk is constant along ϕI. This
can be interpreted geometrically as the flow map of ∂1 is a transvection along the geodesic
ϕI, see [O’N83, Definition 2.29]

jk = 0. Equivalently, ∂mΓ i

∂1∂mΓ i

99

there exists a unique solution u = (u1, (cid:126)u ) to (3.6.1) that exists for all time t ≥ 2. Furthermore,
we have the following uniform estimates:

−1/2

|Liu| (cid:46) τγ t
|(cid:126)u| (cid:46) τγ t
|∂tu| (cid:46) τ

−3/2
−1+γ t

−1/2.

3(cid:88)

i=1

|u1| +

By standard local existence theory we can assume that for sufficiently small initial

data, the solution u of (3.6.1) exists up to Σ2. The breakdown criterion for wave and
Klein–Gordon equations implies that so long as we can show that |(cid:126)u|, |Lu|, |∂tu| remain
bounded on Στ for all τ > 2, we can guarantee global existence of solutions. See [Sog08,
Chapter 1, Theorem 4.3]. Proposition 3.5.1 implies that a sufficient condition for global

existence are a priori estimates on the second order energies. The general approach is

that of a bootstrap argument:

1. We will assume that, up to time τmax > 2, that the energies Ek(τ) of the solution u

and its derivatives Lαu verify certain bounds.

2. Using Proposition 3.5.1, this gives L

∞

bounds on u, and its derivatives of the form

Lαu and ∂tLαu.

3. We can then estimate the nonlinearity using these L

∞

estimates, which we then

feed back into the energy inequality (3.5.4) to get an updated control on Ek(τ) for all
τ ∈ [2, τmax].

4. Finally, show for sufficiently small initial data sizes, that the updated control im-

proves the original control, whereupon by the method of continuity the original
bounds on Ek(τ) must hold for all τ ≥ 2, implying the desired global existence.

100

Since the Lorentz boosts Li commute with the d’Alembertian [Li,(cid:3)m] = 0, after applying
(3.5.4) to Lαu = (Lαu1, Lα (cid:126)u ) we see that we need to estimate the integrals

Lα((cid:3)mu1)∂tLαu1 +(cid:104)Lα((cid:3)m(cid:126)u − (cid:126)u ), ∂tLα (cid:126)u(cid:105) dvolΣτ

(3.6.3)

(cid:90)

Στ

for all tuples α with elements drawn from {1,2,3} and length ≤ 2.

From the structure of (3.6.1), when |α| = 0 we see a complete cancellation of the quadratic

terms in (3.6.3):

−2(cid:104)(cid:126)u, ∂t (cid:126)u(cid:105)∂tu1 + 2(cid:104)∂tu1(cid:126)u, ∂t (cid:126)u(cid:105) = 0.

(3.6.4)
Although this cancellation is unique to the case of |α| = 0, for |α| = 1,2 we do see a can-
cellation of all of the top order derivative quadratic terms. We see for any tuple α with
elements drawn from {1,2,3}
(cid:68)
(cid:12)(cid:12)(cid:12)−(cid:104)(cid:126)u, Lα∂t (cid:126)u(cid:105)∂tLαu1 + Lα∂tu1(cid:104)(cid:126)u, ∂tLα (cid:126)u(cid:105)(cid:12)(cid:12)(cid:12)
(cid:88)

(cid:12)(cid:12)(cid:12)(cid:104)Lγ (cid:126)u, Lβ∂t (cid:126)u(cid:105)∂tLαu1 + Lβ∂tu1(cid:104)Lγ (cid:126)u, ∂tLα (cid:126)u(cid:105)(cid:12)(cid:12)(cid:12) .

(cid:12)(cid:12)(cid:12)(cid:12)Lα(cid:16)− 2(cid:104)(cid:126)u, ∂t (cid:126)u(cid:105)(cid:17)

∂tLαu1 +

Lα(2∂tu1(cid:126)u ), ∂tLα (cid:126)u

(cid:69)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46)

(3.6.5)

+

|β|+|γ|≤|α|
|β|(cid:44)|α|

Using the commutator algebra properties

[Li, ∂t] = −∂xi = −1

t

Li +

xi
t

∂t,

(3.6.6)

we see a cancellation of the top order terms in the first term on the right hand side of

(3.6.5). Consequently, the quadratic terms of (3.6.3) can be estimated schematically as

|Lγ (cid:126)u∂tLβu∂tLαu| +

−1
w
τ

|Lγ (cid:126)uLβu∂tLαu| dvolΣτ

(3.6.7)

(cid:88)

|β|+|γ|≤|α|

(cid:90)

(cid:46)

(cid:88)

|β|+|γ|≤|α|

|β|(cid:44)α

Στ

where we repeatedly used (3.6.6) and (2.1.8). We can now estimate the quadratic terms

of (3.6.3).

101

(cid:69)(cid:12)(cid:12)(cid:12)(cid:12)dvolΣτ

(cid:46)

τ

τ

1

−3/2E2
−3/2E3

· E2
2 + τ

Proposition 3.6.2 (Quadratic energy estimates). Let α (cid:44) 0 be an m-tuple5 with elements
drawn from {1,2,3}. Then
(cid:90)

(cid:68)

(cid:12)(cid:12)(cid:12)(cid:12)Lα(cid:16)− 2(cid:104)(cid:126)u, ∂t (cid:126)u(cid:105)(cid:17)

∂tLαu1 +

Lα(2∂tu1(cid:126)u ), ∂tLα (cid:126)u

Στ

m = 1,
−1E2
· E2 m = 2.
−1. We prove the
estimate for the case m = 1 first. In this case the top ordered derivative terms of (3.6.7)

Proof. Throughout this proof we use the simple inequality w

−1E1 · E2
2 + τ
−1
≤ τ
τ

1

that we need to estimate are of the form

(cid:90)

|L(cid:126)u∂tu∂tLu| + w

−1
τ

|(cid:126)uLu∂tLu| dvolΣτ

Στ

(the estimates for the lower ordered terms will of course be controlled by the top ones).

Here it is understood that L can be any of the boosts Li. For the first term, we estimate

the ∂tu factor by the pointwise estimates in Proposition 3.5.1 and H¨older’s inequality on

the rest of them(cid:90)

|L(cid:126)u∂tu∂tLu| dvolΣτ

−3/2E2(τ)

(cid:46) τ

|L(cid:126)u w1/2

τ

|·|∂tLu w

−1/2
τ

| dvolΣτ

Στ

Στ

−3/2E2(τ)· (E1(τ))2.

≤ τ

For the second term, we control the (cid:126)u factor by the pointwise estimates and use H¨older’s

inequality on the rest

(cid:90)

−1
τ

w

|(cid:126)uLu∂tLu| dvolΣτ

−3/2E2(τ)

(cid:46) τ

|Lu w

−1/2
τ

|·|∂tLu w

−1/2
τ

| dvol

(cid:90)

(cid:90)

Στ

Στ

−3/2E2(τ)· (E1(τ))2.

≤ τ

This concludes the proof for m = 1.

5The case m = 0 does not need to be controlled due to (3.6.4).

102

(cid:90)

For m = 2, the terms from (3.6.7) are
|(cid:126)uLLu∂tLLu|
|LL(cid:126)u∂tu∂tLLu| + w

−1
τ

Στ

+|L(cid:126)u∂tLu∂tLLu| + w

−1
τ

|L(cid:126)uLu∂tLLu| dvolΣτ .

Again, the estimates for all lower ordered terms can be controlled by the estimates of

these. Here it is understood that LL is any arbitrary second order tangential derivative

LiLj. The first two terms are bounded by

−3/2(E2(τ))3

τ

∞
using the same techniques as m = 1 (estimating the lowest ordered terms in L

and the

rest by the energies after using H¨older’s). The other two terms cannot be treated with
the same techniques. Even though |L(cid:126)u| + |Lu| can be bounded by w
E2(τ), this decay
is too weak to improve the bootstrap assumptions that we will make. On the other hand,
we can get stronger decay for the third term above by estimating |L(cid:126)u| ≤ w
E3(τ). This
is not helpful to us because E3(τ) requires square integrability of four derivatives of u
(recall that we want to solve the Cauchy problem for (3.6.1) using data in H3).

−1/2
τ

−3/2
τ

We instead appeal to the interpolated Sobolev estimates in Proposition 3.5.1 with r =

3,6 to control the third and fourth terms above. We see

|L(cid:126)u∂tLu∂tLLu| dvolΣτ =

|L(cid:126)u w1/3

τ

|·|∂tLu w1/6

τ

|·|∂tLLu w

−1/2
τ

| dvolΣτ

(cid:90)

Στ

(cid:90)

Στ

≤ (cid:107)(cid:126)u(cid:107)
≤ τ

(cid:107)∂tu(cid:107)

˚W1,3
˚W1,6
1
1
−1E1(τ)· (E2(τ))2.

(cid:107)∂tu(cid:107)

˚W2,2−1

(3.6.8)

103

Similarly, we see

(cid:90)

−1
τ

w

|L(cid:126)uLu∂tLLu| dvolΣτ

−1

≤ τ

(cid:90)

|L(cid:126)u w1/3

τ

|·|Lu w1/6

τ

|·|∂tLLu w

−1/2
τ

| dvolΣτ

Στ

≤ τ
≤ τ

Στ

(cid:107)u(cid:107)

˚W1,3
1

−1(cid:107)(cid:126)u(cid:107)
˚W1,6
1
−1(E1(τ))2 · (E2(τ)).

(cid:107)∂tu(cid:107)

˚W2,2−1

(3.6.9)

This concludes the proof of the proposition.

Remark 3.6.3. The expression on the right hand side of (3.6.9) would allow us to close

our energy estimates with only a log loss, see Proposition 3.6.6. The borderline terms that

we need to deal with are in fact in (3.6.8).

are

(cid:90)

Estimating the cubic terms in (3.6.1) we identify the integrals that we have to estimate

(cid:16)
Lβ (cid:126)u Lγ (cid:126)u Lσ (cid:126)u + Lβ (cid:126)u · m(dLγu,dLσ u) + Lβ (cid:126)u Lγ (cid:126)u Lσ ∂tu

(cid:17)· ∂tLαudvolΣτ

Στ

for |β|+|γ|+|σ| = |α|. Here we implicitly used that vector fields act on scalars by Lie differ-
entiation, that m is invariant under the Lorentz boosts Li, and that exterior differentiation
commutes with Lie differentiation.

Proposition 3.6.4 (Cubic energy estimates). Let α be an m-tuple with elements drawn from
{1,2,3}. Then
(cid:90)

(cid:12)(cid:12)(cid:12)Lα(((cid:126)u ))3 + Lα(((cid:126)u )2∂tu) + Lα((cid:126)u · m(du,du))

(cid:12)(cid:12)(cid:12)·|∂tLαu| dvolΣτ

(cid:46)

Στ

τ

τ

· E2

2

m

−3E2
−3E4

−2E1/4

0

2 + τ

· E1 · E11/4

2

−3/2E2

1

· E2

2 + τ

+ τ

−5/2E1 · E3

2 m = 2

m = 0,1

104

(cid:90)

(cid:90)

(cid:90)

Στ

(cid:90)

Proof. Let us treat the terms with ((cid:126)u )3 first. When m = 0, we control two of the factors by

the pointwise estimates:

|((cid:126)u )3 · ∂tu| dvolΣτ

−3E2(τ)2

(cid:46) τ

|(cid:126)u w1/2

τ

|·|∂tu w

−1/2
τ

| dvolΣτ

Στ

Στ

−3E0(τ)2 · E2(τ)2,

(cid:46) τ

as desired. For m = 1, the same proof follows by controlling the two factors that are not
differentiated by the pointwise estimates (note that the density is |L(cid:126)u ((cid:126)u )2∂tLu|). When
m = 2, this can again be used to bound the terms of the form

|LL(cid:126)u · ((cid:126)u )2 · ∂tLLu| dvolΣτ

−3E2(τ)4.

(cid:46) τ

For the other cases, we couple the pointwise estimates and the interpolated GNS estimates

of Proposition 3.5.1 to find

(cid:90)

(cid:90)

|(L(cid:126)u )2 · (cid:126)u · ∂tLLu| dvolΣτ

−3/2E2

(cid:46) τ

|L(cid:126)u w1/3

τ

|·|L(cid:126)u w1/6

τ

|·|∂tLLu w

−1/2
τ

| dvolΣτ

Στ

Στ

−3/2E1(τ)2 · E2(τ)2.

(cid:46) τ

Next we control the ((cid:126)u )2∂tu terms. For m = 0, we control one ∂tu and one Klein–

Gordon term by the energy:

(cid:90)

(cid:90)

|((cid:126)u )2∂tu· ∂tu| dvolΣτ

−3E2

(cid:46) τ

|(cid:126)u w1/2

τ

|·|∂tu w

−1/2
τ

| dvolΣτ

Στ

Στ

−3E0(τ)2 · E2(τ)2.

(cid:46) τ

For m = 1, the same technique is used to bound

|(cid:126)u L(cid:126)u ∂tu · ∂tLu|dvolΣτ

−3E1(τ)2E2(τ)2.

(cid:46) τ

Στ

105

When the derivative hits the ∂tu factor we sacrifice some of the decay given by the Klein–
Gordon terms6:

(cid:90)

Στ

(cid:90)

2
Στ

−2E2

|∂tLu w

−1/2
τ

|·|∂tLu w

−1/2
τ

| dvolΣτ

|((cid:126)u )2∂tLu· ∂tLu| dvolΣτ

(cid:46) τ

(cid:46) τ

(cid:46) τ

−2E2(τ)2 ·(cid:107)∂tu(cid:107)2
−3E1(τ)2 · E2(τ)2.

˚W1,2−1

For m = 2 the densities we need to estimate are ∂tLLu multiplied by 7

(cid:126)u LL(cid:126)u ∂tu,

(L(cid:126)u )2∂tu,

(cid:126)u L(cid:126)u ∂tL(cid:126)u,

((cid:126)u )2∂tLLu.

(3.6.10)

For the first density we estimate the undifferentiated terms by the energies as we did for

m = 0 and m = 1 to see (cid:90)

|(cid:126)u LL(cid:126)u ∂tu· ∂tLLu|dvolΣτ

−3E2(τ)4.

(cid:46) τ

Στ

The second density of (3.6.10) is treated with the interpolation inequalities after using

the pointwise estimate to control ∂tu:

(cid:90)

(cid:90)

|(L(cid:126)u )2∂tu· ∂tLLu| dvolΣτ

−3/2E2

(cid:46) τ

|L(cid:126)u w1/3

τ

|·|L(cid:126)u w1/6

τ

|·|∂tLLu w

−1/2
τ

| dvolΣτ

Στ

(cid:46) τ

(cid:46) τ

Στ

−3/2E2(cid:107)(cid:126)u(cid:107)
(cid:107)(cid:126)u(cid:107)
−3/2E1(τ)2E2(τ)2.

˚W1,3
1

(cid:107)∂tu(cid:107)

˚W2,2−1

˚W1,6
1

6Of course, there are lower-ordered terms which appear as a consequence of commut-
∂tu. One can check that the energies of
2. We drop these lower ordered energies

−1
ing the derivative: Li∂tu = ∂tLiu − w
τ Liu + xi
−3E0E1E2
wτ
these commuted terms are bounded by τ
−3E2
1E2
2.
because will of course be controlled by τ

7Again, there are lower ordered terms that rise from commuting Li and ∂t. We drop
these energies because one can check that they will all be controlled by the energies of
((cid:126)u )2 ∂tLLu.

106

(cid:90)

The third density is treated similarly:

|(cid:126)u L(cid:126)u ∂tL(cid:126)u · ∂tLLu| dvolΣτ

−3/2E2

(cid:46) τ

(cid:90)

|L(cid:126)u w1/3

τ

|·|∂tLu w1/6

τ

|·|∂tLLu w

−1/2
τ

| dvolΣτ

Στ

Στ

−3/2E2(τ)·(cid:107)(cid:126)u(cid:107)
˚W1,3
1
−5/2E1(τ)E2(τ)3.

(cid:46) τ

(cid:46) τ

·(cid:107)∂tu(cid:107)

˚W1,6
1

·(cid:107)∂tu(cid:107)

˚W2,2−1

The last case of (3.6.10) is treated by controlling the two Klein–Gordon factors by the

pointwise estimates

(cid:90)

2
Στ

−2E2

|∂tLLu w

−1/2
τ

|·|∂tLLu w

−1/2
τ

| dvolΣτ

|((cid:126)u )2∂tLLu· ∂tLLu|dvolΣτ

(cid:46) τ

(cid:46) τ

(cid:46) τ

−2E2(τ)2 ·(cid:107)∂tu(cid:107)2
−3E2(τ)4.

˚W2,2−1

(cid:90)

Στ

(cid:90)

We finally treat the (cid:126)u · m(du,du) terms. Note firstly that the second equality in (3.6.6)

implies the estimate

|m(dψ1,dψ2)| ≤ 1
τ2

|Lψ1|·|Lψ2| +|∂tψ1|·|∂tψ2|

for any scalars ψ1, ψ2. For m = 0 the pointwise estimates imply

|(cid:126)u · m(du,du)· ∂tu|dvolΣτ

−3E2(τ)2

(cid:46) τ

|(cid:126)u w1/2

τ

|·|∂tu w

−1/2
τ

| dvolΣτ

(cid:90)

Στ

Στ

−3E0(τ)2E2(τ)2.

(cid:46) τ

For m = 1 we similarly see

(cid:90)

|L(cid:126)u · m(du,du)· ∂tLu|dvolΣτ

−3E1(τ)2E2(τ)2.

(cid:46) τ

Στ

107

When the derivative hits the null form factor the density is (cid:126)u · m(dLu,du)· ∂tLu. We can
then use the improved Klein–Gordon decay |(cid:126)u| (cid:46) w
(cid:90)

E2 to estimate

−3/2
τ

(cid:90)
(cid:90)

Στ

τ

−2|LLu|·|Lu| +|∂tLu|·|∂tu|(cid:17) |∂tLu|
(cid:16)
 |LLu|

 |∂tLu|

w1/2
τ

dvol

+

|∂tLu|
w1/2
τ

τw1/2

τ

w1/2
τ

dvol

|(cid:126)u · m(dLu,du)· ∂tLu|dvol (cid:46) τ

−1E2

Στ

(cid:46) τ

(cid:46) τ

−2E2

2
Στ
1E2
2.

−3E2

Replicating the previous estimates, when m = 2,

(cid:12)(cid:12)(cid:12)LL(cid:126)u · m(du,du) + (cid:126)u · m(dLLu,du)

(cid:12)(cid:12)(cid:12)·|∂tLLu|dvol (cid:46) τ

−3E4
2.

(cid:90)

Στ

The remaining term

(cid:90)

Στ

|L(cid:126)u · m(dLu,du)∂tLLu| dvolΣτ

(cid:90)

|L(cid:126)u · m(dLu,du)· ∂tLu|dvol (cid:46) E2

can’t be treated in the same way because the improved decay from the Klein–Gordon
term comes at a loss of one derivative: |L(cid:126)u| (cid:46) w
estimate |L(cid:126)u| (cid:46) w
Proposition 3.5.1 with r = 4:

E3. We must then rely on the weaker
E2 and remedy this loss with the interpolated GNS estimates from

−1/2
τ

−3/2
τ

(cid:90)

(cid:16)

τ

−2|LLu|·|Lu| +|∂tLu|·|∂tu|(cid:17) ∂tLLu

dvol

w1/2
τ

Στ

(cid:46) τ

and the proposition follows.

Στ

−2E1/4

0

· E3/4

1

· E1/4

1

· E3/4

2

· E2.

Using (3.5.5), we have as an immediate corollary of Propositions 3.6.2 and 3.6.4 the

following a priori estimates:

108

−3E2

0E2

2 dτ

Corollary 3.6.5.

E0(τ1)2 − E0(τ0)2 (cid:46)

E1(τ1)2 − E1(τ0)2 (cid:46)

E2(τ1)2 − E2(τ0)2 (cid:46)

τ0

τ1(cid:90)
τ1(cid:90)
τ1(cid:90)

τ0

τ

τ

−3/2E2

1E2 + τ

1E2

2 dτ

−3E2
(cid:17)

(cid:16)

−1E1E2

τ

E1 + E2

+ τ

−3/2E2

2

(cid:16)

E2 + E1E2 + E2
1

(cid:17)

τ0

−3E4

2 + τ

−2E1/4

0

· E1 · E11/4

2

+ τ

dτ.

These estimates imply the following bootstrap estimate.

Proposition 3.6.6. Assume that the initial data satisfy

E2(2) ≤ 


and that for some τmax > 2 the bootstrap assumptions



E0(τ) ≤ δ
E1(τ) ≤ δ
E2(τ) ≤ δτγ

(3.6.11)

(3.6.12)

(3.6.13)

(3.6.14)

(3.6.15)

(3.6.16)

(3.6.17)

hold for all τ ∈ [2, τmax] and some δ < 1, γ (cid:28) 1. Then there exists a constant C depending only

on γ such that the improved estimates

E0(τ) ≤ 
 + Cδ3/2
E1(τ) ≤ 
 + Cδ3/2
E2(τ) ≤ 
 + Cδ3/2τγ

hold for all τ ∈ [2, τmax].

109

Proof. Improving the estimate for E0 follows from (3.6.11) after noting that

τ(cid:90)

τ(cid:90)

∞(cid:90)

−3E0(σ)2 · E2(σ)2 dσ ≤ δ4

σ

−3+2γ dσ ≤ δ4

σ

−3+2γ dσ ≤ Cδ3.

σ

2

2

2

Similarly, the estimate for E1 follows from (3.6.12) because σ
[2,∞) provided that γ < 1/2.

−3/2+γ is integrable for σ ∈

We begin to improve the bootstrap E2 by controlling the first two terms in the right

hand side of (3.6.13), which are bounded by

−1+2γ dσ ≤ Cδ3τ2γ .

σ

δ3

τ(cid:90)

2

∞(cid:90)

The rest of the terms are all bounded by

−3/2+3γ dσ ≤ Cδ3τ2γ ,

σ

δ3

provided that γ < 1/2. We now consider γ fixed once and for all.

2

As a consequence of the improved estimates, if we choose δ ≤ (4C)

−1/2 and then 
 <

δ/4, we we conclude



E0(τ) ≤ 1
2δ
E1(τ) ≤ 1
2δ
E2(τ) ≤ 1
2δτγ

In this case the global existence part of Theorem 3.6.1 follows by a continuity argument,

and the decay estimates follow from an application of the pointwise estimates of Propo-

sition 3.5.1 and these energy bounds.

3.7 Global stability in the setting of SL

In this last section we use the tools from Section ?? to prove stability of the totally
geodesic background ϕI◦ϕS in the case that the target has positive curvature, i.e. (3.4.15)

110

and (3.4.16). With the notations introduced in the previous section, we reduce our atten-

tion to

(cid:3)mu1 = 2(cid:104)(cid:126)u, ∂x1 (cid:126)u(cid:105) + ((cid:126)u )3 + (cid:126)u · m(du,du) + ((cid:126)u )2 · ∂x1u,
(cid:3)mui − ui = −2ui∂x1u1 + ((cid:126)u )3 + (cid:126)u · m(du,du) + ((cid:126)u )2 · ∂x1u,

(3.7.1)

i = 2, . . . , n

With ∂x1 replaced by ∂t on the right hand side, the system above is the same with the
negative curvature case (3.6.1). Employing

∂xi =

Li − xi
t

1
t

∂t,

(3.7.2)

what we can prove is:

Theorem 3.7.1. Under the same assumptions, the results of Theorem 3.6.1 also apply to the

system (3.7.1).

Proof. It suffices to obtain similar estimates as those in Propositions 3.6.2 and 3.6.4, then

the theorem 3.7.1 follows similarly from Corollary 3.6.5 and Proposition 3.6.6. We first

deal with the quadratic terms. We decompose from (3.7.2) the quadratic terms into two

parts, i.e. Q(m) = Q1(m) + Q2(m) with

(cid:69)

−1L1(cid:126)u

(cid:126)u, t

(cid:90)
(cid:90)

Στ

(cid:104)
2Lα(cid:68)
(cid:104)−2Lα(cid:68)

Q1(m) def=

Q2(m) def=

(cid:68)

(cid:69)(cid:105)

−1L1u1), ∂tLα (cid:126)u

∂tLαu1 − 2
(cid:69)

Lα((cid:126)u t

(cid:68)

−1x1∂t (cid:126)u

(cid:126)u, t

∂tLαu1 + 2

Lα((cid:126)u t

−1x1∂tu1), ∂tLα (cid:126)u

dvolΣτ ;

(cid:69)(cid:105)

dvolΣτ

Στ

for an m-tuple α with entries in {1,2,3}. The Q2 term has the same structure with the
quadratic nonlinearities for the negative curvature case presented in previous section,
−1
τ x1. In particular, the top order terms can
−1x1 only contribute lower order

be canceled. As we will see, the boosts Li acting on t

with the introduction of the factor t

−1x1 = w

terms because of (2.1.8).

111

τ



extra decay of t

On the other hand, the top order of Q1(m) can not be cancelled but we can utilize the

−1. We claim the quadratic terms can be bounded as
|Q(m)| (cid:46)

E2(τ)Em(τ)2(cid:17)
−3/2(cid:16)
(3.7.3)
−1E1(τ)E2(τ)2 + τ
−1, we shall also use the identity t = wτ on the surface
Στ. As Q2 can be dealt with in the same way as Proposition (3.6.2), we only provide the
proof for Q1(m). The case m = 0 is straightforward as we control (cid:126)u using the pointwise
estimates of Proposition 3.5.1 and the rest of the vectors using H¨older’s inequality. For

Besides the inequality w

−3/2E2(τ)3,

−1
τ (cid:46) τ

if m = 0,1;

if m = 2.

τ

,

the case m = 1 we need to estimate
−1
τ

|L(cid:126)u|2 + w

−1
τ

w

|(cid:126)u||LLu| +|Lα(t

(cid:90)

(cid:16)

Στ

(cid:90)

−1)||(cid:126)u||Lu|(cid:17)|∂tLu|dvolΣτ .
(cid:90)

(3.7.4)

Choosing the weights appropriately and applying H¨older’s inequality imply

−1
w
τ

|L(cid:126)u|2|∂tLu|dvolΣτ

−3/2E2(τ)

(cid:46) τ

w1/2
τ

|L(cid:126)u|w

−1/2
τ

|∂tLu|dvolΣτ

Στ

∞
Here we also bounded L

Στ

−3/2E2(τ)E1(τ)2.

(cid:46) τ

-norm of L(cid:126)u through Proposition 3.5.1. With a use of the defi-

nition of Li we have

−1) = −xi
t2
which on the surface Στ admits an upper bound w
−3/2E2(τ)

−1)||(cid:126)u||Lu||∂tLu|dvolΣτ

|Lα(t

(cid:90)

Li(t

(cid:46) τ

(cid:90)

−1
τ , and hence implies

−1/2
w
τ

|Lu|w

−1/2
τ

|∂tLu|dvolΣτ

Στ

∞
Here we also applied the L

Στ

−3/2E2(τ)E1(τ)2.

(cid:46) τ

-bound of (cid:126)u in Proposition 3.5.1. In a similar way the second

term in (3.7.4) admits the same upper bound which furthermore implies (3.7.3) for m = 1.

(cid:90)

It remains to establish (3.7.3) for m = 2, in which case Q1(m) can be bounded by

−1
τ (|(cid:126)u||LLLu| +|LL(cid:126)u||Lu| +|L(cid:126)u||LLu|)|∂tLLu|dvolΣτ + l.o.t,

w

Στ

112

|Lα(t

−1)| (cid:46) t

−1

(cid:90)

Στ

−3/2E2(τ)3.
(cid:90)

(cid:90)

(cid:90)

Στ

(cid:90)

where the lower order terms are those that show up when L acts on t

−1 resulting

∞
for any m-tuple α. It suffices to bound the top order terms. Bounding L

the aid of Proposition 3.5.1 implies the first term can be bounded by
−1/2
τ

|(cid:126)u||LLLu||∂tLLu|dvolΣτ

−3/2E2(τ)

|LLLu|w

−1/2
τ

−1
w
τ

(cid:46) τ

w

|∂tLLu|dvolΣτ

(3.7.5)

-norm of (cid:126)u with

Στ

(cid:46) τ

The last two terms can be dealt with by using the interpolation Sobolev inequalities in

Proposition 3.5.1. In particular, the second term can be bounded as

−1
τ

w

|LL(cid:126)u||Lu||∂tLLu|dvolΣτ

−1

(cid:46) τ

|LL(cid:126)u|w1/3

τ

|Lu|w1/6

τ

|∂tLLu|w

−1/2
τ

dvolΣτ

(cid:107)∂tu(cid:107)

˚W2,2−1

˚W1,6
1

(cid:46) τ

(cid:46) τ

Στ

˚W2,3
1

−1(cid:107)(cid:126)u(cid:107)
(cid:107)u(cid:107)
−1E1(τ)E2(τ)2.
(cid:90)

|L(cid:126)u|w1/3

−1

τ

In a similar manner, the third term admits upper bound

−1
τ

w

|L(cid:126)u||LLu||∂tLLu|dvolΣτ

(cid:46) τ

|LLu|w1/6

τ

|∂tLLu|w

−1/2
τ

dvolΣτ

Στ

Στ

−1E1(τ)E2(τ)2,

(cid:46) τ

which competes the proof of Claim (3.7.3).

The cubic terms can be dealt with similarly. In particular, the cubic terms in (3.7.1)
by employing (3.7.2) can be decomposed into two parts, writing as C(m) = C1(m) +C2(m)
with

(cid:105)· ∂tLαudvolΣτ ;

((cid:126)u )2 · t

−1L1u

((cid:126)u )3 + (cid:126)u · m(du,du) + ((cid:126)u )2 · t

−1x1∂tu

(cid:105)· ∂tLαudvolΣτ .

(cid:90)
(cid:90)

Στ

Lα(cid:104)
Lα(cid:104)

C1(m) =

C2(m) =

Στ

for an m-tuple α with entries in {1,2,3}. Again the second term C2(m) admit similar struc-
ture of cubic terms for the negative case and hence has the same bound as in Proposition

113

3.6.4. Here we recall Li acting on t
terms by (2.1.8). The first item C1 can be dealt with by utilizing the extra decay of t
We claim

−1
τ x1 on Στ, only contribute lower order
−1.

−1x1, or w

|C1(m)| (cid:46)

−3E2(τ)4,
∞
For m = 0, the estimate above is a direct result of L

τ

−3E2(τ)2Em(τ)2,
−2E2(τ)3E1(τ) + τ

τ
(cid:16)|L(cid:126)u||(cid:126)u||Lu| +|(cid:126)u|2|LLu| +|(cid:126)u|2|Lu|(cid:17)|∂tLu|dvolΣτ .

if m = 0,1;

if m = 2.

H¨older’s inequality. For m = 1, utilizing (3.7.5) and t = wτ on Στ we need estimate

Στ
∞
We bound the L

−1
-norm of Lu and (cid:126)u as in Proposition 3.5.1, apply w
τ

≤ τ

−1 and dis-

tribute the weight appropriately, arriving at an upper bound:
−1/2
w
τ

|LLu| + w

|L(cid:126)u| + w

−3E2

−1/2
τ

−1/2
τ

w1/2
τ

τ

|Lu|(cid:17)

|∂tLu|dvolΣτ .

(3.7.6)

bound of (cid:126)u in Proposition 3.5.1 and

(cid:90)

−1
τ

w

(cid:90)

(cid:16)

2
Στ

Applying H¨older’s inequality implies the estimate for m = 1 in (3.7.6). Lastly, for m = 2

we need bound(cid:90)

(cid:16)|LL(cid:126)u||(cid:126)u||Lu| +|L(cid:126)u|2|Lu| +|(cid:126)u|2|LLLu|(cid:17)|∂tLLu|dvolΣτ +l.o.t.

−1
w
τ

Utilizing L
−2E2

τ

(cid:16)

−1 yields an upper bound
−1
≤ τ
bound of (cid:126)u and Lu in Proposition 3.5.1 and w
τ
−1/2
|∂tLLu|dvolΣτ +l.o.t.
τ

|LLLu|(cid:17)

|LLu| + w1/2

|L(cid:126)u| + τ

−1/2
w
τ

−1/2
τ

−1w

w

τ

Στ
∞

(cid:90)

2
Στ

Then (3.7.6) for the case m = 2 follows directly by H¨older’s inequality.

114

CHAPTER 4

THE MEMBRANE EQUATION

4.1 Introduction

Let M be a connected oriented (d + 1)-manifold which is immersed in1 R1,1+d through
the map Φ : M → R1,1+d. The immersion Φ is assumed to be timelike, in that the pullback
metric g def= Φ
m on M is Lorentzian. A relativistic membrane shall refer to the image Φ(M)

∗

provided that any of the following equivalent conditions hold:

• the map Φ is a formal critical point to the volume functional

(cid:90)

S[Ψ ] def=

dvol(Ψ

∗

m);

M

• the induced mean curvature vector H vanishes identically;

• the components of Φ satisfy the equation

(cid:3)g(xµ ◦ Φ) = 0,

where (cid:3)g is the Laplace-Beltrami operator on (M, g).

The assumption that Φ is an immersion implies that one can, locally, describe the

(cid:17)
membrane Φ(M) as a graph
equations of motion (cid:3)g(xµ ◦ Φ) = 0 take the divergence form
 = 0.

 mµν∂νφ
(cid:112)

t, x1, . . . , xd, φ(t, x1, . . . , xd)

(cid:16)
d(cid:88)

∂
∂xµ

µ,ν=0

1 + m(dφ,dφ)

. In this coordinate system the

(4.1.1)

1We clarify that R1,1+d is the Minkowski space with one time dimension and (d+1)-space

dimensions.

115

This equation is variously known as the membrane equation, the timelike minimal/maximal

surface equation, or the Lorentzian vanishing mean curvature flow. This is due to the inter-
pretation that the graph of φ in R1,d × R (cid:27) R1,1+d is an embedded timelike hypersurface
with zero mean curvature.

Solutions to (4.1.1) model extended test objects (world sheets), in the sense that the

case where d = 0 reduces to the geodesic equation which models the motion of a test par-
ticle in R1,1. (The membrane equation can also be formulated with codimension greater

than one; see [AAI06, Mil08].) The membranes can also interact with external forces

which manifests as a prescription of the mean curvature; see [AC79,Hop13,Kib76,VS94]

for some discussion of the physics surrounding such objects, and see [Jer11, Neu90] for

rigorous justifications that membranes represent extended particles.

Our interest in the membrane equation arose, however, mainly due to it being an

exceptional model of a quasilinear wave equation that is highly non-resonant. The ex-

ploration of resonant conditions in wave equations proceeded, historically, through two

fronts. In the case of 1 spatial dimension, it has long been understood that hyperbolic sys-

tems with resonance (Lax’s “genuinely nonlinear condition”) lead to shock formation in

finite time [Lax64,Lax73,Joh74]. For higher spatial dimensions, in the small-data regime,

resonance has to compete with the dispersive decay enjoyed by wave equations. By now

it is well understood that quasilinear wave equations enjoy small-data global existence in
dimension d ≥ 4, and also in dimensions d = 2,3 when versions of Klainerman’s null con-
dition are satisfied [Kla80, Kla82, Kla84, Ali01a, Ali01b]. More recently the two fronts

have met, where small-data shock formation for resonant quasilinear wave equations

have been studied in spatial dimensions 2 and 3 [Ali01a, Ali01b, Chr07, Spe16, LS18].

For a recent review of the current understanding of small-data global existence versus

shock formation in quasilinear waves, please see [HKSW16].

In a recent paper, Speck, Holzegel, Luk, and Wong studied the stability of plane-

symmetric shock formation for quasilinear wave equations with resonance, under initial

116

data perturbations that breaks the plane-symmetry [SHLW16]. More precisely, they start

with a background simple-plane-symmetric solution to a quasilinear wave equation that

is genuinely nonlinear, such that it forms a shock singularity in finite time. Such back-

ground solutions can be extracted from, for example, the late-time evolution of any small

compactly supported initial data; we however allow our background solution to be of

arbitrary “size”. We were able to show that the shock formation is stable under arbitrary

initial data perturbations that breaks the simple-plane-symmetry, provided that the per-

turbation is small compared to the background solution.

A natural follow-up question is: when genuine nonlinearity fails, in particular when there

exists simple-plane-symmetric global solutions to the quasilinear wave equation, is the global

existence stable under small, symmetry-breaking initial data perturbations?

Returning to the membrane equation, we note that the equation is highly non-resonant.

It satisfies a stronger null condition than is typical of quasilinear waves in 2 or 3 dimen-

sions. This was explicitly exploited to show global well-posedness of the small-data prob-

lem first by Brendle [Bre02] when d = 3 and then by Lindblad [Lin04] in d = 2 and d = 1.

The d = 1 case is surprising as, there being no dispersive decay for the one-dimensional

wave, any resonance, even arbitrarily high order, can lead to finite-time blow-up. Wong

explored this case in more detail geometrically [Won17b] and enlarged the class of initial

data for which global existence holds.

Our focus on the membrane equation in this chapter then is due to the fact that (i)

as a consequence of [Lin04] and [Won17b], there exists robust families of global plane-

symmetric solutions to the membrane equation, and (ii) the null geometry of such solu-

tions are well understood by the analyses of [Won17b]. We remark that, while not explic-

itly stated, following the same method of proof of the main theorem in [Won17b], one

can show that the global simple planewave solutions described below in Section 4.2.1 are

automatically stable under plane-symmetric perturbations that are not necessarily simple.

We state and prove our main result in dimension d = 3; as described in the previous

117

paragraph, the result is effectively known in d = 1. Our proof also works in all dimensions
d ≥ 3 thanks to the improved dispersive decay of solutions to the linear wave equation
in higher spatial dimensions. Our proof however doesn’t work in d = 2 due to certain

technical losses of decay (see Remark 4.5.1 below). In [LZ19] the authors were able to

prove a similar result in d = 2 with weaker asymptotic control; see Remark 4.1.4 for

further discussion.

One should note, at this juncture, that the non-resonance of the membrane equation
is only effective at preventing a certain type of singularity formation. Indeed, far away

from the nearly-simple-planewave regime that we consider in the present chapter, singu-

larities are known to arise from regular initial data. In the case where d = 1 these were

analyzed by Nguyen and Tian [NT13] and Jerrard, Novaga, and Orlandi [JNO15]; while

their analyses concentrate on the case with spatially periodic domain, by finite speed of

propagation the same singularity formation can be localized and placed in our context.

Analogues of [NT13, JNO15] in higher spatial dimensional backgrounds were studied by

Wong in [Won18b]. In these cases the singularities are not of shock-type, but rather ap-

pear due to the degeneration of the principal symbol of the evolution.

4.1.1 Our main result and discussions

The answer to the question asked in the previous section is in the affirmative: we show

that simple-planewave solutions to the membrane equation are stable under small initial

data perturbations. The precise version of our main theorem is Theorem 4.5.8; there we

state the result as a small-data global existence result for the corresponding perturbation

equations, after a nonlinear change of independent variables that corresponds to a gauge

choice. Here we state a slightly less precise version in terms of the original variables.

Theorem 4.1.1. Fix the dimension d = 3. Let Υ denote a smooth simple-plane-symmetric
solution to (4.1.1) with finite extent in its direction of travel. Fix a bounded set Ω ⊂ R3.
There exists some 
0 > 0 depending on the background Υ and the domain Ω, such that for

118

any (ψ0, ψ1) ∈ (H5(R3)∩ C
problem to (4.1.1) with initial data

∞
0 (Ω))× (H4(R3)∩ C

∞
0 (Ω)) with (cid:107)(ψ0, ψ1)(cid:107) < 
0, the initial value

φ(0, x) = Υ (0, x) + ψ0(x),

∂tφ(0, x) = ∂tΥ (0, x) + ψ1(x)

has a global solution that converges in C2(R3) to Υ as t → ±∞.

Remark 4.1.2 (Finite extent in the direction of travel). We ask that Υ essentially represent
a travelling “pulse”. For example, taking plane-symmetry to mean constant in the x2 and
x3 variables, Υ would be a function of t = x0 and x1 alone. We ask that for any fixed t the
function Υ vanishes for all sufficiently large x1. We make heavy use of this finite extent

property in the course of the proof (see Lemma 4.3.13).

Remark 4.1.3 (Simplicity). By a simple planewave solution we refer to a solution that
is not only constant in the x2 and x3 variables, but one such that the differential dΥ is

null with respect to the dynamic metric. In other words, a simple planewave solution is

one that propagates along only one (and not both) of the characteristic directions of the

nonlinear wave equation.

The assumption of simplicity is only to keep the argument simple (pun intended). In

fact, assuming finite extent of the initial data for the plane-symmetric background, auto-

matically by the sharp Huygen’s principle for one dimensional waves, after a finite-length

of time the background will decouple into two spatially disjoint simple planewaves trav-

elling in opposite directions. By Cauchy stability of the finite-time initial value problem,

we see that the theorem for the simple planewave background also implies the theorem

for general, globally existing plane-symmetric backgrounds such as those demonstrated

to exist in [Lin04, Won17b].

We note here, however, that another feature of simplicity is that simple-planewave

solutions exist for arbitrary pulse profile (see Section 4.2.1 below). The same is not the

case for non-simple planewave solutions: large interacting waves can form finite-time

singularities.

119

Remark 4.1.4 (Dimensionality). The theorem above is stated for d = 3. The same ar-
guments can be used to prove stability for all dimensions d ≥ 3 (in fact the arguments
can be significantly further simplified when d ≥ 5). One needs to modify the degree of
regularity required. When d = 3 the data is taken to be small in H k × H k−1 with k = 5.
When d ≥ 4 is even we will need k = d + 3, and when d ≥ 5 is odd we will need k = d + 2.
Compare to the discussion in Section 4.4 below.

As mentioned before in this introduction, the d = 1 analogue of the result essentially

follows from the arguments in [Won17b]. This leaves the case d = 2, which received at-
tention from Liu and Zhou [LZ19]. Aside from minor technical differences in how we
approach the energy and pointwise estimates, a difference appears in how we linearize

around the background solution. In the present chapter we use the geometric normal

graphical gauge (see below) adapted to the background traveling wave, while in [LZ19]

they used the gauge adapted to the trivial solution. Our gauge has the advantage that the

perturbation equations contain no linear potential from the background; the price paid

being the appearance of nonlinear contributions of lower order whose null structure are

less apparent. In [LZ19], they were so far able to show global existence for the pertur-
bation equations but only C0 (and not C1) convergence to the background. The lack of

higher-derivative convergence can be attributed, at least in part, to their gauge choice.
Based on our own work we have high hopes that in fact C2 convergence can be proven
to hold, though at present there are some technical difficulties for even showing global

existence using a direct extension of our method; see also Remark 4.5.1.

Our main theorem is not a straight-forward small-data global existence result for a

quasilinear wave equation. The equations satisfied by the perturbations around large
solutions generally include coefficients contributed by background, the effects of which

must also be captured. In our problem, to leading order the perturbation equation looks

like

(cid:48)(cid:48)

(cid:3)mφ + φΥ

(∂t + ∂x1)2φ + Υ

(cid:48)(cid:48)

(∂tφ + ∂x1φ)2 = 0.

(4.1.2)

120

Here (cid:3)m is the flat wave operator, and the background pulse is assumed to be travelling
in the +x1 direction, so has compact support in the (t − x1) variable.

The first thing to notice is that the linearized equation is the linear wave equation on

Minkowski space. This is a special geometric feature of simple-travelling wave solutions

to the membrane equation. To expose this special linear structure, one needs to make

an appropriate gauge choice involving a nonlinear change of variables adapted to the

background Υ , which essentially re-writes our perturbation equations as a graph in the
normal bundle of Υ , interpreted as a submanifold of R1,1+d. It is well-known that the

membrane equation has good structure in such “normal graphical gauge”: in this formu-

lation the linearized equation can be expressed as the geometric wave operator adapted

to the induced Lorentzian metric on the background Υ , plus possibly a potential term.

This gauge was also used, for example, in [DKSW16].

In view of this special geometric feature, we do not need to develop special methods to

(cid:48)(cid:48)

perform the linear analysis. On the other hand, the function Υ

is non-decaying and has

support within the “wave zone”; this significantly complicates the analysis of the nonlin-

ear terms, especially since these nonlinearities are not in the shape of classical null forms.

This is in contrast with the analyses in [DKSW16] where the stability of another “large

data” solution to the membrane equation was considered. The background solution in

that case is the static catenoid solution. The nontrivial catenoid background introduced

a low-frequency correction to the linear evolution (in fact giving an exponentially grow-

ing mode). But as the background is asymptotically flat, the high-frequency evolution,

especially in the wave-zone where it is the most delicate when it comes to the nonlinear

interactions, is entirely captured by classical null structures. In particular the nonlin-
earities do not introduce new difficulties beyond the adjustments made for the modified
linear evolution. Another difference with our work and [DKSW16] is that they prove that

the catenoid is globally stable under axially symmetric codimension one initial perturba-

tions, whereas we prove that our planewave solution is globally stable under an open set

121

of symmetry breaking perturbations. Their symmetry assumptions on the perturbations

are there to avoid the issue of trapped geodesics on the catenoid.

In the present chapter, on the other hand, the focus is entirely on the nonlinearity, with

the main difficulty arising precisely from the non-decaying background Υ

. At this point

(cid:48)(cid:48)

it may be worth drawing comparison to another large-data (semi-)global existence result

for the membrane equation. In [WW17], the authors studied the membrane equation

with initial data given as a small perturbation of an out-going “short-pulse”. The (semi-

)global existence (note that by their choice of initial data, the result in [WW17] is not

time-symmetric!) mechanism in this case is essentially still the classical null condition of

Klainerman. The strong non-resonance condition of the membrane equation means that

the “large” short-pulse background does not interact with itself; and in fact the pulse

itself decays like the solution to the linear wave equation. Putting this together with

the fact that the nonlinearities in (4.1.1) are cubic, this means that heuristically we can

understand the result of [WW17] as very similar to the large data stability result for

the wave maps system proven in [Sid89], which also required the “background geodesic

solution” to be one with finite (weighted) energy, and hence decays like finite energy

solutions to the linear wave equation. These types of systems can be modeled by the

quasilinear system

(cid:3)mψ1 = 0,

(cid:3)mψ2 = m(∇ψ1,∇m(∇ψ2,∇ψ2)) + m(∇ψ2,∇m(∇ψ1,∇ψ2)).

Even when ψ1 is a “large” solution, it contributes enough decay that the nonlinearities for
the second equation decay at an integrable rate. Together with the fact that the nonlin-

earity is quadratic in ψ2, we can upgrade the smallness and close the bootstrap. Note that
the decay of ψ1 is crucial, as, in the second term of the nonlinearity we see components
like

(∂t + ∂r)2ψ1 · (∂tψ2 − ∂rψ2)2.

122

This is a resonant interaction in ψ2, whose contribution is significantly ameliorated by
−1 (or better) in
the fact that (∂t + ∂r)2ψ1 should decay like t
R1,2. If we were to replace the ψ1 factor by a generic bounded function in R1,3 (or a
t in R1,2) this term will lead to finite-time blow-up.
function decaying no faster than 1/

−3/2 (or better) in R1,3 or t

√

Returning to our equation (4.1.2), we see that we have precisely this type of resonant
interaction with a non-decaying coefficient. Instead of coefficient decay, we need to ex-
ploit a different aspect of the null structure of the original membrane equation (4.1.1).
has compact support in the (t− x1) variable. The res-
What we will use is the fact that Υ
onant interacting terms (∂tφ + ∂x1φ) represent waves traveling in directions transverse2
to the level sets of t − x1. In particular, we expect that the resonant interaction to only
take place for a bounded length of time (for each wave packet). Our main mechanism

(cid:48)(cid:48)

would therefore be something similar to that which drives Shatah’s space-time resonance

arguments [Sha10], but captured in a purely physical space manner.

Of course, we have to pay a price for this non-decay. This manifests in us having to

use a polynomially-growing energy hierarchy when using the vector field method. In

fact, our higher order energies, starting with the second (controlling the third derivatives
in L2), will grow in time, with each additional order differentiation growing one order

faster in time. One should compare to classical applications of the vector field method

where all but the top-order energies are bounded in time, with the top-order typically

exhibiting no worse than a log growth. The upshot of this energy hierarchy is that we lose

strong peeling properties of the solutions. (See Remark 4.5.9.)

4.1.2 Outline of the chapter

The remainder of this chapter is organized as follows: we first discuss the background

planewave solutions Υ . These solutions are introduced in Section 4.2.1. Their basic ge-

2In (1 + 1)-dimensions, the linear wave equation can be expressed as (∂t − ∂x1)(∂tϕ +
∂x1ϕ) = 0, and since (∂t − ∂x1)(t− x1) = 2, one sees that ∂tϕ + ∂x1ϕ is a bonafide traveling
wave transverse to the level sets of t − x1.

123

ometric properties and our gauge choice for studying the perturbations are described in

Section 4.2.2.

We next discuss the basic analytic tools used in our arguments; in Section 4.3.1 we re-

call the global Sobolev inequalities of Chapter 2 adapted to the geometry of the planewave

background, in Section 4.3.2 we develop a weighted vector field algebra to help simplify

our analyses of the nonlinear terms using more schematic notations.
In Section 4.4 we study the semilinear model problem (cid:3)mφ = Υ

(∂uφ)2, obtained
from dropping the quasilinearity from (4.1.2). This model problem turns out to capture
already the majority of the difficulty one faces when analyzing the full problem. We prove
small-data global wellposedness for the semilinear model in all dimensions ≥ 3. There are
certain additional technical difficulties for studying the quasilinear model (4.1.2) in d = 2

(cid:48)(cid:48)

due to the fact one expects even the first order energy exhibits polynomial growth there,

and the loss seems too strong to overcome with the methods described in this chapter;

therefore we also omit a detailed treatment of the d = 2 semilinear model.

The remainder of the chapter is devoted to studying the full quasilinear problem in

d = 3, and stating and proving a more precise version of Theorem 4.1.1. In Section 4.5 we

perform first some preliminary computations casting the equations for the perturbation

φ and its higher order derivatives in schematic form to prepare for analysis. As many of

the computations are long and involved, we delegate sketches of the arguments separately

to the Appendix. At the end of the section we state our Main Theorem 4.5.8. As usual,

we will prove our Main Theorem by a bootstrap argument for our energy hierarchy. In

Section 4.6 we define our energy quantities, outline our main energy estimate, state our

bootstrap assumptions, and derive some immediate consequences that do not involve the

equations of motion. Section 4.7 is devoted to proving a priori estimates for our equations

of motion, based on the bootstrap assumptions. These are combined in Section 4.8 to

show that the bootstrap assumptions can be improved, and thereby hold for all time and

global existence follows.

124

4.2 The background solution

In this section we first exhibit simple planewave solutions to the membrane equation,

and describe their geometry. These solutions are traveling waves and exist for all time;

our goal is to analyze their stability under small non-plane-symmetric perturbations. To

do so we recast the stability problem as a small-data Cauchy problem for the perturba-

tion. In the second part of this section we exploit the geometric interpretation of the

solutions as minimal submanifolds of higher dimensional Minkowski space to make a

convenient choice of gauge, and derive the corresponding perturbation equations. The

gauge choice allows us to simplify the analysis of the linearized dynamics. As the mem-

brane equation itself is a quasilinear wave equation, when linearizing around a fixed

nontrivial background solution, typically the background contributes to the linearized

dynamics (e.g. in [DKSW16] where the background contributes a potential term leading

to generic instability of the system). For the membrane equation in Minkowski space,

however, it is known [CB76] that the potential term in the linearized dynamics for per-

turbations parametrized by the normal bundle is given by the double contraction of the

extrinsic curvature of the embedding of the background solution. For simple planewaves,

this potential term vanishes [Won17b]. The gauge choice below makes this explicit and

shows that the perturbed system can be described by a quasilinear perturbation of the lin-

ear wave equation on Minkowski space, with the background solution only appearing as
coefficients of the nonlinearity.

125

4.2.1 Simple planewave solutions to the membrane equation
Let Υ ∈ C
by3

(R;R) be arbitrary. One easily sees that the function ˚φ : R1+d → R defined

∞

˚φ(t, x1, x2, . . . , xd) = Υ (t + x1)

(4.2.1)
(t + x1) d(t + x1) and hence m(d ˚φ,d ˚φ) ≡ 0 and
solves (4.1.1), seeing as d ˚φ(t, x1, . . .) = Υ
˚φ ≡ 0. The simple planewave background will be interpreted as the graph of ˚φ in
mµν∂2
µν
R1,d+1, the (d +2)-dimensional Minkowski space equipped with the standard Minkowski
metric M. That is to say, we consider the embedding R1+d (cid:44)→ R1,d+1, given by

(cid:48)

(t, x1, . . . , xd) (cid:55)→ (t, x1, . . . , xd, ˚φ(t, x))

with the first component fixed as the timelike one. The m(dφ,dφ) (cid:44) −1 implies that the
induced metric on the graph of φ is Lorentzian and non-degenerate. By the analysis

of [Won17b] this induced metric is flat; this fact can also be seen through the following

explicit computations.

Denoting the above embedding by Φ, the induced metric can be in fact given by the

line element

∗

Φ

M = ds2 = (−1 + (∂t ˚φ)2) dt2 + 2∂t ˚φ∂x1 ˚φ dt dx1 + (1 + (∂x1 ˚φ)2) d(x1)2

+ d(x2)2 +··· + d(xd)2.

Using that ∂tφ(t, x) = ∂x1φ(t, x) = Υ

(t + x1), we see that if we define

(cid:48)

u def= t + x1,
(cid:20)
t − x1 −(cid:82) x1+t

u def= 1
2

0

(cid:21)

;

(cid:48)

(Υ

)2(τ) dτ

(4.2.2)

3We’ve made the choice to have our background travelling waves move “to the left”,
i.e. as a function of t + x1. Note that for the analyses in [SHLW16] the simple waves move
“to the right”. We beg those readers familiar with the previous work to indulge us and
mentally reorient the space-time and relabel the function u as needed.

126

that the line element can be alternatively written as the Minkowski metric in standard

double-null form

m = ds2 = −2 du du + d(x2)2 +··· + d(xd)2.

(4.2.3)
The functions u and u solve the eikonal equation m(∇u,∇u) = m(∇u,∇u) = 0. For the
subsequent analyses we will parametrize using the coordinates {u, u, x2, . . . , xd}.

Remark 4.2.1. Note that there are two Minkowski metrics involved in the construc-
tion: (1) The metric on the ambient space R1,d+1, which is denoted by M. (2) The in-

duced Minkowski metric on the planewave background given by double-null coordinates
(u, u, ˆx) ∈ R1,d, denoted by m.

For completeness, we note that the change of variables can be inverted:

t = 1

x1 = 1

(cid:82) u
(cid:82) u
2u + u + 1
2
2u − u − 1

2

0 (Υ

0 (Υ

(cid:48)

)2(τ) dτ,
(cid:48)

)2(τ) dτ.

(4.2.4)

For convenience, we note that relative to this coordinate system, our simple planewave

solution is given by the embedding

(u, u, ˆx) (cid:55)→ (t, x1, ˆx, Υ (u)) ∈ R1,d+1

(4.2.5)

where t and x1 are given as functions of u, u by (4.2.4), and for convenience we denote by
ˆx = (x2, . . . , xd).

We finish this subsection by computing the extrinsic curvature (second fundamental

form) of the embedding (4.2.5). The change of variables (4.2.4) implies that the vector

(cid:16)1

2

(cid:48)

(u)2),

(1 + Υ

1
2

(1− Υ

(cid:48)

(u)2),0, . . . ,0, Υ

(cid:48)

(cid:17)

(u)

.

fields

∂u = (1,−1,0, . . . ,0),

∂u =

Denote by n : R1,d → R1,d+1 the unit normal vector field (with respect to the Minkowski
metric on R1,d+1) of the embedding (4.2.5) given by
(cid:48)

(cid:48)

n(u, u, ˆx) = (−1)d−1(−Υ

(u),0, . . . ,0,−1).

(u), Υ

(4.2.6)

127

The expression (4.2.6) can be computed from

nα = (M

−1)ακ
κ,β,γ,σ2,...,σd−1(∂u)β(∂u)γ(∂x2)σ2 ···(∂xd−1)σd−1,

where α, κ, β, γ, σ2, . . . , σd−1 ∈ {0,1, . . . , d−1} and 
κ,β,γ,σ2,...,σd+1 is the anti-symmetric sym-
bol normalized by 
0,1,2,...,d−1 = 1. The second fundamental form can then be computed
to equal

II = (−1)d−1Υ

(cid:48)(cid:48)

(4.2.7)
(We use the convention II(∂u, ∂u) = (cid:104)∂un, ∂u(cid:105)M.) Notice that II is indeed trace-free with
respect to the induced metric as a consequence of and additionally the double contraction

(u) du2.

II : II with respect to the induced metric also vanishes, both a consequence of the eikonal

equation.

4.2.2 The gauge choice and the perturbed system

Small perturbations of the embedding (4.2.5) reside within a tubular neighborhood of

the background. We parametrize the perturbations as a graph within the normal bundle,

analogously to the analysis in [DKSW16]; that is, we look for embeddings of the form

(u, u, ˆx) (cid:55)→ (t, x1, ˆx, Υ (u)) + φ(u, u, ˆx)· n(u, u, ˆx)

(4.2.8)
where φ : R1+d → R is the height of the graph, and n is the unit normal as defined in
(4.2.6). The induced metric for this perturbation will be denoted by g; it is given by the
pull-back of the Minkowski metric M on R1,d+1 by the embedding (4.2.8)

g = m + dφ⊗ dφ− 2φΥ

(cid:48)(cid:48)

du ⊗ du.

(4.2.9)

Its corresponding volume element can be computed to be

(cid:112)|g| dudud ˆx

dvolg =

where

|g| def= 1 + m(∇φ,∇φ) + 2φΥ

(cid:48)(cid:48)

(∂uφ)2.

(4.2.10)

128

We note that g is a perturbation of the Minkowski metric m with terms both quadratic and

linear in φ. For later computations it is helpful to also record the perturbations truncated

◦
to the linear terms, which we will denote by
g
◦
g def= m− 2φΥ

(cid:48)(cid:48)

du ⊗ du.

(4.2.11)

The inverses of g and

coordinates (u, u, ˆx). For

◦
g can be computed explicitly in the double null (relative to m)
◦
g one finds

◦
−1 = m
g

−1 + 2φΥ

(cid:48)(cid:48)

∂u ⊗ ∂u.

(4.2.12)

Note that this implies

(4.2.13)
◦
g +dφ⊗dφ, we can apply the Sherman-Morrison formula [SM50] to obtain

Using that g =

|g| = 1 +
(cid:16)◦
−1 · ∂φ

◦
−1(∇φ,∇φ).
g
(cid:17)⊗(cid:16)◦

g

(cid:17)

−1 · ∂φ

g

−1 =

g

◦
−1 − 1|g|
g

Notation 4.2.2 (Index raising and lowering).

quently needs to lower or raise indices with respect to any of g / g
−1. We will adopt the following conventions

m

.

(4.2.14)

In the computations to follow, one fre-
◦
−1, or m /
g

◦
g /

−1,

• The unadorned musical operators (cid:91) / (cid:93) are used for lowering and raising indices

with respect to the Minkowski metric m of the background simple-planewave solu-

tion.

• Implicitly lowered / raised indices are always with the Minkowski metric m, so ∂jφ

refers to mjk∂kφ.
• When it is clear from the context, we will sometimes omit the index −1 denoting
−1(∇φ,∇φ) since
inverses for brevity. For example, we write m(∇φ,∇φ) instead of m
−1. Simi-
∇φ are naturally covariant and so we will need the contravariant metric m
larly, if we write gµν∂νφ it should be interpreted as (g

−1)µν∂νφ.

129

• Index manipulations with respect to the dynamical metrics g and

◦
g will always be

adorned. So for example we will write
◦
−1 · ∂φ,
g

◦
g(cid:93)φ =
∂

∂g(cid:93)φ = g

−1 · ∂φ

with corresponding index notation

◦
g(cid:93)jφ =
∂

◦
gjk∂kφ,

∂g(cid:93)jφ = gjk∂kφ.

With the notation announced above, we can equivalently write

−1 =

g

◦
−1 − 1|g|∂
g

◦
g(cid:93)φ⊗ ∂

◦
g(cid:93)φ.

For the embedding (4.2.8) to have vanishing mean curvature (i.e. satisfy the membrane

equation), it must be a formal stationary point of the volume functional φ (cid:55)→(cid:82)

dvolg. The
perturbation equations satisfied by φ can be derived as the corresponding Euler-Lagrange

equations, as shown below.

Denoting by L =

(cid:112)|g| =

(cid:112)
1 + m(∇φ,∇φ) + 2φΥ (cid:48)(cid:48)(∂uφ)2 the Lagrangian density, the
(cid:32)

(cid:33)

(cid:33)

(cid:32)

(cid:32)

(cid:33)

corresponding Euler-Lagrange equation is

δL
δφ

δL
δφu

∂
∂u

δL
δφu

∂
∂u

δL
δφ ˆx

∂
∂ ˆx

+

=

(4.2.15)
where we use the subscript on φ to denote partial differentiation. Expanding m(∇φ,∇φ) =
−2∂uφ∂uφ + (∂ ˆxφ)2 we compute
(φu)2

(cid:48)(cid:48)

+

δL
δφ
δL
δφu

= L−1Υ
= L−1(−φu)

δL
δφ ˆx
δL
δφu

= L−1φ ˆx
= L−1(−φu + 2Υ

(cid:48)(cid:48)

φφu).

So the Euler-Lagrange equation reads

∂µ

Observe that by (4.2.14) we have

(cid:33)

(cid:32) ◦
gµν∂νφL

= L−1Υ

(cid:48)(cid:48)

(φu)2.

(4.2.16)

∂g(cid:93)φ =

◦
g(cid:93)φ.

1|g|∂

130

This implies that we can rewrite (4.2.16) as

(cid:3)gφ = |g|−1Υ

(cid:48)(cid:48)

(φu)2;

(4.2.17)

here (cid:3)g refers to the Laplace-Beltrami operator of the metric g, given in local coordinates
by

(cid:16)(cid:112)|g|gµν∂νf

(cid:17)

.

1(cid:112)|g|∂µ

(cid:3)gf =

As the metric g depends on the first jet of the unknown φ, the principal part of the

(4.2.17) may be different from gµν∂2

µνφ. For our equation, this turns out not to be an
issue, as can be seen when we take the first coordinate partial derivatives of (4.2.16).

With the aid of the relation (4.2.14) between g

∂

◦
g(cid:93)µφL

 = ∂µ

∂g(cid:93)µ∂λφ(cid:112)|g|

 + ∂µ

∂λ

−1 and
◦
gµν∂νφ

(cid:112)|g|

◦
−1 we obtain
g

◦
g(cid:93)µφ
∂
|g|3/2 ∂λ

− 1
2

∂λ∂µ

 .

◦
gρσ ∂ρφ∂σ φ

Noticing that the derivatives ∂λ
second, we see that the principal term are all captured in the first term on the right in the

◦
g depends only on the first derivatives of φ, and not the

above identity.

We can simplify the identity further. Notice that

◦
−1 = ∂λ(2φΥ
g

(cid:48)(cid:48)

) ∂u ⊗ ∂u

∂λ

this implies

∂λ

∂µ

◦
gµν∂νφ

(cid:112)|g|

◦
gρσ ∂ρφ∂σ φ

◦
− 1
g(cid:93)µφ

∂
∂
|g|3/2 ∂λ
2
◦
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
− ∂µ
g(cid:93)µφ
|g|1/2
|g|−1/2Υ (cid:48)(cid:48)(∂uφ)2

 = 2∂u

(cid:48)(cid:48)

∂λ(φΥ
(cid:112)|g|
)(∂uφ)2 − ∂



)∂uφ
◦
g(cid:93)µφ
|g|1/2 ∂µ

(cid:32) 1|g|∂λ(φΥ

(cid:33)

.

(cid:48)(cid:48)

)(∂uφ)2

(cid:48)(cid:48)

1|g|∂λ(φΥ

131

So we conclude

|g|− 1

2

− 2
d−1 (cid:3) ˜g∂λφ = ∂λ
− 2∂u

(cid:48)(cid:48)

(cid:16)|g|−1/2Υ
(φu)2(cid:17)
 +|g|−3/2Υ
∂λ(φΥ
(cid:112)|g|

)∂uφ

(cid:48)(cid:48)

(cid:48)(cid:48)

∂λ(φΥ

(cid:48)(cid:48)

)(∂uφ)4
(cid:48)(cid:48)

(cid:32) 1|g|∂λ(φΥ

(cid:33)

)(∂uφ)2

,

(4.2.18)

+|g|−1/2∂

◦
g(cid:93)µφ∂µ

where we have introduced the conformal metric

˜g = |g|− 2

d−1 · g

(4.2.19)

where d is, recall, the number of spatial dimensions. The conformal metric ˜g has its

Laplace-Beltrami operator as

(cid:3) ˜gf = |g| 1

2 + 2

d−1 ∂µ

 1(cid:112)|g|gµν∂νf



which has the same principal part as (cid:3)g.

Remark 4.2.3. Observe that (4.2.17) and (4.2.18) are geometric quasilinear wave equa-
tions that linearize to the linear wave equation on R1,d. The quadratic nonlinearities in-
clude, as can be seen, the resonant semilinear interaction (∂uφ)2 as well as the weakly
resonant quasilinear interaction φ(∂2

uuφ).

That we will be able to prove global existence for this equation (and not suffer from

shock formation in finite time) is due to the background Υ

which accompanies the ap-

(cid:48)(cid:48)

perance of such resonant terms, and localizes the resonant interactions to the region
t ≈ −x1; one can think of Υ
uuΥ , exposing the null condition that was present in
the original membrane equation (4.1.1). However, as the background function Υ has

as ∂2

(cid:48)(cid:48)

non-compact (in the ˆx direction) support, and is non-decaying (in time), the improved

decay we obtain due to this space-time localization is weaker than in classical studies of

nonlinear waves with null condition. Such issues and their ramifications are discussed in

more detail in Section 4.4 where we examine a semilinear model that captures the main
analytical difficulties.

132

4.3 Preliminary L2 analysis

We will analyze (4.2.17) using the vector field method (VFM) adapted to hyperboloidal

foliations. As we saw in the analysis of the wave maps equation in Chapter 3, this variant

of the VFM allows us to derive a priori estimates using only the ∂t-multiplier (see the
discussion following (2.2.8)) and not the Morawetz K multiplier (see Remark 2.2.4). To
efficiently handle the coefficients Υ

present in (4.2.17) using only the Lorentz boosts as

(cid:48)(cid:48)

commutators, we will develop in the second part of this section a weighted vector field al-

gebra. The combination of these techniques will be first illustrated in a model semilinear

problem in Section 4.4, before we state and prove the main result of this chapter.

4.3.1 Global Sobolev estimates on the double-null coordinates

We begin by adapting the global Sobolev estimates of Chapter 2 to the gauge choice of

the perturbation φ. More precisely, we record the estimates on hyperbolas in Minkowski

space as described by the double-null coordinate system (u, u, ˆx) with the metric (4.2.3).

Consider the set I + def= {u > 0, u > 0,2uu −| ˆx|2 > 0}. This set corresponds to the interior
of the future light cone emanating from the origin in Minkowski space. On this set, we can

define the time function

τ def=

(cid:113)
2uu −| ˆx|2.

(4.3.1)

Notation 4.3.1. The level set of τ will be denoted by Στ. The Riemannian metric induced
on Στ by the Minkowski metric m will be denoted ητ. The geometric metric g also induces
a symmetric bilinear form on Στ, we will denote it by hτ. When hτ can in principle be
Lorentzian or degenerate, in our application it will turn out to be always Riemannian.

Remark 4.3.2. It is straightforward to check that (Στ, hτ) as described above is isometric
to the hyperboloid defined in (2.1.1) equipped with the metric (2.1.4).

133

We introduce also the hyperbolic radial function ρ within this forward light-cone I +

(cid:33)

(cid:32)

u + u√
2τ

by

ρ def= cosh

−1

.

(4.3.2)

We note that relative to the Minkowski metric, the unit normal to Στ is given by (using
an abuse of notation)

−(dτ)(cid:93) =

(u∂u + u∂u + ˆx· ∂ ˆx) .

1
τ

(4.3.3)

Relative to the perturbed metric g, the unit normal to Στ takes the form

−

(cid:112)|g(dτ,dτ)| = −(dτ)(cid:93) + 2( u

(cid:113)
1− 2( u

(dτ)g(cid:93)

(cid:48)(cid:48)

∂u − ◦
τ )φΥ
τ )2φΥ (cid:48)(cid:48) +|g|−1[

g(dφ,dτ)∂g(cid:93)φ

◦
g(dφ,dτ)]2

.

(4.3.4)

We define the following vector fields:
1√
2

T =
(∂u + ∂u);
L1 = u∂u − u∂u;
(u + u)∂ ˆxi +

1√
2

Li =

1√
2

ˆxi(∂u + ∂u),

(4.3.5)

(4.3.6)

(4.3.7)

i = 2, . . . , d.

It is straightforward to check that T is the future timelike vector field ∂t of Minkowski
space when expressed in the double-null coordinates. Similarly, Li in (4.3.6) –(4.3.7) are the
Lorentzian boosts xi∂t + t∂xi when expressed in (u, u, ˆx). Hence, these vector fields are all
Killing with respect to the Minkowski metric. Note that the Li (where i = 1, . . . , d) are also
all tangential to Στ. If α is an m-tuple with elements drawn from {1, . . . , d} (namely that
α = (α1, . . . , αm) with αi ∈ {1, . . . , d}), we denote by Lα the differential operator

f (cid:55)→ LαmLαm−1 ··· Lα2Lα1f .

By |α| we refer to its length, namely m.

The global Sobolev inequality adapted to the double-null coordinates on Στ reads:

134

Theorem 4.3.3 (Global Sobolev inequality in double-null coordinates). Let (cid:96) ∈ R be fixed.
For any function f defined on I +, we have, for any (u, u, ˆx) ∈ I +,

0 cosh(ρ0)1−d−(cid:96) (cid:88)

−d

(cid:90)

|f (u, u, ˆx)|2 (cid:46)d,(cid:96) τ

cosh(ρ)(cid:96)|Lαf |2 dvolητ0

.

|α|≤(cid:98) d
2

(cid:99)+1

Στ0

The quantities τ0 and ρ0 appearing on the right of the inequality are given as

τ0 = τ(u, u, ˆx),

ρ0 = ρ(u, u, ˆx).

Proof. The estimate follows immediately from Theorem 2.1.1 when noting that the weight

function wτ (see (2.1.3)) in the weighted Sobolev space W(cid:98)d/2(cid:99)+1,2

(cid:96)

is exactly (u + u)/

2 in

√

the double-null coordinates.

Remark 4.3.4. Note that by the definition of the function ρ, the coefficient in Theorem

4.3.3 can be written as

−d
0 cosh(ρ0)1−d−(cid:96) = τ(cid:96)−1

0

τ

4.3.2 A weighted vector field algebra

(cid:32)

u + u√
2

(cid:33)1−d−(cid:96)

.

In classical arguments using the vector field method, one typically commutes the equa-

tion with the generators of the Poincar´e group, which consists of the

• translation vector fields ∂t, ∂xi ;

• rotations xi∂

xj − xj∂xi ;

• Lorentz boosts t∂xi + xi∂t.

These vector fields form, under the Lie bracket, an R-algebra.

In applying the Sobolev inequality of the previous section, we intend to only commute
with the Lorentz boosts Li. This subset does not form an R-algebra under the Lie bracket.

135

However, they form an algebra with coefficients drawn from a space of weights. For

convenience, we introduce the y-coordinates

y0 =

,

y1 =

u + u√
2

u − u√

2

yi = ˆxi (i ≥ 2).

,

(4.3.8)

(cid:110) y1
Definition 4.3.5. We denote by W∗ the (commutative) ring of polynomial expressions in
y0 , . . . , yd
, with R coefficients. This ring can be graded according
y0 term in the polynomial expression, we denote by Wi the corre-

the d + 1 variables
to the degree of the 1

y0 , 1
y0

(cid:111)

sponding set of homogeneous elements.

y0 )5( y2

∈ W0, while ( 1

Remark 4.3.6. By way of clarification and for example, we will have that the expression
y1
y0
Remark 4.3.7. Notice that within the light cone I +, we have that the functions (for all
i = 1, . . . , d)

y0 ) ∈ W5.

y0 )( y4

(cid:12)(cid:12)(cid:12) ≤ 1

(cid:12)(cid:12)(cid:12) yi

y0

are uniformly bounded.

Now, observe that for i, j ∈ {1, . . . , d},

(cid:17)
(cid:17)

(cid:16) 1
Li(cid:16) 1

T

y0

y0

= −(cid:16) 1

(cid:17)2

,
y0
· yi
= − 1
y0 ,
y0

Furthermore,

(cid:16) yi
Li(cid:16) yj

T

y0

y0

(cid:17)
(cid:17)

= − 1
· yi
y0 ,
y0
= δij − yi
y0

· yj
y0 .

[Li, T ] = − 1

yi
y0 T ,

yi

y0 Lj − yj
y0 Li.

Together these implies that the set of vector fields of the form c0T +(cid:80) ciLi where the cµ are

y0 Li +

[Li, Lj] =

taken from W∗ form not only an R-Lie algebra, but also an algebra over the ring W∗, with
multiplication being the Lie bracket. We will denote this algebra by A∗. The following

proposition follows immediately from the computations above.

136

Proposition 4.3.8. For i ∈ Z+, define

def=

A0

cjLj | cj ∈ W0

 d(cid:88)

j=1

 ,

c0T +

d(cid:88)

j=1

Ai =

cjLj | c0 ∈ Wi−1, cj ∈ Wi

 .

Then A∗ is graded, with Li ∈ A0, and T ∈ A1. In particular, given elements Xa ∈ Aa, Xb ∈ Ab
and f ∈ Wc, we have that

[Xa, Xb] ∈ Aa+b,

f Xa ∈ Aa+c.

Remark 4.3.9. We remark that we also have the following commutator relation

[Li,[Lj, T ]] = δijT ∈ A1

as expected.

Using A∗, we can build an algebra of differential operators which we label by B

∗,∗∗ .

Consider terms of the form

f X1X2X3 . . . Xk

(4.3.9)
where f ∈ W∗ and Xα ∈ {Li, T}. They are differential operators that act on functions
defined on I + in the usual way. Using the computations above we see that terms of such
∗,∗∗ as
form are closed under composition of differential operators. Hence we define B
the set of finite sums of terms of the form (4.3.9), with addition defined normally and

∗,∗∗
composition as multiplication; B

is obviously a W∗-module.

∗,∗∗
In exactly the same way as A∗, the algebra B

is graded. We will use its lower index

to record this grading.

Definition 4.3.10. The weight of a term of the form (4.3.9), where f is a monomial, is
defined by the number of times T appears among the Xα, plus the number of times 1/y0

appears in the monomial f . The degree of a term of the form (4.3.9) is defined as the

number k. The T -degree of a term of the form (4.3.9) is the number of times T appears
∗,∗∗ of elements with weight
among the Xα. By Bk,s

w we refer to the set of finite sums in B

w and degree at most k, and T -degree at most s.

137

Remark 4.3.11. The set Bk,s
w is well-defined due to Proposition 4.3.8. One needs to
check that, for example, f X1X2 and f X2X1 + f [X1, X2], which are equal as differential
operators, have the same degrees and weight. Proposition 4.3.8 implies that for Xα ∈
{Li, T}, the terms making up [X1, X2] always have the same weight as X1X2, and with
same or lower T -degree.

For example, given any m-tuple α, the operator Lα ∈ Bm,0

, while we can identify

0

∗,∗
w . The following
. The set Aw are the set of degree (exactly) 1 elements in B

W∗ = B0,0∗
proposition follows immediately from the definition and Proposition 4.3.8.
Proposition 4.3.12. If A ∈ Bk,s

w , and B ∈ Bk

, then

(cid:48)

(cid:48)
(cid:48)
,s
w

(cid:48)

(cid:48)
w+w

;

1. AB ∈ Bk+k
(cid:48)
,s+s
2. [A, B] ∈ Bk+k
We remark finally that if f = f (u) is a function defined within the light cone I +, then

(cid:48)−1,s+s
(cid:48)

w+w

(cid:48)

.

T f =

1√
2

(cid:48)

f

(u),

and L1f = uf

(cid:48)

(u).

In particular, if f is smooth and supported within a slab u ∈ (a, b), then both T f and L1f
are functions of u alone that are smooth and supported within u ∈ (a, b). Similarly, we see
that for i ≥ 2

Lif =

ˆxif

(u).

(cid:48)

1√
2

To estimate functions of this form, we will use the following lemma.
Lemma 4.3.13. Fix f = f (u) a smooth function supported in u ∈ [a, b]. Then on the set I +,
for any m-tuple α, we have

|Lαf | (cid:46) (1 + u)m/2 · 1{u∈[a,b]}.

The implicit constant depends on the numbers a, b, the degree m, the dimension d, as well as
(cid:107)f (cid:107)

Cm.

138

Proof. Observe that if i, j ∈ {2, . . . , d},

and that

Li ˆxj =

1√
2

δij(u + u)

√

2 ˆxi.

Li(u + u) =

So we have that up to a universal structural constant depending only on the dimension d

and the degree m,

|Lαf | (cid:46)(cid:16)

1 +|u|m +|u + u|m/2 +| ˆx|m(cid:17)·(cid:107)f (cid:107)

Cm · 1{u∈[a,b]}.

As on the set of interest, u ∈ [max(a,0), b], we have that |u| < b. Furthermore, on I + by
definition we have 2uu > | ˆx|2. The boundedness of u implies that | ˆx| (cid:46) √
u. The desired

bound follows.

4.3.3 Generalized energy

In Subsection 2.2.1, for a multiplier vector field X and a solution to the linear wave equa-
tion (cid:3)mφ = 0, we defined the X-energy of φ along spacelike hypersurfaces of Minkowski
space (see (2.2.7)). This was done through the energy-momentum tensor Q[φ], a symmet-

(cid:1)-tensor adapted to the Minkowski metric m. Motivated by the quasilinear nature of

ric(cid:0)0

2

the perturbed metric g (see (4.2.9)), we now generalize this notion of energy to arbitrary

metrics.

In the flowing discussion, let g denote any Lorentzian metric. Then we define the

energy-momentum tensor associated to φ and with respect to g,

Q[φ;g] def= dφ⊗ dφ− 1
2

g(dφ,dφ)g.

(4.3.10)

Given a multiplier vector field X, we define the corresponding X-energy current with re-

spect to g to be the vector field

(X)J α[φ;g] def= (g

−1)αβQβγ[φ]Xγ .

(4.3.11)

139

(cid:90)

(cid:90)

Στ

(cid:90)

d(cid:88)

τ2 cosh ρ

i=1

Στ

1

†
†

The following identity is the analogue of the divergence identity (2.2.8) with respect to g,

which follows from standard computations:

(cid:16)(X)J [φ;g]
(cid:17)

divg

= (cid:3)gφXφ +

Qαβ[φ;g]LXgαβ.

1
2

(4.3.12)

Specializing now to the cases where g is either the Minkowski metric m or the perturbed

metric g, we define the following T -energies along Στ:

Eτ[φ]2 def= Eτ[φ; m]2 def= 2

Q[φ; m](T ,−(dτ)(cid:93)) dvolητ ,
1(cid:112)|g(dτ,dτ)|Q[φ; g](T ,(−dτ)g(cid:93)) dvolhτ ,

Στ

Eτ[φ; g]2 def= 2

(4.3.13)

(4.3.14)

We emphasize that the one corresponding to the Minkowski metric can be written explic-

itly as

Eτ[φ]2 =

(Liφ)2 +

1

cosh ρ

(T φ)2 dvolητ ,

(4.3.15)

see the discussion immediately after the definitions of T and Li (4.3.5) – (4.3.7), Lemma

2.2.5, and its proof via the identity (A.2.17).

Integrating the divergence identity (4.3.12) with m and g in place of g and T in place

of X between two level sets τ0 < τ1 of τ one obtains4 the energy inequalities

(cid:12)(cid:12)(cid:12)(cid:3)gφ· T (φ)
(cid:12)(cid:12)(cid:12) +

(cid:12)(cid:12)(cid:12)Q[φ; g] :g LT g

(cid:12)(cid:12)(cid:12) dvolg,

Eτ1[φ; g]2 −Eτ0[φ; g]2 (cid:46)
Eτ1[φ]2 −Eτ0[φ]2 (cid:46)

τ∈[τ0,τ1]

τ∈[τ0,τ1]

|(cid:3)φ· T φ| dvolm .

(4.3.16)

(4.3.17)

The following proposition is how one obtains pointwise control for terms appearing on

the right hand side of (4.3.16) and (4.3.17).

4Note that the future oriented normals Στ with respect to m and g are (4.3.3) and
(4.3.4), respectively. These appear when applying the divergence theorem to the left hand
side of (4.3.12) through the definitions (4.3.11), (4.3.13), and (4.3.14).

140

Proposition 4.3.14. For any function f defined on I +, we have, for any (u, u, ˆx) ∈ I +,

|f (u, u, ˆx)| (cid:46)d τ1− d

2 cosh(ρ)1− d

2

|Lif (u, u, ˆx)| (cid:46)d τ1− d

2 cosh(ρ)1− d

2

|T f (u, u, ˆx)| (cid:46)d τ

− d
2 cosh(ρ)1− d

2

|α|≤(cid:98) d
2

(cid:99)

(cid:88)
(cid:88)
(cid:88)

|α|≤(cid:98) d
2

Eτ[Lαf ],

Eτ[Lαf ],

(cid:99)+1
Eτ[Lαf ].

|α|≤(cid:98) d
2

(cid:99)+1

Proof. This follows immediately when Propositions 2.2.7 (with M = 0) and 2.2.9 are ex-

pressed in double-null coodinates through Theorem 4.3.3.

Remark 4.3.15 (Remark 2.2.6 expressed in double-null coordinates.). A feature of the

energy (4.3.15) is its anisotropy. The classical energy estimates of wave equations control
integrals of |∂tφ|2 +|∇φ|2 where all components appear on equal footing. Here, however,
the transversal (to Στ) derivative T φ has a different weight compared to the tangential
derivatives Liφ. Noting that by their definitions, T has unit-sized coefficients with ex-
pressed relative to the standard coordinates of Minkowski space. The coefficients for
Li (within the light cone I +) have size ≈ t. Therefore an isotropic analogue would be ex-
pected to contain integrals of 1
t2 (Liφ)2 along with integrals of T φ. Noting that t = τ cosh ρ
this indicates that an isotropic analogue would contain, instead of the integral given in

(4.3.15), the integral(cid:90)

Στ

(cid:88)

1

τ2 cosh(ρ)3

(Liφ)2 +

1

cosh(ρ)

(T φ)2 dvolητ .

In other words, the integral for Liφ in the energy has a better ρ weight than would be ex-

pected from an isotropic energy, such as that controlled by the standard energy estimates.

This improvement reflects the fact that the energy estimate described in this sec-

tion captures the peeling properties of linear waves within the energy integral itself. It

is well-known that derivatives tangential to an out-going light-cone decay faster along the

141

light-cone, than derivatives transverse to the light-cone. As asymptotically hyperboloids

approximate light-cones, we expect the same peeling property to survive. Indeed, the

energy inequality (4.3.16) shows that we can capture this in the integral sense.

4.4 A semilinear model

Before stating and proving our main results, we will illustrate both our method of
proof and the main difficulties encountered in the simpler setting of a semilinear problem.

Recall that the small-data global existence problem for the membrane equation (4.1.1) in
dimension d ≥ 3 follows from a direct application of Klainerman’s vector field method,
after noting that the equation of motion is a quasilinear perturbation of the linear wave

equation with no quadratic nonlinearities, see the expository book [Sog08]. In particu-

lar, Klainerman’s null condition plays no role in establishing this result. As indicated

in Remark 4.2.3, the perturbation problem for simple planewaves introduces resonant

quadratic terms to which Klainerman’s null condition does not directly apply. On the

other hand, as observed in that same remark, there is a hidden null structure from which

we can expect to recover some improved decay rates.

The main difficulty however is that Klainerman’s null condition is built upon the ex-

pected decay rates corresponding to solutions to the linear wave equations with strongly

localized initial data. In particular, the heuristic for the null condition is based on the
expectation that, for generic first derivatives of such a solution, ∂φ decays like t(1−d)/2;
while for “tangential” (to an outgoing null cone foliation) derivatives, the corresponding

derivatives decays like t

−d/2.

In our setting, however, one of the waves in the interaction is a simple planewave
which does not decay at all. This reduces the effectiveness of the null structure in improv-
ing decay. As will be shown this difficulty manifests already in the model problem to be

discussed in this section. For example, when applying the vector field method one studies

the equations of motion satisfied by higher derivatives of the solution. After commuting

142

the equation with the Lorentz boosts, one sees that when the boost hits on Υ
, we obtain
a coefficient that, while still localized to t ≈ −x1, is growing in time. On an intuitive level
one can interpret this as a transfer of energy from the (infinite energy) background sim-

(cid:48)(cid:48)

ple planewave to the perturbation. The null structure in our context then serves to cap

the rate of this energy transfer, ensuring (in our case) global existence of the perturbed

solution.

The specific semilinear model problem we consider takes (4.2.17) and drops from it

the quasilinearity. That is to say, we consider the small-data problem for the semilinear

wave equation

(cid:48)(cid:48)

(cid:3)φ = Υ

(u)(φu)2

(4.4.1)

on R1,d, where (cid:3) is the usual wave operator corresponding to the Minkowski metric m.

To approach this problem using a vector field method, one commutes (4.4.1) with the

Lorentz boosts to derive equations of motions for higher order derivatives. The energy

estimates for these higher order derivatives are then combined with the global Sobolev
decay estimates for the solution. The main difficulty one encounters

inequality to get L

∞

here, however, is when the vector fields hit on the background Υ

(cid:3)Lαφ = Lα(Υ

(cid:48)(cid:48)

)(φu)2 + . . . ,

where Lα(Υ

(cid:48)(cid:48)

) can have growing L

∞

norm.

(cid:48)(cid:48)

. We have

This potential growth of the coefficients is the main technical complication in this
), assuming for convenience Υ ∈
problem. The best uniform estimate we have for Lα(Υ
∞
0 and the initial data for φ is compactly supported, is via Lemma 4.3.13, which gives
C

(cid:48)(cid:48)

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
(cid:3)Lαφ ≈ (1 + u)m/21{u∈[a,b]} · (1 + u + u

−d/2∂φ

)

where we have made the optimistic assumption that φu decays like (1 + u + u)
would be the case for a linear wave.

−d/2, as

∂uφ

143

At this point, two different complications present themselves. First, one may naively

hope that the (higher order) energies always stay bounded, in analogy with the linear
case. This hope is rapidly dashed when we examine the energy estimate for |α| = d. After
commuting with d derivatives, we see that

(cid:3)Lαφ ≈ ∂φ

with no decay! Even assuming that we can prove the boundedness of the lowest order
energy (which controls ∂φ in L2), the best we can obtain is then that energy for Lαφ
grows linearly in time. This first difficulty can be overcome with a modified bootstrap

scheme where the expected (polynomial in time) energy growth is incorporated into the

assumptions.

Remark 4.4.1. Several remarks are in order concerning this energy growth:

1. This growth is different from what appears in typical applications of the vector field

method to nonlinear wave equations with null-condition satisfying nonlinearities in

d = 3. In those cases, the equation takes the schematic form

(cid:3)φ = LφLφ

−d/2 and Lφ is a
where Lφ is a “good” derivative that is expected to decay like t
“bad” derivative that is expected to decay like t(1−d)/2. When considering the energy
estimates for the top order derivatives, one must face the possibility of needing to

control

(cid:3)∂αφ = ∂αLφ· Lφ + . . . .

To close the energy estimate, one must estimate ∂αLφ in L2 and thereby bound Lφ

by 1/t, whereupon the time integration gives a small energy growth of the top

∞
in L

order derivatives.
This difficulty is already largely avoided in hyperboloidal energy methods, exploit-

ing the anisotropic inclusion of “good” versus “bad” derivatives in the energy (see

144

Proposition ?? and Remark 4.3.15), and is not the cause of the energy growth in our

argument.

2. That the two energy growths are distinct can be seen in the fact that for the classical

applications of the vector field method, the energy growth occurs only for the high-

est order derivatives used in the argument. The more derivatives one uses in the

bootstrap, the more levels of energy that remain bounded. In our case, the energy

growth starts appearing at a fixed (depending on the dimension d) level of deriva-

tives, regardless of how many derivatives is used in the bootstrap argument. The

reason for this is because there are terms in the equation that do not enjoy the boost
symmetries, and every time you differentiate them with a boost, one gets another

growth of u. (see Lemma 4.3.13).

3. Similarly, this energy growth is also different from the µ-degeneracy of the highest

orders of energies (and the associated “descent scheme”) that appears in the study

of formation of shocks [Chr07] (see also discussion in [HKSW16]).

The second difficulty is more sinister. To close the energy estimate, and estimate φu
∞
, we need to commute with at least d/2 derivatives in order to make use of Sobolev,
in L
implying that m > d/2. But then the coefficients on the right hand side are of size (1 +
−d/4+
, which is not integrable when d = 2,3,4. This seemingly prevents us from even
u)
closing any bound for |∂φ|. Take for example the case d = 3.

∞

control on |∂φ| of the type (1+ u + u)λ, the coefficients in the equation
• Assuming L
for LLφ grows like (1 + u)1−λ, This implies that, even assuming the lowest-order
energy remains bounded, the energy for LLφ grows like (1 + u + u)2−λ.

∞
• In the best case, we expect that the growth of the LLφ energy means a weakened L
control on |∂φ|, to the tune of (1 + u + u)2−λ−3/2, with the power (−3/2) coming from
the global Sobolev inequalities.

145

• Thus, we see that at every iteration one would increase the growth rate of |∂φ| by
(1 + u + u)1/2.

To handle this difficulty, we will make use of the hyperboloidal foliation and its as-

sociated sharp global Sobolev inequalities. In particular, the anisotropy discussed in

Remark 4.3.15 allows us to exploit an additional vestige of the null structure of the mem-
−1 decay in the most difficult
brane equation to gain, effectively, an additional (1 + u + u)
terms and close the argument also in d = 3 and 4. This is accomplished by essentially
“borrowing” a weight from the |∂uφ| term when we put it in L2, using the fact that the
term we are trying to control is also a “good derivative” and benefits from the anisotropic

energy. The vestigial null structure is explained in Remark 4.5.3 below.

Remark 4.4.2. This improvement is not sufficient for the d = 2 case, even at the heuris-
−1. As the stability of
planewaves is trivial in d = 1 (using either the integrability of the membrane equation in

tic level, due to logarithmic divergences when integrating (1 + s)

this case, or via an easy modification of the arguments in [Won17b]), we have reasons to

expect that the stability result also holds for d = 2. This turns out to be indeed the case, if

we factor in the additional improvements we used in the more detailed analyses for the

quasilinear problem in Section 4.5. See also Remark 4.5.1.

Note that these difficulties are essentially due to the fact that the background function
Υ (u), while being a solution to the linear wave equation (cid:3)Υ (u) = 0, is not one that is
associated to localized initial data. Hence its derivative with respect to Lα has worse

decay rates. (In fact, it grows in time.)

Concerning this semilinear model, we will study the initial value problem for (4.4.1)
with initial data prescribed on the hypersurface {y0 = 2} (here y is defined as in (4.3.8)),

φ|

y0=2 = φ0, ∂y0φ|

y0=2 = φ1.

The remainder of this section is devoted to proving the following theorem.

146

(cid:48)(cid:48)

(u) is smooth and has compact support in u. Con-

Theorem 4.4.3. Let d ≥ 3 and assume Υ
sider the initial value problem for (4.4.1) where φ0 and φ1 are smooth compactly supported
functions on B(0,1) ⊂ Rd. Let s = d if d is odd, and s = d + 1 if d is even. Then provided
(cid:107)φ0(cid:107)
Hs is sufficiently small, the initial value problem has a global-in-time solu-
tion.

Hs+1 + (cid:107)φ1(cid:107)

4.4.1 Preliminaries

Using the standard local existence theorem with finite-speed of propagation we can as-

sume the solution exists up to at least Σ2. Furthermore, by finite speed of propagation,

the solution must vanish when (cid:118)(cid:117)(cid:116) d(cid:88)

|yi|2 > |y0 − 2| + 1.

In particular, this implies

i=1
√

2(u + u) ≤ τ2 + 1

(4.4.2)

on the support of φ.

By the blow-up criterion for wave equations, it suffices to show (cid:107)φ(cid:107)

W 1,∞(Στ ) < ∞ for
every τ ∈ (2,∞), see [Sog08, Chapter 1, Theorem 4.3]. The general approach, which we
will take also for studying the quasilinear problem, is that of a bootstrap argument.
1. We will assume that, up to time τmax > 2, that the energy Eτ of the solution φ and

its derivatives Lαφ verify certain bounds.

2. Using Proposition 4.3.14, this gives L

∞

bounds on φ, and its derivatives of the form

Lαφ and T Lαφ.

3. We can then estimate the nonlinearity using these L

estimates, which we then
feed back into the energy inequality (4.3.16) to get an updated control on Eτ for all
τ ∈ [2, τmax].

∞

147

4. Finally, show for sufficiently small initial data sizes, the updated control improves

the original control, whereupon by the method of continuity the original bounds on
Eτ must hold for all τ ≥ 2, implying the desired global existence.

Before implementing the bootstrap in the following two sections (one each for the cases

d being odd or even), we record first basic pointwise bounds on the nonlinearity. For

estimating the nonlinearity, we observe that

This allows us to decompose

√
2u
u + u

∂u =

T − 1
u + u
(cid:104)

(cid:48)(cid:48)

Υ

(u)(φu)2 =

1

(u + u)2

√
(

2uT − L1).

(4.4.3)

L1 =

1

u + u

A(u)(L1φ)2 + B(u)L1φ· T φ + C(u)(T φ)2(cid:105)

where A, B, C are all compactly supported smooth functions of u. By Proposition 4.3.8 we

can rewrite LαT φ as

(cid:88)

1

|β|≤|α|

u + u

cβLβφ +

(cid:88)

|γ|≤|α|

(cid:48)
γ T Lγ φ
c

LαT φ =

(cid:48)
γ

∈ W0 and hence are bounded. Additionally on the region τ ≥ 2 that we
where cβ, c
are interested in, u + u is bounded from below. So finally using Lemma 4.3.13 on the
coefficients A, B, C above, we obtain the following uniform pointwise bound on the region
{τ ≥ 2}

(cid:12)(cid:12)(cid:12)(cid:12)Lα(cid:104)

(cid:48)(cid:48)

Υ

(u)(φu)2(cid:105)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46)
(cid:12)(cid:12)(cid:12)(cid:12)(L

(cid:88)
k+(cid:96)1+(cid:96)2≤|α|
≤(cid:96)1+1φ)(L

(1 + u)

k
2

≤(cid:96)2+1φ) + (T L

−2 · 1{u∈supp Υ (cid:48)(cid:48)}·
≤(cid:96)1φ)(T L

(cid:12)(cid:12)(cid:12)(cid:12) .

≤(cid:96)2φ) + (L

≤(cid:96)1+1φ)(T L

≤(cid:96)2φ)

(4.4.4)
≤(cid:96)φ terms of the form Lβφ with β an m-

Notation 4.4.4. Here we denote schematically by L
tuple with m ≤ (cid:96).

148

By Proposition 4.3.14 we can replace the term with the smallest of (cid:96)1, (cid:96)2 using an

(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) τ1− d
(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) τ1− d
(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) τ
(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) τ

2

2 cosh(ρ)1− d
2 cosh(ρ)1− d
2 cosh(ρ)1− d
− d
− d
2 cosh(ρ)1− d

2Eτ[L
2Eτ[L
2Eτ[L
2Eτ[L

≤(cid:96)1+(cid:98) d
≤(cid:96)1+(cid:98) d
≤(cid:96)1+(cid:98) d
≤(cid:96)1+(cid:98) d

2

2

2

(cid:99)+1φ]·(cid:12)(cid:12)(cid:12)(cid:12)L
(cid:99)+1φ]·(cid:12)(cid:12)(cid:12)(cid:12)T L
(cid:99)+1φ]·(cid:12)(cid:12)(cid:12)(cid:12)L
(cid:99)+1φ]·(cid:12)(cid:12)(cid:12)(cid:12)T L

≤(cid:96)2+1φ
≤(cid:96)2φ
≤(cid:96)2+1φ
≤(cid:96)2φ

(cid:12)(cid:12)(cid:12)(cid:12) ,
(cid:12)(cid:12)(cid:12)(cid:12) ,
(cid:12)(cid:12)(cid:12)(cid:12) ,
(cid:12)(cid:12)(cid:12)(cid:12) .

energy integral:

≤(cid:96)1+1φ)(L
≤(cid:96)1+1φ)(T L
≤(cid:96)1φ)(L
≤(cid:96)1φ)(T L

(cid:12)(cid:12)(cid:12)(cid:12)(L
(cid:12)(cid:12)(cid:12)(cid:12)(L
(cid:12)(cid:12)(cid:12)(cid:12)(T L
(cid:12)(cid:12)(cid:12)(cid:12)(T L
(u)(φu)2(cid:105)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46)
( 1 + u(cid:124)(cid:123)(cid:122)(cid:125)
)2− d
≈τ cosh(ρ)

≤(cid:96)2+1φ)
≤(cid:96)2φ)
≤(cid:96)2+1φ)
≤(cid:96)2φ)
(cid:88)
k+(cid:96)1+(cid:96)2≤|α|
(cid:96)1≤(cid:96)2
2 ·Eτ[L

This allows us to condense (4.4.4) as

(cid:12)(cid:12)(cid:12)(cid:12)Lα(cid:104)

(cid:48)(cid:48)

Υ

(1 + u)

k
2

≤(cid:96)1+(cid:98) d

2

(cid:99)+1φ]

−2 · 1{u∈supp Υ (cid:48)(cid:48)}·
(cid:34)

1

τ cosh(ρ)

|L

≤(cid:96)2+1φ| +

(cid:35)

.

(4.4.5)

1

cosh(ρ)

|T L

≤(cid:96)2φ|

(u +u) ≈ (1+u) using the support properties of Υ

Here we used that τ cosh(ρ) = 1√
. Next
2
we note that 2uu ≥ τ2 in I +. On the support of Υ
this means u (cid:38) τ2. On the other hand,
from (4.4.2) we also get u (cid:46) 1 + τ2. This allows us to replace (1 + u) by (1 + τ)2 in (4.4.5).
Observe next that since Li is Killing with respect to m, we have that firstly LT m = 0

(cid:48)(cid:48)

(cid:48)(cid:48)

and secondly

(u)(φu)2(cid:105)(cid:12)(cid:12)(cid:12)(cid:12).

(cid:48)(cid:48)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:3)Lαφ

(cid:12)(cid:12)(cid:12)(cid:12) ≤(cid:12)(cid:12)(cid:12)(cid:12)Lα(cid:104)
†

Υ

So from the energy identity (4.3.16) we get

Eτ1[Lαφ]2 −Eτ0[Lαφ]2 (cid:46)

|(cid:3)Lαφ|·|T Lαφ| dvolm .

τ∈[τ0,τ1]

Applying (4.4.5) we finally arrive at our fundamental a priori estimate

Eτ1[Lαφ]2 −Eτ0[Lαφ]2 (cid:46)
τ1(cid:90)

(cid:88)
k+(cid:96)1+(cid:96)2≤|α|
(cid:96)1≤(cid:96)2

τ0

(1 + τ)k−d ·Eτ[Lαφ]·Eτ[L

≤(cid:96)2φ]·Eτ[L

≤(cid:96)1+(cid:98) d

2

(cid:99)+1φ] dτ.

149

Ek(τ) = sup
σ∈[2,τ]

≤kφ].

(4.4.6)

To simplify notation, let us write

Our a priori estimate reads

Ek(τ)2 − Ek(2)2 (cid:46)

(cid:88)
(cid:96)1≤(cid:96)2

(cid:96)0+(cid:96)1+(cid:96)2=k

2

Eσ [L
τ(cid:90)

s(cid:96)0−dEkE

(cid:96)1+(cid:98) d

2

(cid:99)+1

E(cid:96)2ds.

(4.4.7)

In the remainder of this section we will discuss the bootstrap scheme that allows us to
control Ek, for all k ≤ d + 1 when d is even and all k ≤ d when d is odd, for all time τ ≥ 2.
Note that the implicit constant in (4.4.7) depends only on the dimension d, the order
k of differentiation, and properties of the background function Υ , and is in particular

independent of φ.

4.4.2 Bootstrap for d ≥ 6 even
When d ≥ 6 is even, we will denote by m the value d/2. Note that m ≥ 3. We will assume
a uniform bound on the initial data

Ek(2) ≤ 
,

k ≤ d + 1.

(4.4.8)

Our bootstrap assumption is that for some δ > 
 and for all 2 ≤ τ ≤ ˜τ,

k ≤ d − 2

δτk−(d−1) ln(τ) d − 1 ≤ k ≤ d + 1

.

(4.4.9)

We note that under (4.4.7) this system is closed: if (cid:96)0 + (cid:96)1 + (cid:96)2 ≤ d + 1 and (cid:96)1 ≤ (cid:96)2, then
(cid:96)1 ≤ m. This means that (cid:96)1 +(cid:98)d/2(cid:99) + 1 ≤ 2m + 1 = d + 1. Our goal is to show that the boot-
strap assumptions (4.4.9) can be used to prove improved versions of themselves, under a

smallness assumption on δ and 
.

Under our bootstrap assumptions, we can expression every term of the form

s(cid:96)0−dEk(s)E(cid:96)1+m+1(s)E(cid:96)2(s) = w(cid:96)0,(cid:96)1,(cid:96)2(s)δ3,

150

δ

Ek(τ) ≤

noting that (cid:96)1 ≤ (cid:96)2 by assumption and (cid:96)0 + (cid:96)1 + (cid:96)2 = k ≤ d + 1. Observing that at most one
of (cid:96)0, (cid:96)1 + m + 1, and (cid:96)2 can be ≥ d under these conditions, we tabulate upper bounds for
the weight functions w(cid:96)0,(cid:96)1,(cid:96)2(s) in Table 4.1. From this table, we see immediately that
(d ≥ 6, even) List of admissible (cid:96)0, (cid:96)1, (cid:96)2 values as well as the correspond-
Table 4.1:
ing upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2. The value of “—” means any value compatible with the
prescribed columns. The shaded rows are those with non-integrable upper bounds for
w(cid:96)0,(cid:96)1,(cid:96)2.

s

s

s

s
s

w(cid:96)0,(cid:96)1,(cid:96)2(s) ≤ Comment
−2
=⇒ (cid:96)0 ≤ k.
s2−d ln(s)
=⇒ (cid:96)0 ≤ 2.
−d ln(s)
=⇒ (cid:96)0 = 0.
−2 ln(s)
s3−d ln(s)2
=⇒ (cid:96)2 ≤ m + 1, (cid:96)0 ≤ 3
s1−d ln(s)2
=⇒ (cid:96)0 ≤ 1
−d ln(s)2
−1 ln(s)
−1 ln(s)
s4−d ln(s)3
s2−d ln(s)2
s2−d ln(s)2
s ln(s)
ln(s)
s5−d ln(s)3
s3−d ln(s)3
s4−d ln(s)2
s3 ln(s)

=⇒ (cid:96)2 ≤ m + 2, (cid:96)0 ≤ 4
=⇒ (cid:96)0 ≤ 2

=⇒ (cid:96)2 ≤ m + 3, (cid:96)0 ≤ 5
=⇒ (cid:96)2 ≤ m + 2, (cid:96)0 ≤ 3

s

k

≤ d − 2
≤ d − 2
≤ d − 2
d − 1
d − 1
d − 1
d − 1
d − 1
d
d
d
d
d

d + 1
d + 1
d + 1
d + 1

d

(cid:96)1
< m− 2
m− 2
m− 1
m− 2
m− 1
—
—
m− 2
m, m− 1

(cid:96)0
(cid:96)2
—
—
—
—
—
—
≤ d − 2 ≤ m− 3 ≤ d − 2
—
—
—
—
d − 1
—
d − 1
—
≤ d − 2 ≤ m− 3 ≤ d − 2
—
—
—
—
d, d − 1
—
d, d − 1
—
≤ d − 2 ≤ m− 3 ≤ d − 2
—
—
—
—
d, d ± 1
—
d, d ± 1
—
(cid:90)

—
—
m− 2
m, m− 1

—
—

Ek(τ)2 − Ek(2)2 (cid:46) δ3 whenever k ≤ d − 2. Furthermore, using that for p > −1
(p + 1)2 sp+1 (cid:46) sp+1 ln(s)2,

sp+1 ln(s)−

sp ln(s) ds =

p + 1

1

1

and

(cid:90)
s
we conclude that for k = d − 1, d, d + 1

−1 ln(s) ds =

ln(s)2,

1
2

Ek(τ)2 − Ek(2)2 (cid:46) δ3τ2(k−(d−1)) ln(τ)2.

151

Thus for δ sufficiently small (depending on the implicit constants in the inequalities

above), we have that our bootstrap assumptions (4.4.9) together with initial data assump-

tions implies

Ek(τ) ≤

k ≤ d − 2

2δ2
2δ2τ2(k−(d−1)) ln(τ)2 d − 1 ≤ k ≤ d + 1

.

(4.4.10)

(cid:113)
(cid:113)




2 + 1


2 + 1

(cid:113)
(cid:113)



By choosing 
 sufficiently small relative to δ, we can guarantee

Ek(τ) ≤

k ≤ d − 2

3
4δ
4δτk−(d−1) ln(τ) d − 1 ≤ k ≤ d + 1
3

,

(4.4.11)

thereby closing the bootstrap and proving global existence.

4.4.3 Bootstrap for d ≥ 5 odd
When d ≥ 5 odd, we will take our bootstrap assumption to be
k ≤ d − 2
k = d − 1, d

δτk−(d−1) ln(τ)

Ek(τ) ≤

δ

.

(4.4.12)

That we can close with one fewer derivative is due to (cid:98)d/2(cid:99) < d/2 in this case. Define m =
(cid:98)d/2(cid:99) for convenience; note that 2m = d − 1. By our assumption then (cid:96)1 ≤ (cid:96)2 =⇒ (cid:96)1 ≤ m,
and hence (cid:96)1 + m + 1 ≤ d, allowing the system to close.

The bootstrap argument here is largely similar to the case d ≥ 6 even. In Table 4.2 we
record upper bounds for the weight functions w(cid:96)0,(cid:96)1,(cid:96)2(s), and omit the straightforward
remainder of arguments.

4.4.4 Bootstrap for d = 4

We will assume a uniform bound on the initial data
k ≤ 5,

Ek(2) ≤ 
,

152

(4.4.13)

(d ≥ 5, odd) List of admissible (cid:96)0, (cid:96)1, (cid:96)2 values as well as the correspond-
Table 4.2:
ing upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2. The value of “—” means any value compatible with the
prescribed columns. The shaded rows are those with non-integrable upper bounds for
w(cid:96)0,(cid:96)1,(cid:96)2.

k

≤ d − 2
≤ d − 2
d − 1
d − 1
d − 1
d − 1
d
d
d
d

(cid:96)1
≤ m− 2
m− 1
m− 1
—
—

(cid:96)0
(cid:96)2
—
—
—
—
≤ d − 2 ≤ m− 2 ≤ d − 2
—
—
d − 1
—
d − 1
—
≤ d − 2 ≤ m− 2 ≤ d − 2
—
—
d, d − 1
—
d, d − 1
—

m, m− 1

—
—

s

−2

w(cid:96)0,(cid:96)1,(cid:96)2(s) ≤ Comment
=⇒ (cid:96)0 ≤ k.
s1−d ln(s)
=⇒ (cid:96)0 ≤ 1.
−2 ln(s)
s2−d ln(s)2
=⇒ (cid:96)0 ≤ 2
−d ln(s)2
−1 ln(s)
−1 ln(s)
s4−d ln(s)3
s2−d ln(s)2
s ln(s)

=⇒ (cid:96)0 ≤ 3

s

s
s

s

with an additional bootstrap assumption for some δ > 

k ≤ 2
3 ≤ k ≤ 5

δτk−3+γ

Ek(τ) ≤

δ

.

(4.4.14)

The number γ is assumed to be (cid:28) 1 and arbitrary; in particular we will throughout take
γ < 1
3. The smallness of γ will impact the smallness of the initial data allowed: the
smaller the γ the smaller the initial data needs to be. We consider γ as fixed once and for

all.

We argue similarly to the case when d ≥ 6, and record in Table 4.3 the corresponding
weight functions w(cid:96)0,(cid:96)1,(cid:96)2. Note, however, an additional complication arises since d/2 +
1 = 3 = d − 1 in this setting (which is why instead of a logarithmic growth of energy Ed−1,
we see a small polynomial growth).

Based on the weights derived in the table, we see clearly that, by (4.4.7) we have

Ek(τ)2 − Ek(2)2 (cid:46)

k ≤ 2

k = 3,4,5

δ3τ2γ+2k−6

δ3

153

Table 4.3:
(d = 4) List of admissible (cid:96)0, (cid:96)1, (cid:96)2 values as well as the corresponding upper
bounds for w(cid:96)0,(cid:96)1,(cid:96)2. The value of “—” means any value compatible with the prescribed
columns. The shaded rows are those with non-integrable upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2.
(Recall that 3γ < 1 by fiat.)

(cid:96)0

(cid:96)1

0
3
4

0
0
0

0
≤ 2

k
(cid:96)2
≤ 2 — 0 —
2
1
≤ 2
3
3 — 1 —
3
3
0
3
4
0
4 — — 1
4 — — 2
4 — — 3, 4
5
0
5 — — 1
5 — — 2
5 — — 3,4,5

1
0

5

0

w(cid:96)0,(cid:96)1,(cid:96)2(s) ≤

sγ−2
sγ−3
s2γ−2
s2γ−2
s3γ−4
s2γ−1
s2γ+1
s2γ
s2γ−1
s3γ−2
s2γ+3
s2γ+2
s2γ+1
s3γ

and hence taking δ sufficiently small and 
 even sufficiently smaller will allow us to close

the bootstrap and obtain global existence.

Remark 4.4.5. Applying Proposition 4.3.14 we see that the corresponding solution has

the following decay rates:

−1,
|φ| (cid:46) (y0)
|Liφ| (cid:46) (y0)γ−1,
|T φ| (cid:46) (y0)γ−1τ

|LiLjφ| (cid:46) (y0)γ ,
−1.
|T Liφ| (cid:46) (y0)γ τ
−3/2 is due to our not
The difference between the decay rate for |φ| and the expected (y0)
using the Morawetz K multiplier (see [Won17c]) and purely technical. The remaining

−1,

modifications are due to the equation. We see at the first derivative level the decay rates
shown are modified from the standard linear rate by (y0)γ, while at the second derivative

154

level the decay rates are worse by a factor of (y0)γ+1. (For linear waves in d = 4, |LiLjφ|
(cid:48)(cid:48)
−1.) This worsened decay is a consequence of the background Υ
should decay like (y0)

that appears in the equation.

Remark 4.4.6. Notice that we do not make use of fractional Sobolev spaces. In the in-
teger setting, to close the L2–L

Sobolev estimate, in 4 dimensions we need to take 3

∞

expect the equation satisfied by L

≤3φ to have a right hand side growing like (1 + u)

derivatives. Returning to the schematics described in the introduction of this section, we
−1/2.
Our bootstrap assumptions (as well as was shown in Table 4.3) indicate, on the other
−1+
 (remember
hand, that the inhomogeneity can take a coefficient growing like (1 + u)
3 is fixed and arbitrary). This gain of effectively a power of 1/2 is due to our use
that γ < 1
the derivative

of an anisotropic energy (see Remark 4.3.15) and that on the support of Υ

(cid:48)(cid:48)

∂u is well-approximated by a “tangential derivative”.

4.4.5 Bootstrap for d = 3

We close this section by recording the bootstrap argument for d = 3. Here the bootstrap

assumptions will be taken to be

Ek(τ) ≤

δτk−2+γ

k = 0,1

k = 2,3

.

(4.4.15)

Here again γ (cid:28) 1 is fixed to be < 1
similarly to the case d = 4 we see that the bootstrap assumptions imply

3. The weight bounds are shown in Table 4.4. Arguing

δ

δ3

Ek(τ)2 − Ek(2)2 (cid:46)

k ≤ 1

k = 2,3

δ3τ2γ+2k−4

and hence for sufficiently small δ and 
, the bootstrap argument closes and we have global

existence.

155

Table 4.4:
(d = 3) List of admissible (cid:96)0, (cid:96)1, (cid:96)2 values as well as the corresponding upper
bounds for w(cid:96)0,(cid:96)1,(cid:96)2. The value of “—” means any value compatible with the prescribed
columns. The shaded rows are those with non-integrable upper bounds for w(cid:96)0,(cid:96)1,(cid:96)2.
(Recall that 3γ < 1 by fiat.)

(cid:96)2 w(cid:96)0,(cid:96)1,(cid:96)2(s) ≤

(cid:96)1

2

(cid:96)0

k
≤ 1 — — —
2
0
2 — — 1
2
2
3
0
3 — — 1
3 — — 2,3

0
3

0

0
0

sγ−2
s2γ−1
s2γ−2
s3γ−3
s2γ+1
s2γ
s3γ−1

∞
For convenience we record here the corresponding L

decay rates relative to the y

coordinates. These can be obtained by applying Proposition 4.3.14 to the bootstrap as-

sumptions above.

−1/2,
|φ| (cid:46) (y0)
|Liφ| (cid:46) (y0)γ−1/2,
|T φ| (cid:46) (y0)γ−1/2τ

−1,

|LiLjφ| (cid:46) (y0)γ+1/2,
|T Liφ| (cid:46) (y0)γ+1/2τ

−1.

Remark 4.4.7. An examination of Tables 4.1, 4.2, 4.3, and 4.4 shows that, exactly as

discussed in the introduction to this section, the nonlinear terms that cause the main
difficulty are those where the commutator vector fields hit entirely on the background

planewave Υ

. This shows that even if we start by considering initial data with higher

(cid:48)(cid:48)

degree of regularity, the loss of decay will always appear in the energy Ek starting from
k = d − 1.

Remark 4.4.8. In the arguments given above, when d is odd we only commuted with up

to d vector fields, and when d is even we used d+1 vector fields. It is fairly straightforward

to check, in fact, that for initial data of higher regularity, the higher regularity is preserved

in the solution. However, for each additional derivative the energy growth speeds up by

156

another factor of τ. So for example, in dimension d = 3 the higher energy E11(τ) will have
controlled growth like τ9+γ in our bootstrap scheme.

4.5 Commuted equations

We now return to the membrane equation. As discussed in Section 4.2.2, to handle the

quasilinearity it is convenient to consider not just (4.2.17) but also the prolonged system

(4.2.18) for its first derivatives. As seen in Section 4.4 previously, we will prefer to work
with the weighted vector field derivatives Liφ instead of the coordinate partials ∂λφ. In
this section we will first write down the corresponding propagation equations for Liφ.

While the arguments in Section 4.4 sums up neatly our approach toward the semi-

linear inhomogeneity in the equation, the quasilinear nature of (4.2.17) introduces addi-
tional complications. Whereas in the semilinear case the commutation relations [Li,(cid:3)m] =
0 hold, in the quasilinear case [Li,(cid:3)g] are generally non-vanishing second order differen-
tial operators, whose coefficients depend on the unknown φ itself. In the second part of

this section we perform these basic commutation computations and systematically record

the additional terms that arise which would also need to be controlled.

In the final part of this section, we give a statement of our main stability theorem

for simple planewave solutions to the membrane equation. We will state and prove our

theorem in the most critical case d = 3. Returning to the results of Section 4.4, we see
that when d ≥ 5 the solution φ to the semilinear equation is such that φ and its first order
weighted derivatives Liφ, T φ all enjoy pointwise decay at rates identical to the solution

to the linear wave equation. For the corresponding quasilinear problem the dynamical

metric g also has fast decay toward m, and the quasilinearity poses almost no additional

complications compared to the semilinear case.

As already discussed in the introduction to Section 4.4, in lower spatial dimensions
even the semilinearity causes additional difficulties compared to d ≥ 5; this requires,
in particular, that the decay rates of even the lowest order derivatives Liφ and T φ be

157

modified from their expected linear rates. In the quasilinear setting, this causes additional

complications. In three dimensions, in particular, the appearance of terms of the form

(cid:48)(cid:48)

Υ

(u)φ∂2

uuφ

in (4.2.18) is potentially troublesome. Based on purely the linear peeling estimates, which

follows from applying Proposition 4.3.14 to a solution of the linear wave equation, and

which would give (on the support of Υ

(cid:48)(cid:48)

)

|φ| (cid:46) (y0)

|∂uφ| (cid:46) (y0)

−3/2,

|∂2
uuφ| (cid:46) (y0)

−5/2,

−1/2,
(cid:48)(cid:48)

one may naively expect that Υ

uuφ has similar decay properties as the semilinear
(u)(φu)2 that we already treated. However, if we instead examine the

(cid:48)(cid:48)

(u)φ∂2

nonlinearity Υ

decay rates proven in Section 4.4 (which we should not expect to be better), we have

−1/2,
|φ| (cid:46) (y0)
(cid:48)(cid:48)

|∂uφ| (cid:46) (y0)

−3/2+γ ,

|∂2
uuφ| (cid:46) (y0)

−3/2+γ ,

making the decay for Υ

(u)φ∂2

uuφ slower by a factor of y0 compared to the semilinear

term.

This potential difficulty is significantly ameliorated in d ≥ 5; doing a similar anal-
ysis using the proven decay rates in Section 4.4 shows that the difference between the

(cid:48)(cid:48)

(u)φ∂2

quasilinear Υ
uuφ term and its semilinear counterpart, when d = 5,6 is merely a
factor of ln(τ) which does not impact the bootstrap argument; and when d ≥ 7 no differ-
ence is present. Hence, both for brevity of presentation and clarity of argument, we shall
concentrate the remainder of this chapter on the most difficult case d = 3. The higher
dimensional cases can all be handled similarly; with the difference being mainly one of

bookkeeping.

The overcoming of this potential difficulty with the Υ

uuφ terms in dimension
d = 3 relies, unsurprisingly, on the “null structure” of the equation. In Section 4.4 for
brevity of argument the derivatives L1φ and Liφ for i = 2, . . . , d are estimated isotropically.

φ∂2

(cid:48)(cid:48)

However, the equations that they satisfy are not the same: recalling that the worst term

158

of the inhomogeneity arises from when the weighted vector fields hit the background Υ

(cid:48)(cid:48)

,

we expect

(cid:3)L1φ ≈ (L1Υ

(cid:48)(cid:48)

)(φu)2

in the semilinear argument. However, a direct computation shows that

(cid:48)(cid:48)

L1Υ

= uΥ

(cid:48)(cid:48)(cid:48)

is again a smooth function with compact support in u. In particular, while for i = 2, . . . , d
we have the growing weights as described in Lemma 4.3.13, this loss is not seen by pure L1
derivatives. Therefore we expect L1φ to actually enjoy better decay compared to Liφ for
i (cid:44) 1. Finally, returning to the difficult term φuu, we see that the ∂u derivative lies in the
span of T and L1 (see also (4.5.8) and Remark 4.5.3); hence we will expect that ∂2
uuφ to
decay faster than the generic tangential second derivative, allowing us to eventually close

our estimates.

Remark 4.5.1. In d = 2 this observation is in fact enough to allow us to close the energy
estimate for the semilinear model. However, additional difficulties come up in the analysis

of the full quasilinear problem that cannot be treated using only this method, hence we

omit its discussion below. For the semilinear problem (4.4.1), let us denote by Ek the
k-th order energies for φ, and Fk the k-th order energies for L1φ (analogously to how we
proceed in Section 4.6 below for the quasilinear problem in d = 3). This way of treating
the equations for φ and L1φ separately allows us to close the global-existence bootstrap

in a manner similar to that described in Section 4.4 with the energy bounds

E0, F0 (cid:46) δ,
E1, F1 (cid:46) δτγ ,
E2, F2 (cid:46) δτ1+γ ,
E3 (cid:46) δτ2+γ .

159

For the quasilinear problem, this scheme breaks down when dealing with the T T φ deriva-

tives that crop up.

In dimension d ≥ 3, the cubic and higher nonlinearities are essentially harmless, even
combine

∞

with the slightly reduced decay rates. (In the linear case the terms placed in L
to decay at least as fast (y0)

3 can be easily absorbed.) This fact allows us
to essentially ignore all “null structure” when handling the cubic and higher order terms,

−3/2; a loss of γ < 1

which allows us to significantly simplify the bookkeeping involved.

4.5.1 The perturbed system, restated

Our goal this section is to derive the evolution equations for Liφ. Some of the computa-

tions are lengthy and not entirely transparent: they are recorded in Appendix A.3.1. We

start with (4.2.17) which we re-write as
◦

(cid:112)|g|∂µ

gµν∂νφ(cid:112)|g| = Υ

We expand the left hand side as

(cid:16)

(cid:17)

(cid:112)|g|◦

(cid:48)(cid:48)

∂uφ

+

(cid:3)mφ + 2∂u

φΥ

g(dφ,d|g|−1/2)

(cid:48)(cid:48)

(φu)2.

(cid:16)

(cid:48)(cid:48)

φΥ

∂uφ

(cid:17)− 1

2|g|

(cid:16)

◦
g

dφ,d

(cid:16)◦

g(dφ,dφ)

(cid:17)(cid:17)

.

Notice, on the other hand, that

(cid:16)

(cid:17)− 1|g|

(cid:16)

◦
g

(cid:48)(cid:48)

∂uψ

(cid:3)gψ = (cid:3)mψ + 2∂u

φΥ

g(dφ,dψ) +

◦
g(d|g|,dψ).

1
2|g|

(4.5.1)

= (cid:3)mφ + 2∂u

dφ,d

g(dφ,dψ)

(cid:16)◦
− 1|g|Υ

(cid:17)(cid:17)
(φu)2 · ◦

(cid:48)(cid:48)

160

(cid:3)gXφ = X

g(dφ,d(Xφ)) +

◦
g(d|g|,d(Xφ))

1|g|

Together this implies that, if X is a Killing vector field of the Minkowski metric m, that

(cid:16)

Υ

(cid:48)(cid:48)

(cid:48)(cid:48)

(φu)2(cid:17) − 1|g|Υ
− 2[X, ∂u](φΥ
− 1
2|g|2 X(|g|)

(φu)2 ◦
φu) − 2∂u(X(φΥ
◦
g(dφ,d|g|) +

(cid:48)(cid:48)

(cid:48)(cid:48)
(cid:48)(cid:48)
)φu) − 2∂u(φΥ
◦
−1)(dφ,d|g|)
2|g|LX(
1
(cid:16)
(cid:17)(cid:17)
g
◦
◦
−1)(dφ,dφ)
1
2|g|
g
g
◦
−1 is the Lie derivative of the inverse metric
g

dφ,d
(4.5.2)
◦
−1 by the vector field X. It can be
g

(cid:16)LX(

[X, ∂u]φ)

+

.

Here, LX
given as

LX

◦
−1 = 2X(φΥ
g

(cid:48)(cid:48)

)∂u ⊗ ∂u + 2φΥ

(cid:48)(cid:48)

[X, ∂u]⊗ ∂u + 2φΥ

(cid:48)(cid:48)

∂u ⊗ [X, ∂u].

(4.5.3)

The boxed terms in (4.5.2) are those with quadratic nonlinearity and are the ones for which

the null structure play an important role. The remaining terms on the right hand side all

have cubic or higher nonlinearities, and will be treated more roughly in the estimates.

Later on we will take X to be one of Li; we can compute the commutators (see (4.3.6)

and (4.3.7) for definitions)

[L1, ∂u] = −∂u;
[Li, ∂u] = − 1√
2

1

y0 (Li − yiT ),

i ∈ {2, . . . , d}.

(4.5.4)

(4.5.5)

For convenience, we will introduce the following schematic notations.

(cid:18)

Notation 4.5.2. First, in view of Lemma 4.3.13, we will denote by Pm any finite sum of
terms of the form

Polynomial in {u, ˆx}(cid:19)·(cid:18)

(cid:19)
Compactly supported smooth function of u
such that on I + it is bounded by (1 + u)m/2. Our assumptions imply Υ
computations surrounding the proof of Lemma 4.3.13 imply that

(4.5.6)

= P0. The

(cid:48)(cid:48)

T Pm = Pm ,

L1Pm = Pm ,

LiPm = Pm+1 for i ∈ {2, . . . , d}.

(4.5.7)

We will denote by Wm any element of Wm.

161

With these notations, we can rewrite schematically

Pm∂u = W1Pm

L1 + T

(cid:16)

(cid:17)

.

(4.5.8)

(cid:48)(cid:48)

Remark 4.5.3 (Vestige of null condition). As discussed in Remark 4.2.3, the presence of
(φu)2 helps to ameliorate the resonant interaction. This improvement
is a vestige of the null condition of the original membrane equation. In our reformulation

factor in Υ

the Υ

(cid:48)(cid:48)

here, this improvement is captured in (4.5.8) above. Observe that a generic coordinate

derivative ∂u, ∂u, or ∂ ˆx can be written only as an element of the commutator algebra A1,
which means that the transversal factor T is not accompanied by a decaying weight. From
this one can see that quadratic terms of the form (T φ)2 will serve as a severe obstacle to
global existence. In our setting, however, the P0 weight Υ
provides a spatial localization
and gives an anomalous weighting: the term W1T ∈ A2 and has improved decay and this
improvement is, fundamentally, what allows our argument to close in this chapter.

(cid:48)(cid:48)

Notation 4.5.4. We will frequently denote by B k,s
w an element of Bk,s
w
|g| appears in a higher order term, if is often sufficient to control it as

(cid:48) with w

(cid:48) ≥ w. When

|g| = 1 + (B 1,1

1 φ)2(1 + P0φ),

and similarly we can write
◦
g(dϕ,dψ) = (B 1,1

1 ϕ)(B 1,1

1 ψ)(1 + P0φ).

(4.5.9)

(4.5.10)

Remark 4.5.5. Observe that in (4.5.2), the inhomogeneity depends on up to second order

derivatives of φ. If we decompose nonlinearities, the second order derivatives that appear

are generic, in the sense that derivatives with principal parts T T φ, T Liφ, and LiLjφ all
∞
appear. (Note that {T , Li} span the tangent space R1,d.) To control T Liφ and LiLjφ in L
,
by Proposition 4.3.14 it suffices to control the energies of Lαφ. The term T T φ, however,

is not controlled by these energies. There are two approaches to address this. First is

to enlarge the set of commutators required; instead of only commuting with the boosts

162

Lα, one can commute with also the T vector field. Checking the commutator relations, to
close this argument one would have to commute with all differential operators of the form
LαT k where |α| + k is bounded by some k0. For our problem, it appears slightly simpler
computationally to take the second (essentially equivalent) alternative. By decomposing
(cid:3)g we can solve (4.2.17) for T T φ in terms of T Liφ and LiLjφ and lower order derivatives.
This implies T T Lβφ can be estimated in terms of T Lγ φ and Lαφ where |γ| ≤ |β| + 1 and
|α| ≤ |β| + 2. See Appendix A.3.1 for the details of this computation.

Notation 4.5.6. We will denote by G = G (φP0, B 1,0
1 φ) an arbitrary smooth func-
tion of its arguments. In particular, |g| = G in this notation, as well as |g|−1 = G when the
φ, B 1,0

1 φ are all sufficiently small.

0 φ, and B 1,1

0 φ, B 1,1

It is convenient to simplify (4.5.2) a bit more.
With the aid of these schematic notations, we find that L1φ satisfies

(cid:3)gL1φ = P0W2 ·(cid:20)

(L1φ + T φ)2 + T φ(L1L1φ + T L1φ)

+ φ(L1φ + T φ) + (φ + L1φ)(L1L1φ + T L1φ + T T φ)

(cid:21)

+ G (B 1,1

1 φ)(B 1,1
1 φ)3(B 2,1

1 L1φ)(B 2,2
1 φ) + G P0(B 1,1

+ G P0(B 1,1

2 φ) + G P0(B 1,0

0 φ)(B 1,1

1 φ)2(B 2,2

1 φ)5(B 2,2

2 φ) + G P0(B 1,0

2 φ)
0 φ)(B 1,1

1 φ)4

+ G φW1P1(B 1,1

1 φ)2(B 2,1
+ G φW1P1(B 1,1

1 φ) + G φ2W2P2(B 1,1
1 φ)5(B 2,2

2 φ) + G W1P1(B 1,0

1 φ)
0 φ)(B 1,1

1 φ)4(B 2,1

1 φ)3 (4.5.11)

The first brackets capturing all the quadratic nonlinearities and the cubic and higher non-

163

linearities are described schematically after. For i (cid:44) 1, the term Liφ satisfies the equation

(cid:3)gLiφ = P0W1(φ + L1φ + T φ)(B 1,1

1 L1φ + B 2,2

2 φ + B 1,1

1 φ)

+ W2(P0Liφ + P1φ)(L1L1φ + T L1φ + T T φ + L1φ + T φ)

+ P1W2(φ + L1φ + T φ)(L1φ + T φ)
1 φ)(B 2,2
0 φ)(B 1,1
1 φ) + G P0(B 1,1
2 φ) + G P0(B 1,0

2 φ) + G P0(B 1,0
1 φ)5(B 2,2

1 φ)2(B 2,2

2 φ)

0 φ)(B 1,1

1 φ)4

+ G (B 1,1

1 φ)(B 2,1
1 φ)3(B 2,1

+ G P0(B 1,1

+ G φW1P1(B 1,1
+ G φW1P1(B 1,1
+ G φP1(B 1,1

1 φ)2(B 2,1
1 φ)5(B 2,2

1 φ) + G φ2W2P2(B 1,1
2 φ) + G W1P1(B 1,0

1 φ)4(B 2,1
1 φ)
0 φ)(B 1,1
1 φ)3

1 φ)2(B 2,2

2 φ) + G φW1P2(B 1,1

1 φ)3 + G P1(B 1,1

1 φ)4.

(4.5.12)

Note that the cubic and higher-order terms are schematically represented largely in the
same way, with the main differences coming in the quadratic terms. The key observation,

as already mentioned in the introduction to this section, is that the quadratic terms in
the equation for L1φ do not see the growing weight term, and therefore behaves like φ

instead of a generic Lφ term. This improvement then also propagates into the analysis of

the quadratic terms of equation (4.5.12) of the general L derivatives.

For convenience, we record (4.2.17) here in the schematic notation.

(cid:3)gφ = G P0W2(L1φ + T φ)2.

(4.5.13)

4.5.2 Commutator relations

To use the vector field method, we will be commuting our equations with the Li deriva-
tives. More precisely, we study the wave equations satisfied by B k,0
0 (L1φ, Liφ) by writing

(cid:3)g(B k,0

0 Lφ) = B k,0

0 ((cid:3)gLφ) + [B k,0

0 ,(cid:3)g](Lφ).

Note that after applying (4.5.11) and (4.5.12) the right-side does not contain principal
terms. Differentiation of the schematic relations in (4.5.11), (4.5.12), and (4.5.13) are

164

straightforward. To implement our strategy, we need to compute the commutators [X,(cid:3)g]
acting on a smooth scalar ψ, where X = L1 or Li. We merely record the results here, and

defer the actual computation to A.3.2.

[X,(cid:3)g]ψ = P0W1(B 1,1

+ P0W1(B 1,0

0 φ)(B 1,1

1 φ)(L1ψ + T ψ) + P0W1(φ + L1φ + T φ)(B 1,1
1 ψ)
0 L1φ + B 1,0
0 T φ)(L1ψ + T ψ)

1 T ψ) + P0W2(B 1,0

1 L1ψ + B 1,1

+ P1W2(φ + L1φ + T φ)(L1ψ + T ψ) + P1W2φ(L1L1ψ + T L1ψ + T T ψ)

+ (XG )·(cid:104)
+ G ·(cid:104)

+ P0(B 1,1

+ (B 1,1
+ P1(B 1,1
+ P1W1(B 1,0

(B 1,1
1 φ)(B 2,2
1 φ)3(B 1,1

2 φ)(B 1,1
1 ψ) + P1W1φ(B 1,1

1 ψ) + (B 1,1

1 φ)(B 2,2

(B 2,1
1 φ)(B 2,1

1 φ)(B 2,2

2 φ)(B 1,1

1 ψ) + (B 1,1
2 ψ) + P0(B 1,1
1 ψ) + P1W1φ(B 1,1

1 φ)(B 3,2
1 φ)2(B 2,1
1 φ)(B 2,1

1 φ)3(B 1,1

(cid:105)

1 φ)2(B 1,1

2 ψ)

1 φ)2(B 2,2
1 ψ)
2 φ)(B 1,1
1 φ)(B 1,1
1 φ)(B 1,1

1 ψ)
1 ψ)
1 ψ)

(cid:105)

0 φ)(B 1,1

1 φ)2(B 1,1

1 ψ) + P2W1φ(B 1,1

1 φ)2(B 1,1

1 φ)

.

(4.5.14)

Notice that the quadratic terms (linear in both φ and ψ) are listed explicitly, as we expect
to need to use the null structure to extract sufficient decay. The cubic and higher terms

(which are at least quadratic in the background φ), are listed purely schematically.

Remark 4.5.7. Now and in the sequel, HO1 constitutes the cubic and higher order terms
that arise in the right hand side of (4.5.11), see also Appendix A.3.1.4. Similarly, HOi for
equation (4.5.12), see also Appendix A.3.1.5. A key thing to note about the commutator
relation (4.5.14) is that, with ψ = LαL1φ for some multi-index α, every cubic and higher

as a term that appears in an L

term that appears in the schematic decomposition above can be obtained, schematically,
≤|α|+1 derivative of HO1. And similarly with ψ = Lαφ every
≤|α|
cubic and higher term in the schematic decomposition is a term that appears in an L
derivative of HOi. (The only difference being our schematic treatment of the purely cubic
term; see Remark A.3.2.). Thus we will not separately treat the cubic and higher terms

that arise from the commutator in our analyses later, and absorb it as part of the general

165

discussion of higher order terms.

≤|α|

Similarly, with ψ = Lαφ all the quadratic terms that appear in (4.5.14) can be obtained

from L
derivatives hitting on QNi, which are defined as the quadratic inhomogeneity
of (4.5.12). However as we can see in the case ψ = LαL1, the final quadratic commutator
≤|α|+1 derivative
term of the form P1W2φ(L1L1ψ+T L1ψ+T T ψ) cannot be obtained as an L
of QN1, which are defined as the quadratic homogeneity of (4.5.11) (notice the differing
weights P0 and P1). These turn out to be the most delicate terms in the analysis, and in
Section 4.7.2.4 will be the main terms to saturate the polynomial growth in the energy

estimates.

4.5.3 Statement of the main theorem

Our main theorem asserts that when the initial planewave Υ has bounded width, then

this travelling wave solution is stable under small compactly supported perturbations.

By rescaling and translating we can assume the perturbation is supported in the unit ball
B(0,1) ⊂ R3 on the spatial slice {y0 = 2}.

Theorem 4.5.8. Let d = 3 and assume Υ (u) is such that Υ

(cid:48)(cid:48)

has compact support in u. Con-

sider the initial value problem for (4.2.17), where the dynamical metric is given by (4.2.9). We
assume the initial data is prescribed on the spatial slice {y0 = 2} by

φ|

y0=2 = φ0, ∂y0φ|

y0=2 = φ1,

where φ0, φ1 ∈ C
depend on Υ and on γ) such that whenever

∞
0 (B(0,1)). Then for any γ > 0 there exists some 
0 > 0 (which we allow to

(cid:107)φ0(cid:107)

H5 +(cid:107)φ1(cid:107)

H4 ≤ 
0

the solution φ exists for all time y0 ≥ 2. Furthermore, we have the following uniform bounds

166

on the solution and its derivatives:

|φ| +|L1φ| (cid:46) (y0)

−1/2,
|T φ| +|T L1φ| (cid:46) τγ−1(y0)
|B 1,0
0 φ| +|B 1,0
0 L1φ| +|T B 1,0
|B 2,0
0 φ| +|B 3,1

1 φ| (cid:46) τ1+γ(y0)

−1/2.

−1/2,

0 φ| +|T T φ| (cid:46) τγ(y0)

−1/2,

Remark 4.5.9. Observe that in particular, the coordinate derivatives (with respect to y)
up to second order all decay uniformly as y0 (cid:37) ∞. As will be clear from the proof, if
the initial data has higher regularity the regularity persists for the solution. This can

be extended to show that (the details of the proof we omit here) that arbitrary order
−1/2+γ. Peeling, however,
coordinate derivatives of the solution decay uniformly like (y0)
doesn’t hold to arbitrary orders, unlike the case of the linear wave. If we denote by ¯∂

a derivative that is tangential to out-going Minkowski light-cones, our results are only
compatible with these outgoing tangential derivatives ¯∂βφ being uniformly bounded by
(y0)

−3/2+γ for all orders |β| ≥ 2.

4.6 Energy quantities and bootstrap assumptions

The remainder of this chapter is devoted to proving Theorem 4.5.8. In view of the

robust local existence theory for nonlinear wave equations, the strategy we will take is

that of a standard bootstrap argument. In this section we will set the notations for the

basic energy quantities and perform some preliminary analyses on them, having also

introduced the main bootstrap assumptions.

167

4.6.1 The energy quantities defined; bootstrap assumptions

Recall from (4.3.14) the energy quantity

Eτ[ψ; g]2 = 2

(cid:90)

Στ

1(cid:112)|g(dτ,dτ)|Q[ψ; g](T ,(−dτ)g(cid:93)) dvolhτ
†
(cid:12)(cid:12)(cid:12) dvolg .
(cid:12)(cid:12)(cid:12) + 2
(cid:12)(cid:12)(cid:12)(cid:3)gψ · T (ψ)
(cid:12)(cid:12)(cid:12)Q[ψ; g] :g LT g

(4.6.1)

which satisfies the basic energy inequality (4.3.16) for τ0 < τ1

Eτ1[ψ; g]2 ≤ Eτ0[ψ; g]2 +

τ∈[τ0,τ1]

Here ψ will stand for some higher L derivative of the solution φ. One difference between

our quasilinear setting and the semilinear model treated in Section 4.4 is the presence of
the first integrand in the energy inequality. In the semilinear case LT m = 0. The analysis
of the second integrand will occupy Section 4.7, using the equations (4.5.13), (4.5.11),

(4.5.12); we treat the first integrand here.

The integrand can be expanded as
Q[ψ; g] :g LT g = (LT g

−1)(dψ,dψ)− 1
2

−1(dψ,dψ)· g :g LT g.

g

We primarily care about terms that are linear in φ: the terms with higher order depen-

dence on φ we expect to behave better and will estimate very roughly. With that, and

(4.2.14) in mind, schematically
(LT g

−1)(dψ,dψ) = (φ + T φ)P0W2(L1ψ + T ψ)2

+ G(cid:104)

(B 1,1

1 φ)(B 2,2

2 φ) + (B 1,1

1 φ)2(1 + T φP0)

(B 1,1

1 ψ)2.

(cid:105)

And we also have schematically, by (4.2.9), that

g :g LT g = g

−1(dφ,dT φ) + g
1 φ)(B 2,2

= G (B 1,1

−1(du,du)(φ + T φ)P0

2 φ) + G P0(B 1,1

1 φ)2(φ + T φ).

Therefore we can conclude that schematically
Q[ψ; g] :g LT g = (φ + T φ)P0W2(L1ψ + T ψ)2
1 φ)(B 2,2

+ G(cid:104)

(B 1,1

2 φ) + (B 1,1

1 φ)2(1 + T φP0)

(cid:105)

(B 1,1

1 ψ)2.

(4.6.2)

168

We will return to estimating this term in Section 4.6.5

For convenience, for τ ≥ 2 and k a non-negative integer, we will denote by

Ek(τ) def= sup
σ∈[2,τ]
Fk(τ) def= sup
σ∈[2,τ]

Eσ [L
Eσ [L

≤kφ; g],
≤kL1φ; g].

(4.6.3)

(4.6.4)

We will make the following initial data assumption:
E4(2) + F3(2) ≤ 


(AID)
for some 
 ≥ 
0. We can make this assumption as by the standard local-existence ar-
gument for nonlinear wave equations, with the assumptions in Theorem 4.5.8, for suf-

ficiently small 
0 the solution necessarily exists up to Σ2. The continuity of the energy
norms on initial data implies that as 
0 → 0 the quantity E4(2) + F3(2) → 0 also.

As is typical of bootstrap arguments, we will assume there is some T > 2 such that
for every τ ∈ [2, T ] the following bootstrap assumptions hold. We need three parameters:
δ0 > 0 whose size will be fixed in Section 4.6.3 and considered constant afterwards; δ ∈
(0, δ0) which is a smallness parameter we will adjust to close the bootstrap. Without
loss of generality we will assume γ ∈ (0,1/4) is fixed throughout the argument. Our
goal, as usual, is to demonstrate that the bootstrap assumptions below leads to improved
versions of themselves, when δ and 
 are taken to be sufficiently small. This then implies

by standard continuity argument that the assumptions in fact hold for all times τ > 2 and

we obtain global existence.

Our bootstrap assumptions are: First, along Στ we have the uniform bounds

(BA∞)



|φ| ≤ δ0(y0)
|L1φ| ≤ δ0(y0)
|Liφ| ≤ δ0(y0)
|T φ| ≤ δ0(y0)

−1/2;
−1/2;
−1/2τγ;
−1/2τγ−1.

169

Second, we assume that



E1(τ) + F1(τ) ≤ δ;
E2(τ) + F2(τ) ≤ δτγ;
E3(τ) + F3(τ) ≤ δτ1+γ;
E4(τ) ≤ δτ2+γ .

(BA2)

4.6.2 Inequalities on that we use frequently

In the subsequent analysis, we will freely use the control of y0, cosh(ρ), and u afforded by

Lemma 4.6.1. As we will see, these estimates will be an important tool to obtain coercive

control (with respect to Ek, Fk) of terms that arise in the energy estimates. They also have
important consequences when used concurrently with the bootstrap assumptions, see,

for instance, Proposition 4.6.2.
Lemma 4.6.1. The following estimates hold on I + ∩{supp φ}∩{supp P0}

u ≈ y0
y0 ≈ τ2
cosh(ρ) ≈ τ.

(4.6.5)

(4.6.6)

(4.6.7)

√

Proof. Using y0 = (u +u)/

2, (4.6.5) follows because P0 has compact support in u. Under
the assumptions of the initial data in Theorem 4.5.8, finite speed of propagation implies

that

(cid:113)|y1|2 +|y2|2 +|y3|2 ≤ |y0 − 2| + 1 = y0 − 1

on the support of φ. Since τ2 = 2uu − | ˆx|2 = (y0)2 − (y1)2 − (y2)2 − (y3)2, the previous
inequality reads 2y0 ≤ τ2 + 1 and hence y0 (cid:46) τ2 because τ ≥ 2. Since 2uu ≥ τ2 on I + (see
Section 4.3.1),

τ2 ≤ 2uu (cid:46) u ≈ y0

170

by appealing to the support of P0. We have then proved (4.6.6). Finally, (4.6.7) follows
from the identity τ cosh(ρ) = y0 and (4.6.6).

4.6.3 Some first consequences of (BA∞)

The assumptions (BA∞) are not strictly speaking necessary; its presence however helps

jump-start basic geometric comparisons that simplifies especially the energy comparisons

to be taken in the next subsection.

Proposition 4.6.2. The assumptions (BA∞) imply
−1;
|P0φ| (cid:46) δ0τ
−1/2τγ;
|B 1,0
0 φ| (cid:46) δ0(y0)
−1/2τγ−1.
|B 1,1
1 φ| (cid:46) δ0(y0)

And hence

0 φ| and |B 1,1

|G | (cid:46) 1.
Proof. The estimates on |B 1,0
1 φ| are trivial using the assumptions, together
with the fact that y0 ≥ τ by definition. The estimate on |P0φ| follows from the bootstrap
assumption and the estimate (4.6.6) in Lemma 4.6.1. Finally, as γ < 1/2 by assumption,
we see that the three B 1,0
1 , and P0φ all have global uniform bounds, therefore we
must also have global uniform bounds on the arbitrary smooth functions G .

0 , B 1,1

Proposition 4.6.3 (Geometric consequences). The assumptions (BA∞) implies, when δ0 is
sufficiently small, that

1. The Jacobian determinant 1
2

≤ |g| ≤ 2.

2. The hyperboloids Στ are space-like relative to g; in fact g

−1(dτ,dτ) = −1 + O(δ0τ

−5/2).

3. The volume forms dvolητ and dvolhτ are uniformly comparable.

171

4. The quantity cT T from (A.3.3) is comparable to τ2/(y0)2.

Proof. The first claim follows from the fact that

|g| = 1 +

◦
−1(dφ,dφ) = 1 + G (B 1,1
g

1 φ)2.
−1(dτ,dτ). From (4.2.14) we have

For the second claim it suffices to prove bounds on g

that

g

−1(dτ,dτ) = m

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

−1(dτ,dτ)
=−1

+2φΥ

(cid:48)(cid:48)

(∂uτ)2 − 1|g|(

◦
−1(dτ,dφ))2.
g

By definition

and since Υ

(cid:48)(cid:48)

∂uτ =

u
τ

has compact support in u the middle term (cid:46) δ0τ

−3. For the final term we

have schematically
◦
−1(dτ,dφ) = m
g

−1(dτ,dφ) + φΥ

(cid:48)(cid:48)

(cid:88)

i

∂uτ∂uφ
(cid:48)(cid:48) u
τ

Liφ + φΥ

yi
y0τ

1
y0 (L1φ + T φ)

=

τ
y0 T φ +
−1(dτ,dφ)| (cid:46) δ0τγ(y0)

g

and so we see |◦
as fast as (δ0)2τ2γ(y0)

−3/2. This implies that the final term decays at least
−3 and hence for sufficiently small δ0 we have the desired bounds.
For the third claim we first examine (4.3.4), as the induced volume form on Στ is
given by the interior product of the space-time volume form with the unit normal. By
the explicit form of g and the pointwise bounds of Proposition 4.6.3, it suffices that
(dτ)(cid:93) − (dτ)g(cid:93)/
on g(dτ,dτ), it suffices to control

(cid:112)|g(dτ,dτ)| is bounded when measured by m. Due to the above bound

(dτ)g(cid:93) − (dτ)(cid:93) = 2

(cid:48)(cid:48)

u
τ

φΥ

∂u − ◦

g

−1(dφ,dτ)∂g(cid:93)φ.

In terms of the coordinate basis ∂yµ, the coefficients of the right hand side can be worked
out to be bounded by

−2 + (δ0)2τ2γ−1(y0)

−2.

δ0τ

172

This implies the desired conclusion.

The fourth and final claim follows immediately from the definition of (A.3.3).

For conducting the estimates, we will frequently need to swap between the quantities

Eτ[ψ; m]2, Eτ[ψ; g]2, and(cid:90)

(cid:88)|Liψ|2 +

1

cosh(ρ)

|T ψ|2 dvolhτ .

1

τ2 cosh(ρ)

Στ

These three quantities turns out to be comparable if we assume (BA∞) holds with δ0
sufficiently small.

Proposition 4.6.4 (Energy comparison). Assuming (BA∞) holds with δ0, the three energy-
type quantities above are compatible.

Proof. By Proposition 4.6.3, it suffices to compare the terms

Q[ψ; g](T ,(−dτ)g(cid:93)),

Q[ψ; m](T ,(−dτ)m(cid:93)).

We note first that their difference is given by
T ψ[(dτ)g(cid:93) − (dτ)m(cid:93)]ψ − 1
2

We can expand this to be schematically

(cid:20)

T ψ

φP0

−1(dψ,dψ).

T (τ)(g − m)
(cid:21)

◦
−1(dφ,dτ)
(cid:104)
g
y0
τ

u
τ

W1(L1ψ + T ψ) + G

◦
−1(dφ,dψ)
g

+

◦
φP0W2(L1ψ + T ψ)2 + G (
g

−1(dφ,dψ))2(cid:105)

.

Hence we can bound the expression by, using (BA∞) and Proposition 4.6.3,

(cid:46) δ0
τ2y0

P0|T ψ(L1ψ + T ψ)| +

δ0τγ
(y0)3/2

+

|T ψ
δ0y0
τ2

◦
−1(dφ,dψ)|
g

P0W2(L1ψ + T ψ)2 +

◦
−1(dφ,dψ))2.
g

(

y0
τ

(4.6.8)

173

The first term in (4.6.8) can be bounded by
|T ψ|2 +

1

(cid:46) δ0
τ2

cosh(ρ)

δ0
τ2

1

τ2 cosh(ρ)

|L1ψ|2

and the third term by

(cid:46) δ0
τ

1

τ2 cosh(ρ)

|L1ψ|2 +

1

|T ψ|2.

cosh(ρ)

δ0
τ3

Both are bounded obviously by a small multiple of Q[ψ; m](T ,(−dτ)m(cid:93)). We can evaluate
◦
−1(dφ,dψ) = − τ2
g

1 φT ψ + B 1,0

1 ψT φ

(y0)2 T φT ψ + B 1,0

+ B 1,0

1 φB 1,0

1 ψ + φP0W2(L1φ + T φ)(L1ψ + T ψ).

(cid:32)
(cid:32)

0 ψ|.
|B 1,0
(cid:33)
(cid:88)|Liψ|2
(cid:33)
(cid:88)|Liψ|2

.

,

1

1

This implies

|◦
−1(dφ,dψ)| (cid:46) δ0τγ
g
(y0)3/2

|T ψ| +

δ0τγ
(y0)3/2τ

Thus the second term in (4.6.8) can be bounded by
|T ψ|2 +

(δ0)2

(cid:46)

1

τ3−2γ cosh(ρ)2

cosh(ρ)

τ2 cosh(ρ)

and the fourth term by

(cid:46)

(δ0)2

τ3−2γ cosh(ρ)

1

|T ψ|2 +

cosh(ρ)

τ2 cosh(ρ)

Both terms are similarly controlled by a small multiple of Q[ψ; m](T ,(−dτ)m(cid:93)). This im-
plies our proposition.

In Section 4.7 below where we treat the inhomogeneous terms, we frequently need to
estimate weighted L2 integrals along Στ. We can compare such integrals to the energies
by the following Corollary, which follows after noting y0 = τ cosh(ρ).

Corollary 4.6.5. We have the following bounds for L2 integrals of derivatives of φ:

(cid:46) Ek−1(τ),
(cid:46) Fk(τ),

L2(Στ )

(cid:107)(y0τ)
(cid:107)(y0τ)
(cid:107)(y0)
(cid:107)(y0)

−1/2B k,0
0 φ(cid:107)
−1/2B k,0
0 L1φ(cid:107)
L2(Στ )
−1/2τ1/2B k+1,1
φ(cid:107)
−1/2τ1/2B k+1,1
L1φ(cid:107)

1

L2(Στ )

1

L2(Στ )

(cid:46) Ek(τ),

(cid:46) Fk+1(τ).

174

4.6.4 Improved L

∞ bounds from (BA2)

As a consequence of the energy comparison Proposition 4.6.4, we can apply Proposition

∞
4.3.14 with d = 3 to (BA2) and derive the following L

estimates of φ and its derivatives.



δ

(y0)1/2 ,
0 L1φ| (cid:46) δτγ
(y0)1/2 ,

|φ| +|L1φ| (cid:46)
|B 1,0
0 φ| +|B 1,0
|T φ| +|T L1φ| (cid:46) δτγ
(y0)1/2τ
|B 2,0
0 φ| (cid:46) δτ1+γ
(y0)1/2 ,
|B 2,1
1 φ| (cid:46) δτγ
(y0)1/2 ,
1 φ| (cid:46) δτ1+γ
|B 3,1
(y0)1/2 .

,

(4.6.9)

With the aid of (A.3.2), we can also estimate



|T T φ| (cid:46) δτγ−1,
|B 1,0
0 T T φ| (cid:46) δτγ .

(4.6.10)

Here we also made use of Lemma 4.6.1 freely.

Remark 4.6.6. Note that we have estimated (4.6.10) by directly estimating the right hand

side of (A.3.2) using (4.6.9). In particular these were not derived from applying Propo-
sition 4.3.14 to appropriate energy integrals: in fact we have not yet proven any L2 esti-
mates for T T φ and its higher derivatives. It turns out the necessary L2 estimates require

a little bit of work, and we defer their proofs to Lemma 4.7.1.

Remark 4.6.7. Notice that (4.6.9) and (4.6.9) controls up to two derivatives of φ in all

directions, and in particular controls the first derivative of the dynamical metric g. Thus

we can apply the blow-up criterion for quasilinear wave equations and assert that the a
priori estimates guaranteed by our bootstrap argument suffices to prove global existence

of the solution.

175

Remark 4.6.8.

In the bootstrap argument we will be studying energies to the top order

E4 and F3, which corresponds to 3 additional derivatives applied to the equations (4.5.12)
and (4.5.11) respectively. Examining the terms that show up in the nonlinearities, which

depend only on up-to-two derivatives of φ, this means that when performing energy

∞
estimates the highest derivative that we will put into L
only be commuting with B 1,0

∞
Hence between (4.6.9) and (4.6.10) all possible L

terms are captured.

0 derivatives, there will be no T T T φ terms to worry about.

would be three; and as we will

4.6.5 Controlling the deformation tensor term

Now let us return to studying the first integrand in (4.6.1) as promised. First, using

Proposition 4.6.3, the space-time integral with regards to dvolg can be replaced by the
integral with regards to dvolm to which we can apply the co-area formula and decompose
as dvolητ dτ. The same proposition also implies we can replace the hypersurface volume
element and have the integral conducted with respect to dvolhτ dτ.

For the integration along Στ, we will put ψ, which is automatically top order, in the
appropriate weighted L2 space; by Proposition 4.6.4 these L2 integrals can be bounded

by the quasilinear energies. We therefore obtain the following bound

†

τ∈[τ0,τ1]
+

(cid:13)(cid:13)(cid:13)(cid:13)cosh(ρ)G(cid:104)

∞
The terms in L

|Q[ψ; g] :g LT g| dvolg ≤

1

cosh(ρ)

P0(φ + T φ)

Eτ[ψ; g]2

∞(Στ )

(B 1,1

1 φ)(B 2,2

2 φ) + (B 1,1

1 φ)2(1 + P0T φ)

Eτ[ψ; g]2 dτ.

(4.6.11)

∞(Στ )

can be estimated with the help of (4.6.9) and (4.6.10). First we have

τ1(cid:90)
(cid:13)(cid:13)(cid:13)(cid:13)

τ0

(cid:13)(cid:13)(cid:13)(cid:13)L
(cid:105)(cid:13)(cid:13)(cid:13)(cid:13)L
(cid:17) ≤ δτ

−2;

(cid:12)(cid:12)(cid:12) (cid:46) 1

τ

(cid:16) δ

τ

+

δτγ
τ2

P0(φ + T φ)

(cid:12)(cid:12)(cid:12)

1

cosh(ρ)
≈ τ

we used here that

1

cosh(ρ)

−1 by Lemma 4.6.1. Next we have

|cosh(ρ)G B 1,1

1 φB 2,2

2 φ| (cid:46) y0

τ

176

δτγ(cid:112)

y0τ

≤ δ2τ2γ−2

δτγ
τ

after observing Lemma 4.6.1 again. Finally the last term

|cosh(ρ)G (B 1,1

1 φ)2(1 + P0T φ)| (cid:46) y0

δ2τ2γ
y0τ2

τ

1 +

(cid:16)

(cid:17) (cid:46) δ2τ2γ−3.

δτγ(cid:112)

y0τ

Hence, with our assumption that γ < 1/4 we have that

†

τ∈[τ0,τ1]

τ1(cid:90)

τ0

|Q[ψ; g] :g LT g| dvolg (cid:46)

−3/2Eτ[ψ; g]2 dτ.

δτ

(4.6.12)

Note the integrable power in τ: the deformation tensor term does not cause any difficulty

in the analysis.

4.7 Controlling the inhomogeneity

In this section we focus our attention on estimating the second term in the energy

estimate (4.6.1), given by the integral†

By virtue of the geometric comparison Proposition 4.6.3 and the energy comparison

Proposition 4.6.4, we can bound this by

(cid:12)(cid:12)(cid:12)(cid:3)gψ · T ψ

(cid:12)(cid:12)(cid:12)dvolg .

τ∈[τ0,τ1]

(cid:13)(cid:13)(cid:13)(cid:13)(cid:114)
τ1(cid:90)

τ0

(cid:13)(cid:13)(cid:13)(cid:13)L2(Στ )

E[ψ; g] dτ.

y0
τ

(cid:3)gψ

We will take ψ here one of {φ, L1φ, LαL1φ, Liφ, LαLiφ}, where α is some multi-index with
length no more than 3, and i ∈ {2,3}.

principle:

To streamline our control for the higher derivative terms, we observe the following

(cid:13)(cid:13)(cid:13)(cid:13)(cid:114)

y0
τ

(cid:13)(cid:13)(cid:13)(cid:13)L2(Στ )

(cid:46) τν =⇒ (cid:13)(cid:13)(cid:13)(cid:13)(cid:114)

B 1,0

y0
τ

(cid:13)(cid:13)(cid:13)(cid:13)L2(Στ )

(expr)

(SP)
Here, (expr) means some polynomial expressions in G , P∗, W∗, and B∗,1∗ φ. We emphasize
that (SP) is a principle meta to our proof, where we will bound each term in the poly-
nomial expression either in some weighted L2 space on Στ or in L

, using the bootstrap

0 (expr)

∞

(cid:46) τν+1.

177

assumptions (BA2) and their consequences (4.6.9) and (4.6.10). The symbol “(cid:46)” in (SP)
should be understood to mean “can be proven as the result of our bootstrap argument to

be bounded by”, and not a factual assertion of a possibly better bound.

Understood this way, (SP) follows simply from the facts that:

• For B∗,1∗ φ terms, in (BA2), each higher derivative brings at most an additional loss

of τ.

• The terms W∗ are invariant under action by L-derivatives.

• As discussed after Notation 4.5.2, B 1,0

0

Pm = Pm+1, which allows it to grow with an

additional factor of u1/2. By Lemma 4.6.1 this can be bounded by τ.

• Finally, observe that

B 1,0

0

G = G ·(cid:104)B 1,0

0 (φP0) + B 2,0

0 φ + B 2,1
1 φ

(cid:105)

by the chain rule. The first and third terms are in fact decaying by (4.6.9), and the
middle term is bounded by δτ1/2+γ, which, since γ < 1/4, is less than a full order of

τ increase in growth.
Occasionally B∗,2∗ φ terms also occur: these are the terms with two T derivatives.
estimates are already captured in (4.6.10) and they can be seen to also obey

∞
Their L

the schematic principle (SP) where higher derivatives lose factors of τ.

We complement the estimates with the following L2 version:

Lemma 4.7.1. For 0 ≤ k ≤ 3, we have

(cid:107)τ5/2(y0)

−3/2B k,0

0 T T φ(cid:107)

(cid:46) δτmax(k−1,0)+γ .

L2(Στ )

Sketch of proof. The proof of this estimate itself is an application of the principle (SP) and

the bootstrap assumptions. Observe first that by (A.3.2) that T T φ can be expanded as

1/cT T times a polynomial expression in G , P∗, W∗, and B∗,1∗ φ to which (SP) can apply.

178

For convenience call this polynomial expression O. From Corollary 4.6.5 combined with
(4.6.9) we have that

(cid:107)τ1/2(y0)

(cid:107)(τy0)1/2B 2,1
2 φ(cid:107)
L2 (cid:46) E1
−1/2B 1,1
1 φ(cid:107)
L2 (cid:46) E0
|(y0)P0φW1| (cid:46) δτ

−1

|(y0)P0(B 1,1
|(y0)(B 1,1
|(y0)(1 + P1φ)W1(B 1,1

1 φ)3| (cid:46) δ3τ3γ−4
1 φ)2| (cid:46) δ2τ2γ−2
1 φ)2| (cid:46) δ3τ2γ−4.

Additionally, we pay attention to the quadratic term

(cid:107)(τy0)1/2P0W2(L1φ + T φ)2(cid:107)

L2 (cid:46) (cid:107)P0(L1φ + T φ)(cid:107)L

∞E0 (cid:46) δ2τ

−1.

Together with the estimate cT T ≈ τ2/(y0)2 implies the Lemma when k = 0. Specifically,
we have that

(cid:107)(τy0)1/2O(cid:107)

L2 (cid:46) E1 + E0δτ

−1.

Similar arguments show that

(cid:107)(τy0)1/2B 1,0

0

O(cid:107)

L2 (cid:46) E2 + δE1.

For this we crucially need Remark A.3.1 which shows that there is no growth arising from

first derivatives of G terms in (A.3.2). (Note that this step requires explicit argument and

not an appeal to the principle (SP).) For higher derivatives we can appeal to (SP).

For higher k, one also needs to estimate derivatives of cT T . We observe the following

schematic computation

B k,0

0 cT T = τ2W2

(cid:104)
1 + B≤k,0

0

(cid:16)

(B 1,1

1 φ)2 +

0 φ)P0(1 + (B 1,1

1 φ)(L1φ + T φ))

(cid:17)(cid:105)

1
τ2 (B 1,0
0 φ)2
1
τ2 (B 0,0

+

179

The inner term, operated on by B≤k,0
δ2τ2γ−2(y0)

0

can be bounded by
−1 + δτ

−3(1 + δ2τγ−4)

through (4.6.9). By the schematic principle we have that for k ≤ 2, B k,0
by τ2W2. And this shows the Lemma up to k ≤ 2.

0 cT T is bounded

For k = 3, we need to consider the case where all derivatives hits on cT T , since all
other terms follow from the principle (SP). In this case we need to essentially estimate

something that is schematically the same as

(cid:13)(cid:13)(cid:13)(cid:13)τ5/2(y0)

−3/2T T φ·(cid:104)

(B 1,1

1 φ)(B 4,1

1 φ)+
−2(B 1,0

τ

0 φ)(B 4,0

0 φ) + τ

−2φP0(L1φ + B 1,1

(cid:105)(cid:13)(cid:13)(cid:13)(cid:13)L2.

1 φ)B 4,1
1 φ
−1/2 with B 4,0

0 φ, to

Here we group τ1/2(y0)
1 φ terms, and τ
bound in L2 by E3. The remaining parts to be controlled in L

−1/2 with the B 4,1

−1/2(y0)
∞

boils down to

τ2(y0)

−1T T φ[B 1,1

1 φ + τ

−1(B 1,0

0 φ) + τ

−2φP0(L1φ + B 1,1

1 φ)]

which can be bounded by

δ2τ1+γ(y0)

−1[τγ−1(y0)

−1/2 + δτ

−5] (cid:46) δ2τ2γ−3/2

and so we see contributes to a lower order term, and the Lemma holds also for k = 3.

This previous Lemma implies we can also extend (SP) to handle also B∗,2∗ φ terms in

the expression.

Remark 4.7.2. Note that (SP) gives the worst case scenario bound on the higher derivatives

of an expression. As one already sees in the proof of the Lemma above, sometimes this

worst case bound is not realized. For example, first derivatives of G do no lose a whole

factor of τ even in the worst case, and as seen in the proof of the Lemma above, sometimes
derivatives of G do not lose decay at all. Similarly, going from φ to B 1,0
entails a τγ loss.

0 φ in L

only

∞

180

is due to the possible presence of the P∗ terms. Each time a B 1,0

However, overall, the schematic principle (SP) cannot be generally improved. This
0 derivative hits P∗ we
necessarily incur a penalty of one factor of τ. This entirely agrees with our semilinear

analysis in Section 4.4 where the highest growth rates always accompanies the terms

when (cid:96)0 is largest (where most derivatives hit on Υ

4.7.1 Higher order nonlinear terms

(cid:48)(cid:48)

).

Proposition 4.7.3. The following bounds hold:
−1/2HO1(cid:107)
−1/2HOi(cid:107)
L2(Στ )
1 L1φ)(B 2,2

(cid:107)(y0)1/2τ
(cid:107)(y0)1/2τ

L2(Στ )

Proof. We focus first on G (B 1,1

1 φ)(B 1,1
side the T T φ term this can be bounded by
(cid:107)(y0)

−1/2G (B 1,1

−1/2τ

1 φ)(B 1,1

1 φ)(cid:107)
1 L1φ)(B 2,1
(cid:46) (cid:107)τ

L2(Στ )
−1(B 1,1

For the T T φ term we need to add a factor of (y0)
have

|c

−1
T T

2 φ| (cid:46) (y0)2
B 2,1
τ2

1
y0

B 2,1

1 φ.

(cid:46) δ3τ3γ−4
(cid:46) δ3τ2γ−3.

(4.7.1)

(4.7.2)

1 φ), the sole cubic term in HO1. Out-

1 φ)(B 1,1

1 L1φ)(cid:107)L

∞E1(τ) (cid:46) δ3τ2γ−4.

τ2 : this is because by Proposition 4.6.3 we

This gives the bound by
(cid:107)τ

−3(y0)(B 1,1

1 φ)(B 1,1

1 L1φ)(cid:107)L

∞δ (cid:46) δ3τ2γ−5

where we again used Lemma 4.6.1. (The differing decay rates of the two terms stems from
the fact that B 1,0

0 φ has additional decay along Στ compared to T φ, but this decay is not

seen when taking L

on Στ.)

∞

The quartic and higher order terms can be treated similarly, the details of which we
omit here, the general idea being to put the highest order derivative terms in L2 and lower

181

∞

ones in L
. This shows that the quartic and higher order terms in HO1 can be bounded by
(cid:46) δ4τ3γ−4 uniformly. (We remark here that as all the remaining terms are multiplied by
a P∗ weight, for their estimates we can consider (y0) ≈ τ2. This means that the anisotropy
between B 2,1
2 φ terms and T T φ terms that showed up in the cubic term estimates can be

avoided.)

1 φ instead of B 1,1

For HOi, the additional cubic term now is a generic B 2,1

1 L1φ, which
means it decays slower by a factor of τ. The additional quartic terms can all be bounded
by (cid:46) δ4τ2γ−3, and our claims follow.
Remark 4.7.4. To estimate B∗,1∗ φ terms using either the energy (and then by the boot-
∞
estimate using the peeling estimates in Proposition
strap (BA2)) or using a straight-up L
4.3.14, we would need any factors of T derivative to be the outermost one. Luckily, com-

mutation reduces the order of derivatives and leaves the weight unchanged (see Propo-

sition 4.3.12), which has the advantage of guaranteeing that the commutator terms have

−1).

faster decay (by τ
Remark 4.7.5. The quartic term bounds for HO1 can be improved from δ4τ3γ−4 to
δ4τ4γ−5, thereby upgrading the overall bound on HO1 to δ3τ2γ−4. This improvement
comes from noting that the term G P0(B 1,0
2 φ) in the definition of HO1
is actually

1 φ)2(B 2,2

0 φ)(B 1,1

G P0(L1φ)(B 1,1

1 φ)2(B 2,2

2 φ).

As for our purposes these types of improvements are not essential, and does not effect the

closing of the bootstrap, we shall not pursue this and myriad other improvements in the

higher order terms.

One should however note that for studying the missing case d = 2, the above indicates

that careful treatment of all quadratic, cubic, and quartic nonlinearities will be likely

necessary.

182

4.7.2 Quadratic terms

Now we consider the quadratic nonlinearities. These terms are a bit more delicate and

we will include more details of the arguments.

4.7.2.1 Zeroth order case

Looking at (4.5.13), we need to estimate

(cid:107)(y0)

−3/2τ

−1/2P0(L1φ + T φ)2(cid:107)

L2(Στ ).

We observe that
(cid:107)(y0)

−3/2τ

−1/2P0(L1φ)2(cid:107)

∞E0(τ) (cid:46) δ2
τ3 .
Here we used that by (4.6.9), our bootstrap assumptions imply |L1φ| (cid:46) δ(y0)
tionally recall that y0 ≈ τ2 in the presence of P∗. Next, we have

−1P0L1φ(cid:107)L

(cid:46) (cid:107)(y0)

L2(Στ )

−1/2. Addi-

(cid:107)(y0)

−3/2τ

−1/2(T φ)2(cid:107)

L2(Στ )

(cid:46) (cid:107)(y0)

−1τ

−1T φ(cid:107)L

∞E0(τ) (cid:46) δ2τγ
τ5 .

We note here for this term the W2 term in the nonlinearity is crucial: without it the de-
−1 which would not have enabled us to close our estimates.
nominator would only have τ

4.7.2.2 First order, ψ = L1φ

Let us now consider QN1 (see (A.3.6)). The terms with (L1φ + T φ)2 and φ(L1φ + T φ) are
−3. For the remaining terms, we
controlled exactly as the zeroth order term case, by δ2τ

see first that
(cid:107)(y0)

−3/2τ

−1/2P0T φ(L1L1φ + T L1φ)(cid:107)

L2(Στ )

(cid:46) (cid:107)(y0)

−1T φ(cid:107)L

∞F0(τ) (cid:46) δ2τγ
τ4 .

Similarly

(cid:107)(y0)

−3/2τ

−1/2P0(φ + L1φ)(L1L1φ + T L1φ)(cid:107)

L2(Στ )

(cid:46) δ2
τ3 .

The final term involves T T φ, for which we can bound
(cid:46) (cid:107)τ

−1/2P0(φ + L1φ)T T φ(cid:107)

−3/2τ

(cid:107)(y0)

−3(φ + L1φ)(cid:107)L

∞E1(τ) (cid:46) δ2
τ4 .

L2(Στ )

183

4.7.2.3 First order, ψ = Liφ

We next consider QNi (see (A.3.9)). There is a loss compared to the QN1 terms, which we
expect. First,
−1/2τ

−1/2P0(φ + L1φ + T φ)(B 1,1
1 L1φ + B 2,2
(cid:46) (cid:107)τ

1 φ)(cid:107)
−1P0(φ + L1φ + T φ)(cid:107)L

2 φ + B 1,1

L2(Στ )
∞ · [F0(τ) + E1(τ)] (cid:46) δ2
τ2 .

(cid:107)(y0)

Next

(cid:107)(y0)

−3/2τ

Finally,

(cid:107)(y0)

−3/2τ

−1/2(P0Liφ + P1φ)(L1L1φ + T L1φ + T T φ + L1φ + T φ)(cid:107)
0 φ + P1φ)(cid:107)L

−1(P0B 1,0

(cid:46) (cid:107)(y0)

L2(Στ )
∞ · [F0(τ) + E1(τ)] (cid:46) δ2
τ2 .

−1/2P1(φ + L1φ + T φ)(L1φ + T φ)(cid:107)

L2(Στ )
(cid:46) (cid:107)(y0)

−1P1(φ + L1φ + T φ)(cid:107)L

∞ · E0(τ) (cid:46) δ2
τ2

4.7.2.4 Higher order cases

By Remark 4.5.7, the higher order derivatives LαLiφ where i = 2,3 can be treated using
(SP). It suffices to consider the higher derivatives of L1φ. Observe that the principle
(SP) can also be applied to the commutator terms: that control of B 1,0
0 ,(cid:3)g]ψ also
gives control of [B 1,0
0 ψ, since the terms of the latter is schematically a subset
of those terms that appears in the former. Hence it suffices to consider the estimates for
[B 1,0

0 ,(cid:3)g]B 1,0

0 [B 1,0

0 ,(cid:3)g]L1φ.
We treat each of the six quadratic terms in [B 1,0

0 ,(cid:3)g]L1φ listed in the schematic de-

composition (4.5.14) below. First, we can estimate

(cid:107)(y0)

−1/2τ

−1/2P0(B 1,1

1 φ)(L1ψ + T ψ)(cid:107)

L2(Στ )

(cid:46) (cid:107)P0(B 1,1

1 φ)(cid:107)L

∞ · F0(τ) (cid:46) δ2τγ
τ2 .

184

Next, we have
(cid:107)(y0)

−1/2τ

−1/2P0(φ + L1φ + T φ)(B 1,1

1 ψ)(cid:107)

(cid:46) (cid:107)τ

−1P0(φ + L1φ + T φ)(cid:107)L

∞ · F0(τ) (cid:46) δ2
τ2 .

L2(Στ )

The third term we estimate by
(cid:107)(y0)

−1/2P0(B 1,1

−1/2τ

1 φ)(B 2,1

1 ψ + T T ψ)(cid:107)
(cid:46) (cid:107)τ

L2(Στ )
−1P0(B 1,1

1 φ)(cid:107)L

∞ · [F1(τ) + E2(τ)] (cid:46) δ2τ2γ
τ3

.

Next we have
−3/2τ
(cid:107)(y0)

−1/2P0(B 1,0

0 L1φ + B 1,0

0 T φ)(L1ψ + T ψ)(cid:107)
−1P0(B 1,0
(cid:46) (cid:107)(y0)

L2(Στ )

0 L1φ + B 1,0

0 T φ)(cid:107)L

∞ · F1(τ) (cid:46) δ2τγ
τ3 .

The fifth term we estimate by
(cid:107)(y0)

−3/2τ

−1/2P1(φ + L1φ + T φ)(L1ψ + T ψ)(cid:107)

L2(Στ )
(cid:46) (cid:107)(y0)

−1P1(φ + L1φ + T φ)(cid:107)L

∞ · F1(τ) (cid:46) δ2
τ2 .

And the final term is estimated by
(cid:107)(y0)

−1/2P1φ(L1L1ψ + T L1ψ + T T ψ)(cid:107)

−3/2τ

L2(Στ )
(cid:46) (cid:107)(y0)

−1P1φ(cid:107)L

∞ · [F1(τ) + E2(τ)] (cid:46) δ2τγ
τ2 .

4.8 Closing the bootstrap

We conclude our proof of Theorem 4.5.8 by putting together the estimates in the pre-

vious sections using (4.6.1). By our control of the deformation tensor (4.6.12), we have

that

Eτ1[ψ; g]2 −Eτ0[ψ; g] (cid:46)

τ1(cid:90)

τ0

Eτ[ψ; g]2 +(cid:107)(y0)1/2τ

−1/2(cid:3)gψ(cid:107)

L2(Στ )

Eτ[ψ; g] dτ.

δ
τ3/2

185

σ(cid:90)

2

2

From our bootstrap assumptions (BA2) and the computations of Section 4.7.2.1, we

From Section 4.7.2.2, we get

E0(σ)2 − E0(2)2 (cid:46)
σ(cid:90)

F0(σ)2 − F0(2)2 (cid:46)

δ3
τ3/2 +

δ3
τ3 dτ (cid:46) δ3.

δ3
τ3/2 +

δ3
τ3 dτ (cid:46) δ3.

δ3
τ3/2 +

δ3
τ2 dτ (cid:46) δ3.

(4.8.1)

(4.8.2)

(4.8.3)

From Section 4.7.2.3, we get

E1(σ)2 − E1(2)2 (cid:46)

2

σ(cid:90)

By applying the principle (SP) and factoring in Remark 4.5.7 this implies for k ≥ 2,

Ek(σ)2 − Ek(2)2 (cid:46)

δ3τ2γ+2(k−2)

τ3/2

δ3τγ+k−2

τ2

+

· τk−1 dτ

(cid:46) δ3σ 2γ+2(k−2)−1/2 + δ3σ γ+2(k−2) ≤ δ3σ 2γ+2(k−2).

(4.8.4)

Now let σ ∈ (2, T ).

get

σ(cid:90)

2

σ(cid:90)

2

σ(cid:90)

2

Finally, from Section 4.7.2.4 and principle (SP) we get

F1(σ)2 − F1(2)2 (cid:46)

δ3
τ3/2 +

δ3
τ2 +

δ3τγ
τ2 dτ (cid:46) δ3.

(4.8.5)

Further applications of the principle (SP) gives us the higher order estimates for k ≥ 2

Fk(σ)2 − Fk(2)2 (cid:46)

δ3τ2γ+2(k−2)

τ3/2

δ3τγ+k−2

τ2

+

· τk−1 +

δ3τ2γ+k−2

τ2

· τk−1 dτ

(cid:46) δ3σ 2γ+2(k−2)−1/2 + δ3σ γ+2(k−2) + δ3σ 2γ+2(k−2) ≤ δ3σ 2γ+2(k−2).

(4.8.6)

With these estimates, the bootstrap closes provided δ, 
 are taken sufficiently small.

We close our discussion with a couple of remarks.

186

Remark 4.8.1. One interesting aspect of our argument is that the semilinear nonlinear-

ities seem to allow closing the bootstrap using only a log(τ) loss instead of τγ. This is

seemingly in contradiction to the discussion in Section 4.4, where τγ losses seems to be

necessary when the dimension d = 3,4. The explanation for this is that in our semilin-
ear analyses we did not separate out the privileged direction L1φ as having better decay
properties. Had we also isolated the direction L1φ and run the argument with separate
energies for generic derivatives and derivatives with at least one L1 vector field, we would

find also that it is possible to close the argument with merely a log(τ) loss at energy level
d−1, with a further τ loss with each additional derivative, analogously to the cases where
d ≥ 5.

As discussed at the start of Section 4.5, one would see additional losses for the full
quasilinear problem were one not to separate out the better direction L1. This is reflected
in the fact that the part where we required the τγ loss in place of a mere log-loss occurs in
Section 4.7.2.4, where we considered the effects of the commutator term [X,(cid:3)g]ψ; note of
course that the commutator term vanishes for our semilinear model problem.

Remark 4.8.2. One may ask whether the higher energy growth is associated to the blow-

up at infinity described by Alinhac [Ali03], and which seems generic for wave equations

with weak-null quasilinearities [Lin08, LR05, DP18]. This seems not to be the case for

several reasons. First among the reasons is that we observe the same higher energy growth

even for the semilinear model considered in Section 4.4. Additionally, our energy growth

is not very severe; when translated back to L

estimates of the coordinate derivatives,

∞

we still observe decay (though at a reduced rate compared to what is available for the

linear wave equation). Finally, examining the leading order correction of the quasilinear
metric is given with the coefficients φΥ
term means
that the slowly decaying coefficients are supported away from future time-like infinity.

du ⊗ du. The localization by the Υ

(cid:48)(cid:48)

(cid:48)(cid:48)

This appears in contrast to the known manifestations of blow-up at infinity where the
null structure of the dynamical metric is significantly different from the Minkowskian

187

one near future time-like infinity.

188

APPENDIX

189

A.1 Tools from analysis

The following are the standard Sobolev inequalities on Rd that we use freely through-

out this work.
Theorem A.1.1 (Sobolev inequalities). For u ∈ C
constant C depending only on p and d such that

∞
c (Rd), for any 1 ≤ p < d, there exists a

(A.1.1)
∞
c (Rd), for k ∈ N and d < kp, there exists a constant depending only on p, k and d

Lp(Rd).

For u ∈ C
such that

(cid:107)u(cid:107)

∞(Rd)

L

(cid:107)∂αu(cid:107)

Lp(Rd).

(A.1.2)

(cid:107)u(cid:107)

≤ C(cid:107)∇u(cid:107)

Ld/(d−p)(Rd)
(cid:88)

≤ C

|α|≤k

Remark A.1.2. In the special case that p = 1, (A.1.2) holds at the end point k = d.

The following is an abstract formulation of the bootstrap principle, adapted from

[Tao06, Proposition 1.21].
Proposition A.1.3 (Abstract bootstrap principle). Let I ⊂ R be a time interval and for each
t ∈ I let C(t) be a boolean valued function. Suppose one can verify the following statements:

1. (Conclusion holds somewhere). There exists a t0 ∈ I such that C(t0) is true.
2. (Conclusion is continuous) If {tn}∞

⊂ I is a sequence converging to t∞ and C(tn) is

n=1

true for all n, then C(t∞) is true.

3. (Conclusion implies better than inclusion.) If C(t0) = 1 for some t0 ∈ I, then there

exists some 
 > 0 such that (t0 − 
, t0 + 
) ⊂ I and C(t) is true on (t0 − 
, t0 + 
).

Then C(t) is true on all of I.

Proof. The proposition is an immediate consequence of the continuity principle: Let X
be a connected topological space with a non-empty, open, and closed subspace Y ⊂ X. Then
Y = X.

190

A.2 Hyperboloidal polar coordinates and related estimates

In this appendix we prove several estimates found in Chapter 2 for which a proof

was not provided. The proofs provided are adapted from [Won17b] and are not original

material. The proofs are technicalIy involved, and since they do not contribute to the

points made in Chapter 2, but for the sake of completeness, we decided to provide them

here.

A.2.1 Geometric setup
We will be doing our computations on a hyperbolic coordinate system. Identify Sd−1
canonically as a submanifold of Rd with coordinates θ. Then let

(τ, ρ, θ) ∈ R+ × R+ × Sd−1

be the coordinates defined by

t = τ cosh(ρ);

x = τ sinh(ρ)θ.

(A.2.1)

(A.2.2)

In this coordinate system the Minkowski metric takes the form
m = −dτ2 + τ2dρ2 + τ2 sinh2(ρ)dθ2,

where dθ2 is the Euclidean metric on Sd−1. The induced Riemannian metric and its
inverse on Στ are

Here ∂θ ⊗ ∂θ is the Euclidean inverse metric on Sd−1. We claim that

(hτ)

(cid:32)

1
τ2

∂ρ ⊗ ∂ρ +

hτ = τ2(dρ2 + sinh2(ρ)dθ2);
−1 =
d(cid:88)

Li ⊗ Li = ∂ρ ⊗ ∂ρ +

i=1

191

(cid:33)

.

1

sinh2(ρ)

∂θ ⊗ ∂θ

t2

r2 ∂θ ⊗ ∂θ.

(A.2.3)

(A.2.4)

(A.2.5)

We compute with the chain rule

∂ρ =

∂t
∂ρ

∂t +

· ∂x

∂x
∂ρ

= τ sinh(ρ)∂t + τ cosh(ρ)θ · ∂x.

(A.2.6)

Then

∂ρ ⊗ ∂ρ = τ2 sinh2(ρ)∂t ⊗ ∂t + τ cosh(ρ)(∂t ⊗ τ sinh(ρ)θ · ∂x + τ sinh(ρ)θ · ∂x ⊗ ∂t)

+ τ2 cosh2(ρ)θ · ∂x ⊗ θ · ∂x
= |x|2∂t ⊗ ∂t + t(∂t ⊗ x· ∂x + x· ∂x ⊗ ∂t) + t2θ · ∂x ⊗ θ · ∂x.

Since x = rθ and x· ∂x = r∂r,

∂ρ ⊗ ∂ρ = r2∂t ⊗ ∂t + tr(∂t ⊗ ∂r + ∂r ⊗ ∂t) + t2∂r ⊗ ∂r.

We compute from θ = x
r

dθ =
dθ ⊗ dθ =

dx− 1
1
r2 xdr;
r
r2 dx⊗ dx− x
1
r2 dx⊗ dx− 1

1

|x|2
r4 dr ⊗ dr

=

r3 (dx⊗ dr + dr ⊗ dx) +
r2 dr ⊗ dr;
∂θ ⊗ ∂θ = r2∂x ⊗ ∂x − r2∂r ⊗ ∂r.
d(cid:88)
= t2∂x ⊗ ∂x + tr(∂r ⊗ ∂t + ∂t ⊗ ∂r) + r2∂t ⊗ ∂t
= t2∂x ⊗ ∂x − t2∂r ⊗ ∂r + tr(∂r ⊗ ∂t + ∂t ⊗ ∂r) + r2∂t ⊗ ∂t + t2∂r∂r
r2 ∂θ ⊗ ∂θ + ∂ρ ⊗ ∂ρ.
t2

t2∂i ⊗ ∂i + txi(∂i ⊗ ∂t + ∂t ⊗ ∂i) + (xi)2∂t ⊗ ∂t

i=1

=

Then we finally compute

d(cid:88)

i=1

Li ⊗ Li =

This finally concludes the proof of (A.2.5). Of course, this also implies

d(cid:88)

i=1

Li ⊗ Li = ∂ρ ⊗ ∂ρ +

cosh2(ρ)
sinh2(ρ)

∂θ ⊗ ∂θ.

(A.2.7)

192

xj − xj∂xi )

(xi∂

xj − xj∂xi )⊗ (xi∂
xj ⊗ ∂

(xi)2∂
∂xi ⊗ ∂xi − r2∂r ⊗ ∂r

xj + (xj)2∂xi ⊗ ∂xi − xixj(∂

xj ⊗ ∂xi + ∂xi ⊗ ∂
xj )

On the other hand we have

(cid:88)

i<j

Ωij ⊗ Ωij =

i<j

(cid:88)
(cid:88)
= r2(cid:88)

i<j

=

i

= ∂θ ⊗ ∂θ
(cid:88)

−1 +

i<j

(A.2.8)

(A.2.9)

These computations help us conclude

−2hτ)

(τ

Ωij ⊗ Ωij = ∂ρ ⊗ ∂ρ +

1

∂θ ⊗ ∂θ + ∂θ ⊗ ∂θ

∂θ ⊗ ∂θ

sinh2(ρ)
1 + sinh2(ρ)

sinh2(ρ)
cosh2(ρ)
sinh2(ρ)

∂θ ⊗ ∂θ

= ∂ρ ⊗ ∂ρ +
= ∂ρ ⊗ ∂ρ +
d(cid:88)
Li ⊗ Li.

=

i=1

≤ d(cid:88)

i=1

−2hτ is the standard metric on hyperbolic space Hd. Let ∇ be the Levi-Civita

Note that τ
connection on Στ relative to the metric hτ. Contracting (A.2.9) with df = ∇f shows

(cid:104)∇f ,∇f (cid:105)

−2hτ

τ

|Lif |2.

(A.2.10)

A.2.2 Proof of global Sobolev pointwise estimate

In this subsection we prove Theorem 2.1.1. The Sobolev inequality on the unit disk im-

plies the following version in a bounded region of hyperbolic space.
Proposition A.2.1. Let f : Hd → R be a function on hyperbolic space, which we represent in

193

|∇kf |2

h sinhd−1(ρ) dθdρ.

(A.2.11)

−2hτ on the τ-slices. For conve-
as

−1hτ

Remark A.2.2. In estimate (A.2.11), h corresponds to τ

nience we denote dvolhτ as the volume form induced by hτ (see (A.2.3)) and dvolτ
the volume form induced by the rescaled metric τ
Corollary A.2.3. Let (cid:96) ∈ R and f be a function on Στ ⊂ R1,d. Then

|f (τ, ρ, θ)|2 cosh(cid:96)+d−1(ρ) (cid:46) τ

sup
ρ< 5
3

cosh(cid:96)(ρ)|Lαf |2 dvolhτ .

−2hτ.
(cid:90)

−d (cid:88)

|α|≤(cid:98) d
2

(cid:99)+1

Στ

polar coordinates (ρ, θ) ∈ R+ × Sd−1. Then
(cid:88)

2(cid:90)

(cid:90)

|f (ρ, θ)|2 (cid:46)

sup
ρ< 5
3

k≤(cid:98) d
2

(cid:99)+1

Sd−1

0

Proof. Applying (A.2.11) and then (A.2.10) we find

|f (τ, ρ, θ)|2 (cid:46)

sup
ρ< 5
3

(cid:46)

(cid:88)
(cid:88)

k≤(cid:98) d
2

(cid:99)+1

(cid:90)
(cid:88)
Στ∩{ρ<2}

k≤(cid:98) d
2

(cid:99)+1

|α|≤k

τ

|∇kf |2
(cid:90)
Στ∩{ρ<2}

−2hτ

dvolτ

−2hτ
|Lαf |2 dvolτ

−2hτ

.

Here we have implicitly used that when ρ < 2, the quantities in the lower-order terms
|∇

have universal bounds, and the Li are close to orthogonal. Finally, we note

Li Lj|

τ

−2hτ

that cosh(ρ) is universally bounded from above and below on ρ < 2 so that we may insert

it to our estimates by paying the price of universal constants.

We finally note that

dvolhτ =

(cid:112)
(cid:113)
det hτ dρ∧ dθ
τ2d sinh2d−2(ρ) dρ∧ dθ
=
= τd sinhd−1 dρ∧ dθ
= τd dvolτ

−2hτ

.

194

(A.2.12)

From this we conclude

|f (τ, ρ, θ)|2 cosh(cid:96)+d−1(ρ) (cid:46)

sup
ρ< 5
3

|α|≤(cid:98) d
2

(cid:99)+1

(cid:88)
−d (cid:88)
−d (cid:88)

|α|≤(cid:98) d
2

(cid:90)
(cid:90)
Στ∩{ρ<2}
(cid:90)
Στ∩{ρ<2}

(cid:99)+1

(cid:46) τ

(cid:46) τ

|α|≤(cid:98) d
2

(cid:99)+1

Στ

cosh(cid:96)(ρ)|Lαf |2 dvolτ

−2hτ
cosh(cid:96)(ρ)|Lαf |2 dvolhτ

cosh(cid:96)(ρ)|Lαf |2 dvolhτ ,

as desired.

On ρ > 1 we can use a different localized Sobolev estimate. The Sobolev inequality on

the half-infinite cylinder states that

|f (ρ, θ)|2 (cid:46)

sup
ρ> 4
3

(cid:88)

∞(cid:90)

(cid:90)

|α|≤(cid:98) d
2

(cid:99)+1

1

Sd−1

|∂αf |2 dθdρ.

(A.2.13)

Proposition A.2.4. Let (cid:96) ∈ R and f be a function on Στ ⊂ R1,d. Then

|f (τ, ρ, θ)|2 cosh(cid:96)+d−1(ρ) (cid:46) τ

sup
ρ> 4
3

cosh(cid:96)(ρ)|Lαf |2 dvolhτ .

Proof. We apply (A.2.13) to the function f cosh

(cid:96)
2 (ρ)sinh

d−1
2 (ρ). Also note that, since

ρ > 1, the hyperbolic functions cosh(ρ), sinh(ρ) are uniformly comparable to eρ. Con-

sequently, the terms that arise from the product rule can be bounded as

−d (cid:88)

(cid:90)

|α|≤(cid:98) d
2

(cid:99)+1

Στ

|∂α(f cosh

(cid:96)
2 (ρ)sinh

d−1
2 (ρ))|2 (cid:46)

(cid:46)

|∂β1f ∂β2(cosh

(cid:96)
2 (ρ))∂β3(sinh

d−1
2 (ρ))|2

(cid:88)

(cid:88)
|β1|+|β2|+|β3|=|α|

|β|≤|α|

|∂βf |2 cosh(cid:96)(ρ)sinhd−1(ρ).

195

Equation (A.2.6) shows that ∂ρ can be written as a linear combination of Li:

=

Contracting (A.2.8) with dθ implies

∂θ =

Then

and

1 ≥ Ωij(θk) =

Ωij =

i<j

0

xj
r

k (cid:44) i, j;
k = i ∨ k = j,

xi
t

Lj − xj
t

Li

r Li (recalling that t ≥ r). This,
show that ∂θ can be bounded by a linear combination of x
and since x/r have bounded derivatives along Στ to all orders away from ρ = 0, we con-
clude that

|∂βf |2 (cid:46)

|Lαf |2.

∂ρ = x· θ∂t + tθ · ∂x
· t∂x

=

x
r

Li.

x
r
x
r

· x∂t +
· d(cid:88)
(cid:88)

i=1

Ωij(θ)Ωij.

(cid:88)

|α|≤|β|

196

∞(cid:90)

sup
ρ> 4
3

|α|≤(cid:98) d
2

We finally conclude with (A.2.13, A.2.12)
|f (τ, ρ, θ)|2 cosh(cid:96)+d−1(ρ) (cid:46) sup
|f (τ, ρ, θ)|2 cosh(cid:96)(ρ)sinhd−1(ρ)
(cid:90)
(cid:88)
ρ> 4
3
(cid:90)
(cid:88)
(cid:90)
(cid:88)
Στ∩{ρ>1}
(cid:90)
(cid:88)
Στ∩{ρ>1}
|Lαf |2 cosh(cid:96)(ρ) dvolτ
(cid:90)
−d (cid:88)

|α|≤(cid:98) d
2

|α|≤(cid:98) d
2

|α|≤(cid:98) d
2

(cid:99)+1

1

Sd−1

(cid:99)+1

Στ

(cid:99)+1

(cid:46)

(cid:46)

(cid:46)

(cid:46)

(cid:99)+1

(cid:46) τ

−2hτ
cosh(cid:96)(ρ)|Lαf |2 dvolhτ ,

|Lαf |2 cosh(cid:96)(ρ) dvolτ

−2hτ

|∂αf |2 cosh(cid:96)(ρ)sinhd−1(ρ) dθdρ

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

|∂αf |2 cosh(cid:96)(ρ) dvolτ

dvol

τ

−2hτ
−2hτ

|α|≤(cid:98) d
2

(cid:99)+1

Στ

as desired.

We are finally ready to prove Theorem 2.1.1.

Proof of Theorem 2.1.1. Combining Corollary A.2.3 and Proposition A.2.4, we conclude

|f (τ, ρ, θ)|2 (cid:46) τ

−d cosh1−(cid:96)−d(ρ)

cosh(cid:96)(ρ)|Lαf |2 dvolhτ

(cid:88)

(cid:90)

|α|≤(cid:98) d
2

(cid:99)+1

Στ

The result follows from the observation that wτ = τ cosh(ρ) and that τ is constant over
Στ.

Theorem 2.1.1 becomes useful when used in conjunction with Lemma 2.2.5 and Propo-

sition 2.2.2. In order to prove Lemma 2.2.5, we provide an intermediate result which is

equivalent to Hardy’s inequality adapted to the hyperbolas Στ. This proof is adapted
from [Won17b].

197

Lemma A.2.5. ( [Won17b, Lemma 5.1]). Let d ≥ 3. For any function u defined on Στ,

(A.2.14)
Proof. First we claim that if f : [0,∞) → R has compact support, though not necessarily
vanishing at zero, then

˚W1,2−1

.

(cid:107)u(cid:107)L2−1

(cid:107)u(cid:107)

≤ 2
d − 2
∞(cid:90)

f (ρ)2 cosh(ρ)sinhα−1(ρ) dρ ≤ 4
α2

(cid:48)

f

∞(cid:90)

0

∞(cid:90)

(cid:16)

0

∞(cid:90)

0

Through the compact support assumption we see

0 =

∂ρ

f (ρ)2 sinhα(ρ)

dρ = α

Then we estimate with H¨older’s inequality
(cid:48)

f 2 sinhα−1(ρ)cosh(ρ) dρ ≤ 2

f (ρ)f

α

(cid:17)

dρ.

(ρ)2 sinhα+1(ρ)
cosh(ρ)
∞(cid:90)

0

0

0

∞(cid:90)
f (ρ)2 sinhα−1(ρ)cosh(ρ) dρ + 2
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
∞(cid:90)
∞(cid:90)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)f (ρ)cosh
 ∞(cid:90)

f 2 cosh(ρ)sinhα−1(ρ) dρ

(ρ)sinhα(ρ) dρ

1
2 (ρ)sinh

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)f

α−1
2

(ρ)

2 ∞(cid:90)
 1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

0

0

(cid:48)

≤ 2

≤ 2

(A.2.15)

(cid:48)

f (ρ)f

(ρ)sinhα(ρ) dρ.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dρ

α+1
2 (ρ)
1
2 (ρ)

sinh

cosh

(cid:48)2 sinhα+1(ρ)
cosh(ρ)

 1

2

.
We have thus proved (A.2.15). We now prove . We choose α = d − 2 so that α − 1 = d − 3
and α + 1 = d − 1. Then we use the trivial inequality sinh(ρ) ≤ cosh(ρ) and (A.2.15) to see

0

0

f

dρ

1

cosh(ρ)

φ2 dvolhτ = τ

1

cosh(ρ)

φ2 sinhd−1(ρ) dθdρ

(cid:90)

Στ

∞(cid:90)
∞(cid:90)

0

−d

−d

Sd−1

(cid:90)
(cid:90)
∞(cid:90)
(cid:90)

0

198

≤ τ

(cid:90)

Sd−1
0
−d
≤ 4τ
(d − 2)2

=

4

(d − 2)2

Sd−1
1

cosh(ρ)

Στ

cosh2(ρ)
cosh(ρ)

φ2 sinhd−3(ρ) dθdρ

(∂ρφ)2 sinhα+1(ρ)
cosh(ρ)

dθdρ

(∂ρφ)2 dvolhτ .

The decomposition (A.2.7) concludes the proof.

We now prove Lemma 2.2.5.

Proof of Lemma 2.2.5. We begin with the proof of (2.2.12). The chain rule dictates

∂t =

∂τ
∂t

∂τ +

∂ρ
∂t

∂ρ +

· ∂θ.

∂θ
∂t

From θ = x/r we see ∂θ

∂t = 0. From t2 −|x|2 = τ2 we find

so ∂τ

∂t = cosh(ρ). Solving ρ = coth
1

∂ρ
∂t

=

1
x·θ
1− t2|x|2

=

τ sinh(ρ)

1− cosh2(ρ)
sinh2(ρ)

2τ cosh(ρ) = 2t = 2
−1( t

∂τ
∂t
x·θ ) we compute

τ,

−1

= τ

sinh(ρ)

sinh2(ρ)− cosh2(ρ)

= −τ

−1 sinh(ρ).

In total this shows

Then we come to

∂t = cosh(ρ)∂τ − τ

−1 sinh(ρ)∂ρ.

(A.2.16)

Q(∂t, ∂τ) = cosh(ρ)Q(∂τ, ∂τ)− τ

−1 sinh(ρ)Q(∂ρ, ∂τ).

Since ∂τ, ∂ρ are m-orthogonal, we immediately find Q(∂ρ, ∂τ) = ∂ρφ∂τφ. It now suffices
to compute Q(∂τ, ∂τ), which we do by using (A.2.4):

Q(∂τ, ∂τ) = (∂τφ)2 − 1
2
(∂τφ)2 +

=

1
2

m(∂τ, ∂τ)m
1
2τ2 (∂ρφ)2 +

−1(dφ,dφ)− 1
2

1

2τ2 sinh2(ρ)

m(∂τ, ∂τ)M2φ2
|∂θφ|2 +

M2φ2.

1
2

199

We finally expose the coercivity by completing the square

Q(∂τ, ∂t) =

+

=

+

=

+

(∂τφ)2 +
(cid:32)
M2φ2 − τ
∂τφ− τ
(cid:32)
∂τφ− τ

cosh(ρ)

2

cosh(ρ)

2

cosh(ρ)

2
cosh(ρ)

2τ2 sinh2(ρ)
cosh(ρ)

2
cosh(ρ)

2τ2 sinh2(ρ)

cosh(ρ)

2τ2 sinh2(ρ)

|∂θφ|2

cosh(ρ)

(∂ρφ)2 +
2τ2
(cid:33)2
−1 sinh(ρ)∂ρφ∂τφ
−1 sinh(ρ)
cosh(ρ)

∂ρφ

+

(cid:32)cosh(ρ)

2τ2

− sinh2(ρ)
2τ2 cosh(ρ)

(cid:33)

(∂ρφ)2

|∂θφ|2 +

cosh(ρ)

2
−1 sinh(ρ)
cosh(ρ)

M2φ2

(cid:33)2

∂ρφ

+

cosh(ρ)

2

M2φ2.

|∂θφ|2 +
d(cid:88)

1

2τ2 cosh(ρ)

(∂ρφ)2

Finally, we recall (A.2.9, A.2.16) to write this as

Q[φ](∂τ, ∂t) =

1

2τ2 cosh(ρ)

i=1

(Liφ)2 +

1

2cosh(ρ)

(∂tφ)2 +

cosh(ρ)

2

M2φ2.

(A.2.17)

Integrating over Στ concludes the proof of (2.2.12).

The only non-trivial estimates in (2.2.13) are the ones concerning (cid:107)φ(cid:107)L2−1

and (cid:107)φt(cid:107)Wk,2−1
The estimate for φ follows from Hardy’s inequality proved in Lemma A.2.5. We stress
that the estimates for φt differ on both sides of the inequality (2.2.12) because on the left
hand side we have integrals of the form

.

1

cosh(ρ)

|∂tLαφ|2 dvolτ,

where on the right hand side we have integrals of the form
|Lα∂tφ|2 dvolτ .

1

cosh(ρ)

(cid:90)

Στ

(cid:90)

Στ

To show that these two are comparable, we induct the commutation relation [Li, ∂t] =
t ∂t, and the fact that xi/t ≈ 1 in the interior light
−∂xi , the algebraic relation ∂xi = 1
(cid:88)

cone to estimate

t Li − xi

(cid:88)

d(cid:88)

1

|LiLβφ|.

|Lα∂tφ| (cid:46)

|∂tLβφ| +

|β|≤|α|

|β|≤|α|−1

i=1

τ cosh(ρ)

200

In the above inequality, LiLβφ def= Liφ when |α| = 1. The terms in the second sum can be
controlled by

1
τ

|LiLβφ|
(cid:88)

(cid:90)

(cid:88)

because cosh(ρ) ≥ 1. Putting these estimates together we find that
|∂tLαφ|2
d(cid:88)

|Lα∂tφ|2 dvolτ (cid:46)

|α|≤(cid:98) d
2

|α|≤(cid:98) d
2

cosh(ρ)

cosh(ρ)

(cid:90)

(cid:99)+1

Στ

(cid:99)+1

Στ

1

1

1

+

τ2 cosh(ρ)

i=1

|LiLαφ|2 dvolτ .

(A.2.18)

By (A.2.17), the right hand side is bounded by(cid:88)

as desired

|α|≤(cid:98) d
2

(cid:99)+1

Eτ[Lαφ]2,

A.3 Various computations for the perturbed system (4.2.17)

A.3.1 Computations supporting Section 4.5.1

A.3.1.1 Verification of (4.5.1)

(cid:3)gψ =

=

=

(cid:17)
(cid:16)(cid:112)|g|gµν∂νψ
1(cid:112)|g|∂µ
(cid:16)(cid:112)|g|◦
(cid:16) 1(cid:112)|g|
(cid:17)− 1(cid:112)|g|∂µ
1(cid:112)|g|∂µ
◦
(cid:48)(cid:48)
g(d|g|,dψ) + (cid:3)mψ + 2∂u(φΥ
(cid:17)
(cid:16) 1(cid:112)|g|
− 1|g| ·(cid:112)|g|∂µ
(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)
·◦
g(dφ,dψ)− 1|g|

gµν∂νφ· ◦
◦
(cid:16)

◦
gµν∂νφ

gµν∂νψ

1
2|g|

∂uψ)

◦
g

=Υ (cid:48)(cid:48)(φu)2

g(dφ,dψ)

(cid:17)

(cid:16)◦

(cid:17)(cid:17)

dφ,d

g(dφ,dψ)

201

A.3.1.2 Verification of (4.5.2)

We start with

LX

(cid:18)(cid:112)|g|∂µ

(cid:19)

◦

gµν∂νφ(cid:112)|g|

(cid:16)

(cid:48)(cid:48)

Υ

(φu)2(cid:17)

.

= X

Now, the left hand side can be written as

[X,(cid:3)m]φ + (cid:3)mXφ + 2[X, ∂u](φΥ

∂uφ)

(cid:16)
2|g|2 X(|g|)

+ 2∂u

1

+

(cid:48)(cid:48)

(cid:17)

φΥ

(cid:17)

(cid:16)
(cid:48)(cid:48)
(cid:48)(cid:48)
[X, ∂u]φ
+ 2∂u(φΥ
∂uXφ)
◦
◦
−1)(dφ,d|g|)− 1
g(dXφ,d|g|)
2|g|LX(
(cid:16)
(cid:16)LX(
2|g|
g
◦
◦
◦
−1)(dφ,dφ)
g(dφ,dXφ)
g
g

(cid:17)(cid:17)− 1|g|

dφ,d

dφ,d

)∂uφ

(cid:48)(cid:48)
X(φΥ
+ 2∂u
◦
g(dφ,d|g|)− 1
(cid:16)

◦
g

− 1
2|g|

(cid:17)

.

Throughout we have used the Leibniz rule for Lie differentiation with respect to tensor
contractions, as well as the fact that Lie derivatives commute with exterior differentiation.
The boxed terms, we notice, are identical to the principal terms in (cid:3)gψ if we set ψ = Xφ.
The formula (4.5.2) follows by rearranging the terms.

A.3.1.3 Control of T T φ terms

As the null structure that we require can all be recovered as discussed in Remark 4.5.3,
for the control of the T T φ terms in terms of other B 2,1∗
precise with the weights. Starting from the equation

terms we do not need to be too

(cid:3)mφ + 2Υ

(cid:48)(cid:48)

∂u(φφu)− 1
2|g|

we first observe

(cid:3)mφ = − τ2

(y0)2 T T φ− d

1

y0 T φ +

(cid:124)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:123)(cid:122)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:125)

LiLiφ− yiLiT φ− yiT Liφ

(y0)2

i=1

.

(cid:48)(cid:48)

(φu)2

◦
g(dφ,d|g|) = Υ
d(cid:88)

=B2,1
2 φ

202

Additionally, the quadratic terms

(cid:48)(cid:48)

∂u(φφu)− Υ

(cid:48)(cid:48)

2Υ

(φu)2 = P0W2(L1φ + T φ)2

+ φP0W2(L1L1φ + L1T φ + T L1φ + T T φ + L1φ + T φ).

The cubic and higher order terms are captured schematically by
◦
g(dφ,d|g|) = W2[τ2(B 1,1

0 φ)2 + φP0(B 1,1

1 φ)2 + (B 1,0

1 φ)(L1φ + T φ)]T T φ

+ [(B 1,1

1 φ)2 + φP0(B 1,1

1 φ)2]B 2,1
2 φ

+ W1(B 1,1

1 φ)3 + P0(B 1,1

1 φ)4 + φP0(B 1,1

1 φ)3 + φP1W1(B 1,1

1 φ)3,

where we took care to isolate the terms with T T φ from other second derivatives.

(A.3.1)

(A.3.2)

1 φ)3(cid:17)

This means that we can re-write

(cid:16)G B 2,1

T T φ =

1
cT T

2 φ + P0W2(L1φ + T φ)2 + P0(B 1,1

1 φ)4

+ φP0W1(B 1,1

1 φ) + G (B 1,1

1 φ)3 + (1 + φP1)W1(B 1,1

where

cT T

def=

τ2
(y0)2

(cid:104)

1 + (B 1,1

1 φ)2 +

1
τ2 (B 1,0
+

0 φ)2
1
τ2 (B 0,0

0 φ)P0(1 + (B 1,1

1 φ)(L1φ + T φ))

(cid:105)

.

(A.3.3)

Remark A.3.1. Note that none of the G factors in (A.3.2) include any B 1,0

0 φ dependence.

A.3.1.4 Verification of (4.5.11)

We note the following very rough estimate for the cubic terms

−1(dψ1,dψ2) = B 1,1

1 ψ1B 1,1

1 ψ2

m

and hence

◦
g(dψ1,dψ2) = B 1,1

1 ψ1B 1,1

1 ψ2(1 + φP0).

203

We also have that

L
L1(

◦
−1)(dψ1,dψ2) = B 1,0
g

0 φP0B 1,1

1 ψ1B 1,1

1 ψ2.

So all the higher-order, non-boxed terms in (4.5.2) can be captured by the sum

HO1

def= G (B 1,1
+ G P0(B 1,1

0 φ)(B 1,1

1 φ)2(B 2,2
2 φ) + G P0(B 1,0

2 φ)
0 φ)(B 1,1

1 φ)4

1 φ)(B 1,1

1 L1φ)(B 2,2

2 φ) + G P0(B 1,0

1 φ)3(B 2,1

1 φ) + G P0(B 1,1

1 φ)5(B 2,2

+ G φW1P1(B 1,1

1 φ)2(B 2,1
+ G φW1P1(B 1,1

1 φ) + G φ2W2P2(B 1,1
1 φ)5(B 2,2

2 φ) + G W1P1(B 1,0

1 φ)
0 φ)(B 1,1

1 φ)4(B 2,1

1 φ)3.

(A.3.4)

(A.3.5)

1 φ)(B 1,1

1 L1φ)(B 2,2

Remark A.3.2. The term G (B 1,1

2 φ) stands out in the expression of
HO1: it is both the only cubic term (all other terms are at least quartic in the unknowns)
and the only term that is not explicitly multiplied by a factor of P∗. In fact, this term
is the only nonlinearity that would remain when Υ ≡ 0, where the equations reduce to
the small-data scenario studied by Lindblad [Lin04], and the nonlinearity is of the double

null form m

−1(dφ,d(m

−1(dφ,dφ))).

We note that instead of writing B 2,1

1 L1φ. This is de-
liberate in order to allow us to exploit certain improvements of decay for the L1φ deriva-

1 φ we have chosen to write B 1,1

tives.

We concentrate on the boxed, quadratic terms in (4.5.2) next. For these terms we need
the additional null structure as seen in (4.4.4), and we write, noting that [L1, ∂u] = −∂u,

204

the following schematic decompositions for the quadratic terms:

(cid:48)(cid:48)

L1(Υ

(φu)2) = P0W2(L1φ + T φ)(L1L1φ + T L1φ) + P0W2(L1φ + T φ)2,
(cid:48)(cid:48)

∂u(φΥ

φu) = P0W2(L1φ + T φ)2

(cid:48)(cid:48)

∂u(L1(φΥ

+ P0φW2(L1φ + T φ) + P0φW2(L1L1φ + T L1φ + T T φ),

)φu) = P0(φ + L1φ)W2(L1φ + T φ)

+ P0(φ + L1φ)W2(L1L1φ + T L1φ + T T φ)
+ P0W2(L1φ + T φ)2 + P0W2(L1φ + T φ)(L1L1φ + T L1φ).

So we can summarize the quadratic nonlinearities schematically as

QN1 = P0W2 ·(cid:20)

(L1φ + T φ)2 + T φ(L1L1φ + T L1φ)

+ φ(L1φ + T φ) + (φ + L1φ)(L1L1φ + T L1φ + T T φ)

(cid:21)

.

(A.3.6)

(A.3.7)

A.3.1.5 Verification of (4.5.12)

In the case where X = Li for i (cid:44) 1, we have that
◦
−1)(dψ1,dψ2) = B 1,0
g

L
Li (

0 φP1B 1,1

1 ψ1B 1,1

1 ψ2.

One can check that the higher-order, non-boxed terms in (4.5.2) are now captured by

HOi = HO1 + G (B 1,1

1 φ)(B 2,1
+ G φP1(B 1,1

1 φ)(B 2,2
1 φ)2(B 2,2

2 φ)
2 φ) + G φW1P2(B 1,1

1 φ)3 + G P1(B 1,1

1 φ)4.

(A.3.8)

The added terms are now the pure cubic term which now can include L derivatives in all

(cid:48)(cid:48)

which generates

directions, and additional quartic terms which arises from X hitting Υ
a P1 instead of P0.

205

The quadratic parts of the nonlinearity can also be expanded schematically. The com-

putations are as follows:

Li(Υ

(cid:48)(cid:48)
y0 (Li − yiT )(φΥ

1

(φu)2) = P0W2(L1φ + T φ)(LiL1φ + LiT φ) + P1W2(L1φ + T φ)2,
(cid:48)(cid:48)

1 φW1(L1φ + T φ) + P1W2φ(L1φ + T φ)

φu) = P0B 1,1

(cid:48)(cid:48)

∂u(Li(φΥ

+ P0φW1(B 1,1

1 L1φ + B 1,1

1 T φ),

)φu) = P0W2(L1φ + T φ)(L1Liφ + T Liφ) + P1W2(L1φ + T φ)2

+ P0W2Liφ(L1L1φ + T L1φ + T T φ + L1φ + T φ)
+ P1W2φ(L1L1φ + T L1φ + T T φ + L1φ + T φ),
1 φ + P0W1φ(L1B 1,1

(cid:48)(cid:48) 1
y0 (Li − yiT )φ) = P0W1(L1φ + T φ)B 1,1

1 φ + T B 1,1

1 φ).

∂u(φΥ

Thus we can collect the quadratic nonlinearities using the schematic expression

QNi = P0W1(φ + L1φ + T φ)(B 1,1

1 L1φ + B 2,2

2 φ + B 1,1

1 φ)

+ W2(P0Liφ + P1φ)(L1L1φ + T L1φ + T T φ + L1φ + T φ)

+ P1W2(φ + L1φ + T φ)(L1φ + T φ).

(A.3.9)

A.3.2 Computations supporting 4.5.2

Observe that, expanding using the standard formula for the Laplace-Beltrami operator

and (4.2.14), we obtain

(cid:3)gψ = (cid:3)mψ +

1
2

1|g|g

−1(d|g|,dψ) + 2∂u(φΥ
(cid:48)(cid:48)
◦
−1(dφ,d(
g

− 1(cid:112)|g|

∂uψ)
◦
−1(dφ,dψ)))− 1|g|Υ
g

1(cid:112)|g|

(cid:48)(cid:48)

206

(φu)2 ◦

g

−1(dφ,dψ).

This implies

[X,(cid:3)g]ψ =

g

(cid:48)(cid:48)

◦
g

1
2

∂uψ) + 2∂u(X(φΥ

−1(dln|g|,dψ)

−1(d(X ln|g|),dψ) +

LXg
1
2
(cid:48)(cid:48)
1(cid:112)|g|
−1(dφ,dψ)))− 1(cid:112)|g|LX(
+ 2[X, ∂u](φΥ
◦
−1(dφ,d(
g
1(cid:112)|g|
1(cid:112)|g|
◦
−1(d(Xφ),d(
g
−1)(dφ,dψ)))− 1(cid:112)|g|
1(cid:112)|g|LX(
◦
−1(dφ,d(
g
◦
−1(dφ,dψ)− 1|g|Υ
− X(
1|g|Υ
g

◦
−1(dφ,dψ))) +
g
◦
g

(φu)2)

1
2

(cid:48)(cid:48)

(cid:48)(cid:48)

|g|−2/3X(|g|)

+

1
2

− 1(cid:112)|g|
− 1(cid:112)|g|

)∂uψ) + 2∂u(φΥ

◦
−1(dφ,dψ)))
g

[X, ∂u]ψ)

1(cid:112)|g|

(cid:48)(cid:48)
◦
−1)(dφ,d(
g
X|g|
◦
−1(dφ,dψ)))
|g|3/2
g
◦
−1(d(Xφ),dψ)))
g

1(cid:112)|g|

◦
−1(dφ,d(
g
◦
−1(dφ,d(
g
◦
−1)(dφ,dψ)
(φu)2LX(
g
(φu)2 ◦
(cid:48)(cid:48)
− 1|g|Υ

g

−1(d(Xφ),dψ).

Except for the three boxed terms, which are linear in both φ and ψ, all remaining terms

are at least quadratic in φ.

The terms that are quadratic and above in φ are generally harmless. We will use the

following rough estimate for the quadratic form g

−1:

−1(dψ1,dψ2) = B 1,1

1 ψ1B 1,1

1 ψ2(1 + φP0 + G (B 1,1

1 φ)2).

g

For X = Li, where i = 1, . . . ,3, the terms quadratic and above in φ can be schematically

captured by the following collection of terms:

(XG )·(cid:104)

(B 1,1

1 φ)(B 2,2

2 φ)(B 1,1
+ P0(B 1,1

1 ψ) + (B 1,1
1 φ)3(B 1,1

1 φ)2(B 2,2

2 ψ)

1 ψ) + P1W1φ(B 1,1

1 φ)2(B 1,1

+ G ·(cid:104)

1 φ)(B 2,2

(B 2,1
1 φ)(B 2,1

1 φ)(B 2,2

2 φ)(B 1,1

1 ψ) + (B 1,1
2 ψ) + P0(B 1,1
1 ψ) + P1W1φ(B 1,1

1 φ)(B 3,2
1 φ)2(B 2,1
1 φ)(B 2,1

1 φ)3(B 1,1

+ (B 1,1
+ P1(B 1,1
+ P1W1(B 1,0

0 φ)(B 1,1

1 φ)2(B 1,1

1 ψ) + P2W1φ(B 1,1

1 φ)2(B 1,1

1 φ)

(cid:105)

1 ψ)
2 φ)(B 1,1
1 φ)(B 1,1
1 φ)(B 1,1

1 ψ)
1 ψ)
1 ψ)

(cid:17)

(cid:105)

(A.3.10)

(A.3.11)

where since X = Li we have that

XG = G ·(cid:16)B 1,0

0 φP0 + φP1 + B 2,0

0 φ + B 2,1
1 φ

.

207

We note, as before, all terms involving P∗ weights are at least cubic in φ.

The terms that are linear in φ in the commutator can also be expanded. For higher
level commutations we do not need to separate between L1 and Li for i (cid:44) 1. So we can
just write [X, ∂u] = B 1,1

1 , which allows us to capture the relevant terms by

(cid:48)(cid:48)

[X, ∂u](φΥ

ψu) = P0B 1,1

1 φW1(L1ψ + T ψ) + P1W2φ(L1ψ + T ψ)

(cid:48)(cid:48)

∂u(X(φΥ

+ P0φW1B 1,1

1 (L1ψ + T ψ),

)ψu) = (P1φ + P0B 1,0

0 φ)W2(L1ψ + T ψ + L1L1ψ + T L1ψ + T T ψ)

1 φW1(L1ψ + T ψ) + P0W2B 1,0

+ P0B 1,1
+ P1W2(L1φ + T φ)(L1ψ + T ψ) + P1W2φ(L1ψ + T ψ),

0 (L1φ + T φ)(L1ψ + T ψ)

[X, ∂u]ψ) = P0W1(L1φ + T φ)B 1,1

1 ψ + P0W1φ(L1B 1,1

1 ψ + T B 1,1

1 ψ).

(cid:48)(cid:48)

∂u(φΥ

Using the commutation relations of Proposition 4.3.12 we can summarize these terms by

P0W1(B 1,1

1 φ)(L1ψ + T ψ) + P0W1(φ + L1φ + T φ)(B 1,1
1 ψ)
1 T ψ) + P0W2(B 1,0
0 L1φ + B 1,0

1 L1ψ + B 1,1

0 φ)(B 1,1

+ P0W1(B 1,0

0 T φ)(L1ψ + T ψ)

+ P1W2(φ + L1φ + T φ)(L1ψ + T ψ) + P1W2φ(L1L1ψ + T L1ψ + T T ψ)

(A.3.12)

which have similar structure to the terms appearing in QN1 and QNi above.

208

BIBLIOGRAPHY

209

BIBLIOGRAPHY

[AAI06]

Paul Allen, Lars Andersson, and James Isenberg, Timelike minimal submani-
folds of general co-dimension in Minkowski space time, J. Hyperbolic Differ. Equ.
3 (2006), no. 4, 691–700. MR 2289611 (2008c:35187)

[AC79]

[AC19]

[AL14]

[Ali01a]

[Ali01b]

[Ali03]

[AW19a]

[AW19b]

[Bre02]

[BZ09]

[CB76]

A. Aurilia and D. Christodoulou, Theory of strings and membranes in an exter-
nal field. I. General formulation, J. Math. Phys. 20 (1979), no. 7, 1446–1452.
MR 538719 (84a:35184)

Leonardo Abbrescia and Yuan Chen, Global stability of some totally geodesic
wave maps, Submitted (2019), arXiv: 1907.07226.

Xinliang An and Jonathan Luk, Trapped surfaces in vacuum arising dynamically
from mild incoming radiation, Adv. Theor. Math. Phys. 21 (2014), no. arXiv:
1409.6270, 1–120.

S. Alinhac, The null condition for quasilinear wave equations in two space dimen-
sions. I, Invent. Math. 145 (2001), no. 3, 597–618. MR 1856402 (2002i:35127)

, The null condition for quasilinear wave equations in two space di-
mensions. II, Amer. J. Math. 123 (2001), no. 6, 1071–1101. MR 1867312
(2003e:35193)

Serge Alinhac, An example of blowup at infinity for a quasilinear wave equa-
tion, Ast´erisque (2003), no. 284, 1–91, Autour de l’analyse microlocale. MR
2003417 (2005a:35197)

Leonardo Abbrescia and Willie Wai Yeung Wong, Global nearly-plane-
symmetric solutions to the membrane equation, Submitted (2019), arXiv:
1903.03553.

, Global versions of Gagliardo-Nirenberg-Sobolev inequality and ap-
plications to wave and Klein-Gordon equations, Submitted (2019), arXiv:
1903.12129.

Simon Brendle, Hypersurfaces in Minkowski space with vanishing mean curva-
ture, Comm. Pure Appl. Math. 55 (2002), no. 10, 1249–1279. MR 1912097
(2003j:58043)

Lydia Bieri and Nina Zipser, Extensions of the Stability Theorem of the
Minkowski Space in General Relativity, vol. 45, American Mathematical Soc.,
2009.

Y. Choquet-Bruhat, Maximal submanifolds and submanifolds with constant
mean extrinsic curvature of a Lorentzian manifold, Ann. Scuola Norm. Sup. Pisa
Cl. Sci. (4) 3 (1976), no. 3, 361–376. MR 0423405 (54 #11384)

210

[CBM96] Yvonne Choquet-Bruhat and Vincent Moncrief, Existence theorem for solutions
of Einstein’s equations with 1 parameter spacelike isometry groups, Quantization,
nonlinear partial differential equations, and operator algebra (Cambridge,
MA, 1994), Proc. Sympos. Pure Math., vol. 59, Amer. Math. Soc., Providence,
RI, 1996, pp. 67–80. MR 1392984

[Chr86]

[Chr91]

[Chr07]

Demetrios Christodoulou, Global solutions of nonlinear hyperbolic equations
for small initial data, Communications on pure and applied mathematics 39
(1986), no. 2, 267–282.

, The formation of black holes and singularities in spherically symmetric
gravitational collapse, Communications on Pure and Applied Mathematics 44
(1991), no. 3, 339–373.

, The formation of shocks in 3-dimensional fluids, EMS Monographs
in Mathematics, European Mathematical Society (EMS), Z¨urich, 2007. MR
2284927 (2008e:76104)

[Chr09]

, The formation of black holes in general relativity, vol. 4, European Math-

ematical Society, 2009.

[CK93]

[CM14]

Demetrios Christodoulou and Sergiu Klainerman, The global nonlinear stabil-
ity of the Minkowski space, S´eminaire ´Equations aux d´eriv´ees partielles (Poly-
technique) (1993), 1–29.

Demetrios Christodoulou and Shuang Miao, Compressible flow and euler’s
equations, Higher Education Press, Beijing, Surveys of Modern Mathematics,
2014. MR 3288725

[DKSW16] Roland Donninger, Joachim Krieger, Jeremie Szeftel, and Willie Wai-Yeung
Wong, Codimension one stability of the catenoid under the vanishing mean cur-
vature flow in Minkowski space, Duke Math. J. 165 (2016), 723–791.

[DP18]

[ES64]

[FJS17]

Yu Deng and Fabio Pusateri, On the global behavior of weak null quasilinear
wave equations, preprint (2018), arXiv:1804.05107.

James Eells and Joseph H Sampson, Harmonic mappings of Riemannian mani-
folds, American journal of mathematics 86 (1964), no. 1, 109–160.

David Fajman, J´er´emie Joudioux, and Jacques Smulevici, A vector field method
for relativistic transport equations with applications, Anal. PDE 10 (2017), no. 7,
1539–1612. MR 3683922

[GML60] Murray Gell-Mann and Maurice L´evy, The axial vector current in beta decay, Il

Nuovo Cimento (1955-1965) 16 (1960), no. 4, 705–726.

[Gra04]

Alfred Gray, Tubes, second ed., Progress in Mathematics, vol. 221, Birkh¨auser
Verlag, Basel, 2004, With a preface by Vicente Miquel. MR 2024928

211

[Gri10]

Viktor Grigoryan, Stability of geodesic wave maps in dimensions d ≥ 3, Int.
Math. Res. Not. IMRN (2010), no. 23, 4544–4584. MR 2739804

[HKSW16] Gustav Holzegel, Sergiu Klainerman, Jared R. Speck, and Willie Wai-Yeung
Wong, Shock formation in small-data solutions to 3D quasilinear wave equations:
An overview, Journal of Hyperbolic Differential Equations 13 (2016), no. 1, 1–
105.

[Hop13]

[H¨or97]

[Jer11]

[JNO15]

[Joh74]

[Kib76]

[Kla80]

[Kla82]

[Kla84]

[Kla85a]

[Kla85b]

[Kla86]

Jens Hoppe, Relativistic membranes, J. Phys. A 46 (2013), no. 2, 023001, 30.
MR 3005897
Lars H¨ormander, Lectures on Nonlinear Hyperbolic Differential Equations,
vol. 26, Springer Science & Business Media, 1997.

R. Jerrard, Defects in semilinear wave equation and timelike minimal surfaces in
minkowski space, Analysis and PDE 4 (2011), 285–340.

R. Jerrard, M. Novaga, and G. Orlandi, On the regularity of timelike extremal
surfaces, Commun. Contemp. Math. 17 (2015), no. 1, 1450048.

Fritz John, Formation of singularities in one-dimensional nonlinear wave propa-
gation, Comm. Pure Appl. Math. 27 (1974), 377–405. MR 0369934 (51 #6163)

T. Kibble, Topology of cosmic domains and strings, J. Phys. A 9 (1976), 1387–
1398.

Sergiu Klainerman, Global existence for nonlinear wave equations, Comm. Pure
Appl. Math. 33 (1980), no. 1, 43–101. MR 544044 (81b:35050)

, Long-time behavior of solutions to nonlinear evolution equations, Arch.

Rational Mech. Anal. 78 (1982), no. 1, 73–98. MR 654553 (84b:35015)

, Long time behaviour of solutions to nonlinear wave equations, Proceed-
ings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw,
1983) (Warsaw), PWN, 1984, pp. 1209–1215. MR 804771

, Global existence of small amplitude solutions to nonlinear Klein-Gordon
equations in four space-time dimensions, Comm. Pure Appl. Math. 38 (1985),
no. 5, 631–641. MR 803252 (87e:35080)

, Uniform decay estimates and the Lorentz invariance of the classical
wave equation, Communications on Pure and Applied Mathematics 38 (1985),
no. 3, 321–332.

, The null condition and global existence to nonlinear wave equations,
Nonlinear systems of partial differential equations in applied mathematics,
Part 1 (1986), 293–326.

[Kri07]

Joachim Krieger, Global regularity and singularity development for wave maps,
Surveys in differential geometry 12 (2007), no. 1, 167–202.

212

[Lax64]

[Lax73]

[Lin04]

[Lin08]

[LM14]

[LR05]

[LS18]

[LZ19]

[Ma19]

[Mil08]

[Neu90]

[Nir59]

[NT13]

Peter D. Lax, Development of singularities of solutions of nonlinear hyperbolic
partial differential equations, J. Mathematical Phys. 5 (1964), 611–613. MR
0165243

, Hyperbolic systems of conservation laws and the mathematical theory of
shock waves, Society for Industrial and Applied Mathematics, Philadelphia,
Pa., 1973, Conference Board of the Mathematical Sciences Regional Confer-
ence Series in Applied Mathematics, No. 11. MR 0350216 (50 #2709)

Hans Lindblad, A remark on global existence for small initial data of the minimal
surface equation in Minkowskian space time, Proc. Amer. Math. Soc. 132 (2004),
no. 4, 1095–1102 (electronic). MR 2045426 (2005a:35203)

, Global solutions of quasilinear wave equations, Amer. J. Math. 130

(2008), no. 1, 115–157. MR 2382144 (2009b:58062)

Philippe G. LeFloch and Yue Ma, The hyperboloidal foliation method, Series in
Applied and Computational Mathematics, vol. 2, World Scientific Publishing
Co. Pte. Ltd., Hackensack, NJ, 2014. MR 3362362

Hans Lindblad and Igor Rodnianski, Global existence for the Einstein vacuum
equations in wave coordinates, Comm. Math. Phys. 256 (2005), no. 1, 43 – 110.

Jonathan Luk and Jared Speck, Shock formation in solutions to the 2D com-
pressible Euler equations in the presence of non-zero vorticity, Invent. Math. 214
(2018), no. 1, 1–169. MR 3858399

Jianli Liu and Yi Zhou, Uniqueness and stability of traveling waves to
the time-like extremal hypersurface in minkowski space, preprint (2019),
arXiv:1903.04129.

Yue Ma, Global solutions of nonlinear wave-Klein-Gordon system in two spatial
dimensions: weak coupling case, preprint arXiv:1907.03516 (2019).

Olaf Milbredt, The cauchy problem for membranes, Ph.D. thesis, Freien Univer-
sit¨at Berlin, 2008.

J. C. Neu, Kinks and the minimal surface equation in Minkowski space, Physica
D 43 (1990), 421–434.
Louis Nirenberg, On elliptic partial differential equations, Annali della Scuola
Normale Superiore di Pisa - Classe di Scienze Ser. 3, 13 (1959), no. 2, 115–162
(en). MR 109940

Luc Nguyen and Gang Tian, On smoothness of timelike maximal cylinders in
three-dimensional vacuum spacetimes, Classical Quantum Gravity 30 (2013),
no. 16, 165010. MR 3094875

[O’N83]

Barrett O’Neill, Semi-riemannian geometry with applications to relativity, 103,
volume 103 (pure and applied mathematics), Academic Press, 1983.

213

[Sha88]

Jalal Shatah, Weak solutions and development of singularities of the SU(2) σ-
model, Communications on Pure and Applied Mathematics 41 (1988), no. 4,
459–469.

[Sha10]

, Space-time resonances, Quart. Appl. Math. 68 (2010), no. 1, 161–167.

MR 2598888

[SHLW16] Jared Speck, Gustav Holzegel, Jonathan Luk, and Willie Wai-Yeung Wong,
Stable shock formation for nearly simple outgoing plane symmetric waves, Annals
of PDE 2 (2016), 10.

[Sid89]

[SM50]

Thomas C. Sideris, Global existence of harmonic maps in Minkowski space,
Comm. Pure Appl. Math. 42 (1989), no. 1, 1–13. MR 973742 (89k:58069)

Jack Sherman and Winifred J. Morrison, Adjustment of an inverse matrix corre-
sponding to a change in one element of a given matrix, Ann. Math. Statistics 21
(1950), 124–127. MR 0035118

[Smu16]

Jacques Smulevici, Small data solutions of the Vlasov-Poisson system and the
vector field method, Ann. PDE 2 (2016), no. 2, Art. 11, 55. MR 3595457

[Sog08]

[Spe16]

[SS98]

[Tao06]

[Vil70]

[VS94]

Christopher Donald Sogge, Lectures on Non-Linear Wave Equations, Second
Edition, vol. 2, International Press, 2008.

Jared Speck, Shock formation in small-data solutions to 3D quasilinear wave
equations, Mathematical Surveys and Monographs, vol. 214, American Math-
ematical Society, Providence, RI, 2016. MR 3561670

Jalal Shatah and Michael Struwe, Geometric wave equations, Courant Lec-
ture Notes in Mathematics, vol. 2, New York University, Courant Institute
of Mathematical Sciences, New York; American Mathematical Society, Provi-
dence, RI, 1998. MR 1674843

Terence Tao, Nonlinear dispersive equations: local and global analysis, no. 106,
American Mathematical Soc., 2006.
Jaak Vilms, Totally geodesic maps, Journal of differential geometry 4 (1970),
no. 1, 73–79.

A. Vilenkin and E. P. S. Shellard, Cosmic strings and other topological defects,
Cambridge University Press, 1994.

[Won17a] Willie Wong, Global existence for the minimal surface equation on R1,1, Proceed-

ings of the American Mathematical Society, Series B 4 (2017), no. 5, 47–52.

[Won17b] Willie Wai Yeung Wong, Small data global existence and decay for two dimen-

sional wave maps, Submitted (2017), arXiv: 1712.07684.

[Won17c]

, Small data global existence and decay for two dimensional wave maps,

Submitted (2017), arXiv:1712.07684.

214

[Won18a]

[Won18b]

, A commuting-vector-field approach to some dispersive estimates, Archiv

der Mathematik 110 (2018), no. 3, 273–289.

, Singularities of axially symmetric time-like minimal submanifolds in
Minkowski space, J. Hyperbolic Differ. Equ. 15 (2018), no. 1, 1–13. MR
3786768

[WW17]

Jinhua Wang and Changhua Wei, Global existence of smooth solution to rela-
tivistic membrane equation with large data, preprint (2017), arXiv:1708.03839.

215