SHOCK FORMATION IN THE BIG BANG

By

Shih-Fang Yeh

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Mathematics—Doctor of Philosophy

2024

ABSTRACT

This thesis aims to construct Big Bang models and to investigate their properties. Under the

warped product spacetime ansatz, we classify all the physical Big Bang models that are spatially

homogeneous and analyze their asymptotes. Furthermore, we prove that the Big Bang models with

a positive blowup time 𝑟∗ > 0 are dynamically unstable under non-homogeneous perturbations in

Chapter 3.

In addition, we also prove the stability of specially relativistic fluids on a fixed Big Bang

spacetime in Chapter 4. This work implies that Euler equations are not sufficient to generate

shocks. One needs the feedback from fluids to metrics, together forming Einstein-Euler equations,

to generate shocks.

Finally, we prove the global existence of the membrane equation on R1,2 × T1 for sufficiently

small, compactly supported initial data in Chapter 5. This is a work independent of the previous

ones. We use the standard vector field method to show that the energy remains small throughout

the time, estabilishing the global existence for the equation.

Copyright by
SHIH-FANG YEH
2024

ACKNOWLEDGEMENTS

I would like to thank my advisor to give me this problem and appropriate reference. Thank Tim

Yang for our casual conversation in November 2022, motivating me to have Chapter 4 although he

is probably not aware. Thank Shun-Ya Chang’s birthday power to let me come up with the crucial

idea on April 24, 2023 and have Chapter 3. Thank Professor Thomas Parker to help me revise many

documents. Thank me in the past to work hard enough, to do well in both academia and industry.

iv

TABLE OF CONTENTS

CHAPTER 1

CONSTRUCT BIG BANG MODELS . . . . . . . . . . . . . . . . . . .
1.1 Classical Big Bang models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
1.2 Warped product Big Bang models

1
1
7

CHAPTER 2

TECHNIQUES FOR SHOCKS . . . . . . . . . . . . . . . . . . . . . . 16
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
.
2.1 Newtonian fluids
2.2 Relativisitic fluids .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
.
2.3 Total variation: John’s technique . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Piontwise blowup: Riccati equation . . . . . . . . . . . . . . . . . . . . . . . 25

.
.

CHAPTER 3

INSTABILITY OF THE BIG BANG . . . . . . . . . . . . . . . . . . . 27
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1
. .
Introduction . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Main Equations . .
3.3 Strategy .
. 33
.
.
3.4 Control of total variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5 Riccati equation for derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 53
. 73
3.6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Initial data .

.
.
.

.

.

.

.

.

.

.

.

.

.

CHAPTER 4

STABILITY OF RELATIVISTIC FLUIDS ON FIXED BIG BANG
SPACETIMES .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .

4.1 The model problem and its homogeneous solutions
4.2 Dynamical Stability of homogeneous 𝜃 ≠ 0 solutions

. 80
. 80
. . . . . . . . . . . . . . 82

CHAPTER 5

5.1 Geometry and Energy .
5.2 Sobolev inequality .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Estimate for derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Energy Comparability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Global Existence

STABILITY OF MEMBRANE EQUATIONS . . . . . . . . . . . . . . 93
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
. 100
. 102
. 112
. 116

.

.

CHAPTER 6

CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

BIBLIOGRAPHY . .

. .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

v

CHAPTER 1

CONSTRUCT BIG BANG MODELS

1.1 Classical Big Bang models

After Einstein proposed his General Relativity theory, people started to use the Einstein field

equations to model the evolution of our universe. In cosmology, it is assumed that there exists a

family of fundamental observers with timelike geodesics (timelines) which span a four-dimensional

spacetime 𝑉 with a Lorentzian metric 𝑔 satisfying the Einstein field equations

𝑅𝑖𝑐 −

1
2

𝑆𝑔 + Λ𝑔 = 𝑇 .

(1.1)

These fundamental observers have timelines orthogonal to space sections.

In other words, the

spacetime 𝑉 = R × 𝑀 and the metric 𝑔 decomposes to

𝑔 = −𝑑𝑡2 + (3)𝑔.

Here 𝑀 is a Riemannian manifold with the metric (3)𝑔 modeling our universe at a fixed time. In

the cosmology scale, it is reasonable to assume the homogeneity and isotropy on 𝑀. Although

there are also homogeneous non-isotropic models, such as Bianchi cosmologies, we only focus on

the homogeneous isotropic models in this section. Intuitively, isotropy means that at each point in

𝑀, every direction looks the same for the observer. Homogeneity means that for every two points

in 𝑀, there is an isometry of 𝑀 that takes one point to the other. This section basically follows [4].

1.1.1 Riemannian manifolds with constant curvature

Mathematically, We interpret isotropy and homogeneity as follows. Recall that the definition

of the sectional curvature of a Riemannian manifold 𝑀 is

𝐾 (𝑃) =

𝑅(𝑋, 𝑌 , 𝑋, 𝑌 )
𝑔(𝑋, 𝑋)𝑔(𝑌 , 𝑌 ) − 𝑔(𝑋, 𝑌 )2

where 𝑋, 𝑌 are two tangent vectors spanning the two sub-plane 𝑃. This quantity characterizes the

geometry for each "direction" 𝑃. For a fixed point in 𝑀, isotropy means that the sectional curvature

𝐾 (𝑃) is independent of the choice of 𝑃. On the other hand, homogeneity means that the sectional

1

curvature 𝐾 is furthermore independent of the choice of the point in 𝑀.

In other words, 𝐾 is

constant throughout the whole manifold 𝑀. Indeed, Schûr proved that isotropy and the Bianchi

identity imply the constancy of 𝐾 on a Riemannian manifold whose dimension is greater than 2.

It is well-known in geometry that a Riemannian manifold with constant curvature 𝐾 is locally

isometric to a sphere of radius 1√

space with pseudo-radius

1√

|𝐾 |

𝐾 (when 𝐾 > 0), the Euclidean space (when 𝐾 = 0), or a hyperbolic
(when 𝐾 < 0). In our context, the Riemannaian manifold 𝑀 is

3-dimensional with the metric (3)𝑔 of the form

(3)𝑔 =

1
|𝐾 |

𝛾𝜖

where

or more conveniently,

𝛾𝜖 =

𝑑𝑟 2
1 − 𝜖𝑟 2

+ 𝑟 2(𝑑𝜃2 + sin2(𝜃)𝑑𝜙2),

𝜖 = 𝑠𝑔𝑛(𝐾),

𝛾+ = 𝑑𝛼2 + sin2(𝛼) (𝑑𝜃2 + sin2(𝜃)𝑑𝜙2),

𝜖 = 1

𝛾0 = 𝑑𝑟 2 + 𝑟 2(𝑑𝜃2 + sin2(𝜃)𝑑𝜙2),

𝜖 = 0

𝛾− = 𝑑𝜒2 + sinh2( 𝜒) (𝑑𝜃2 + sin2(𝜃)𝑑𝜙2),

𝜖 = −1.

Note that in our context, the curvature 𝐾 is constant over 𝑀 but potentially depends on the time 𝑡.

Therefore, the analysis shows that, under the assumptions homogeneity and isotropy of the universe

𝑀, the metric has the following form

𝑔 = −𝑑𝑡2 + 𝑅(𝑡)2𝛾𝜖 ,

where 𝑅 is a function of 𝑡 and 𝜖 = 1, 0, or −1. From now on, we will assume our universe is

𝑀 = S3, R3, or H3 supporting 𝛾+, 𝛾0, 𝛾− respectively. These spacetimes are called Robertson-

Walker spacetimes.

2

1.1.2 Friedman equations

Consider the Einstein field equations for a Robertson-Walker spacetime with the perfect fluid

as the source energy momentum tensor

𝑇 = ( 𝑝 + 𝜌)𝜉 ⊗ 𝜉 + 𝑝𝑔,

where 𝑝 is the fluid pressure, 𝜌 is the fluid density, and 𝜉 is the fluid velocity. Under the isotropy

and homogeneity conditions, we may assume 𝑝 = 𝑝(𝑡), 𝜌 = 𝜌(𝑡), and 𝜉 = 𝜕𝑡. Inserting this energy

momentum tensor back to (1.1), one finds that the Einstein field equations are reduced to:

𝑝 = −

𝜌 =

−

2 (cid:165)𝑅
𝑅

𝜖
𝑅2
3( (cid:164)𝑅2 + 𝜖)
𝑅2

−

(cid:164)𝑅2
𝑅2

+ Λ

− Λ.

These equations are called the Friedmann equations. They describe the evolution of the radius of

our universe.

1.1.3 Big Bang

Assuming Λ = 0, the Friedmann equations can be rearranged to the following form.

(cid:164)𝑅2
𝑅2

(cid:165)𝑅
𝑅

=

1
3

𝜌 −

𝜖
𝑅2

= −

(cid:16)

1
2

𝑝 +

(cid:17)

.

𝜌

1
3

Assuming 3𝑝 + 𝜌 > 0 and assuming the current universe is expanding, General Relativity suggests

that (cid:165)𝑅 < 0, so (cid:164)𝑅 is decreasing. This is surprising: denoting the current time by 𝑡0, then less than
𝑅(𝑡0)
𝐻 (𝑡0) time units ago, we have 𝑅(0) = 0 (see Figure 1.1). This is one motivation for the Big
(cid:164)𝑅(𝑡0)
Bang conjecture. Note that 𝐻, the Hubble constant, is a function of time. We provide more details

(cid:66) 1

for the evolution of the universe in the next section.

1.1.4 Friedmann-Lemaître models

Assume Λ = 0. The above Friedmann equations imply

(cid:164)𝜌 = −

3 (cid:164)𝑅
𝑅

( 𝑝 + 𝜌).

3

Figure 1.1 Big Bang conjecture for classical models.

Now we impose the equation of state

𝑝 = 𝛾𝜌

where 0 ≤

√𝛾 < 1 denotes the sound speed. The models with the equation of state 𝑝 = 𝛾𝜌 and

Λ = 0 are usually called Friedmann-Lemaitˆre models. Cosmologists believe that the early universe

is dominated by radiation with 𝑝 = 1
3

𝜌, while the later universe is described by dust with 𝑝 = 0.

Using this equation of state, one derives the relation between 𝜌 and 𝑅:

𝜌 · 𝑅3(1+𝛾) = 𝑀

where 𝑀 > 0 is a constant. This suggests different asymptotes of 𝜌 as 𝑅 → 0 for the dust and

radiation cases.

• For the dust case, 𝛾 = 0, so 𝜌 ∼ 1
𝑅3 .

• For the radiation case, 𝛾 = 1

3, so 𝜌 ∼ 1
𝑅4 .

4

For the early universe, 𝑅 is close to 0. The above asymptotes imply that the density of radiation is

greater than the density of dust. This explains why cosmologists believe that the radiation is a good

model for the early universe. With this observation, one can simplify the Friedmann equation to

(cid:164)𝑅2 =

𝑀

3

𝑅2−3(1+𝛾) − 𝜖 .

The evolution of the radius of our universe depends on the choice of 𝜖.

• Hyperbolic case, 𝜖 = −1. 𝑅 will increase forever for this case.

• Euclidean case, 𝜖 = 0. 𝑅 keeps increasing with a slower rate compared with the previous

case.

• Elliptic case, 𝜖 = 1. 𝑅 increases initially and decreases later because (cid:165)𝑅 < 0 if 𝜌 > 0.

1.1.5 Einstein static universe

Historically, it was originally believed that 𝜖 = 1 and 𝑅(𝑡) = 𝑅0 is independent of 𝑡 for a

Robertson-Walker cosmological model. In other words, the metric reads

The Friedman equations reduce to

𝑔 = −𝑑𝑡2 + (𝑅0)2𝛾+.

𝑝 = −

1
(𝑅0)2

+ Λ

𝜌 =

3
(𝑅0)2

− Λ.

Since the pressure cannot be negative, Einstein introduced the cosmological constant Λ > 0 to save

this. After a few years, Einstein accepted those cosmological models with 𝑅 changing with time

and abandoned the introduction of the cosmological constant Λ. The red-shift phenomenon can be

explained if the universe is expanding. Later however, cosmologists reintroduced the cosmological

term in a time-dependent form due to the observation that the universe’s expansion is accelerating.

5

1.1.6 De Sitter and anti de Sitter spacetimes

Consider the vacuum Einstein field equations

𝑅𝑖𝑐 −

1
2

𝑆𝑔 + Λ𝑔 = 0

with a cosmological constant Λ. Using a Robertson-Walker spacetime model, one derives a system

of evolution equations for the radius 𝑅:

(cid:165)𝑅 −

Λ
3

𝑅 = 0

(cid:164)𝑅2 −

Λ
3

𝑅2 = −𝜖 .

We have three cases based on the sign of 𝜖.

• 𝜖 = 0. In this case, Λ > 0 for non-trivial solutions. The first equation gives

𝑅 = 𝐴𝑒𝑘𝑡 + 𝐵𝑒−𝑘𝑡

√︃ Λ

3 . From the second equation, 𝐴𝐵 = 0. Therefore, the spacetime metric is of

where 𝑘 =

the form

−𝑑𝑡2 + 𝑒𝑘 ′𝑡 (𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2)

with 𝑘′ = ±𝑘. The spatial metric is Euclidean up to a time-dependent factor.

• 𝜖 = 1. In this case, Λ > 0 for non-trivial solutions. When 𝑅 is time symmetric, the spacetime

metric reduces to

𝑔de Sitter = −𝑑𝑡2 +

cosh2(𝑘𝑡)
𝑘 2

(cid:16)

𝑑𝛼2 + sin2(𝛼) (𝑑𝜃2 + sin2(𝜃)𝑑𝜙2)

(cid:17)

,

the de Sitter spacetime where 𝑘 =

√︃ Λ

3 . The spatial metric is 𝑆3 up to a cosh2 (𝑘𝑡)

𝑘 2

time-

dependent factor. The de Sitter spacetime is conformal to the slice −𝜋 < 𝑡′ < 𝜋 of the

Einstein static universe via the change of variable 𝑡′ = 2 tan−1(𝑒𝑘𝑡).

6

• 𝜖 = −1.

In this case, Λ may be positive, negative, or zero. When Λ = −3 and 𝑅 is

time-symmetric, the spacetime metric becomes

−𝑑𝑡2 + cos2(𝑡)

(cid:16)

𝑑𝜒2 + sinh2( 𝜒) (𝑑𝜃2 + sin2(𝜃)𝑑𝜙2)

(cid:17)

.

The standard anti de Sitter spacetime is an extension of the above spacetime

− cosh2( 𝜒)𝑑𝑡2 + 𝑑𝜒2 + sinh2( 𝜒) (𝑑𝜃2 + sin2(𝜃)𝑑𝜙2)

with 0 ≤ 𝜒 < ∞. The anti de Sitter spacetime is conformal to the Einstein cylinder 0 ≤ 𝛼 < 𝜋
2
via the change of variable 𝛼 = 2 tan−1(𝑒 𝜒) − 𝜋
2 .

1.2 Warped product Big Bang models

It is difficult to deal with the Einstein field equations (1.1) directly. In order to simplify the

equations, one usually put the spherical symmetry assumption on the spacetime metric:

˜𝑔 = 𝑔 + 𝑟 2 (cid:0)𝑑𝜃2 + sin2(𝜃)𝑑𝜙2(cid:1)

where 𝑔 is a metric on a (1+1)-Lorentzian manifold 𝑄, 𝑟 is a function on 𝑄, and 𝜃, 𝜙 are coordinates

on the two sphere S2. Christodoulou and Dafermos are able to make progress using this ansatz

([5], [9]). Based on this, An and Wong proposed warped product spacetimes in [2] as the following

definition. The reason why we put the warped function 𝑟 as the time function is because we want

to use this spacetime to model the Big Bang. Recall that in classical models, the radius 𝑅 is an

increasing function of time from the Big Bang to the current time. In section 1.2.2, we see that this

is a generalization of the Friedmann-Lemaître-Robertson-Walker spacetime.

Definition 1.2.1. A warped product spacetime is a spacetime 𝑄 ×𝑟 𝐹 with the metric

˜𝑔 = 𝑔 + 𝑟 2ℎ,

where (𝑄, 𝑔) is a simply-connected, 2-dimensional Lorentzian manifold, and (𝐹, ℎ) is an 𝑛-

dimensional Riemannian manifold. Here 𝑟 : 𝑄 → (0, ∞) is a positive function on 𝑄. We further

assume that 𝑟 serves as a time function for the spacetime satisfying

⟨𝑑𝑟, 𝑑𝑟⟩𝑔 < 0.

7

Since 𝑄 is simply-connected and 𝑑𝑟 is timelike, by setting 𝑠 = 𝑐𝑜𝑛𝑠𝑡 along the integral curves

for ∇𝑟, one can assume 𝑔 = −𝛼𝑑𝑟 2 + 𝛽𝑑𝑠2 where 𝛼 = 𝛼(𝑟, 𝑠), 𝛽 = 𝛽(𝑟, 𝑠) are functions on 𝑄. Note

that we have the freedom to choose where to set 𝑠 = 0, but this point does not matter throughout

our analysis. This means that our spacetime metric is

˜𝑔 = −𝛼𝑑𝑟 2 + 𝛽𝑑𝑠2 + 𝑟 2ℎ.

1.2.1 Homogeneous solutions

If we put the homogeneity assumption on warped product spacetimes, the Einstein-Euler

equations reduce to

𝜕𝑟

(cid:16) 𝑟 𝑛−1
𝛼

+

𝑆 [ℎ]
𝑛(𝑛 − 1)

𝑟 𝑛−1 −

2Λ
𝑛(𝑛 + 1)

𝑟 𝑛+1(cid:17)

= −

2𝛾𝜌0
𝑛

·

1
𝑟 𝑛𝛾 𝛽 1+𝛾

2

𝜕𝑟 (𝛼𝛽)

1+𝛾
2 =

(1 + 𝛾)2𝜌0
𝑛

· 𝑟 1−𝑛(1+𝛾) · 𝛼 3+𝛾
2 .

(1.2)

(1.3)

with 𝜌 · 𝑟 𝑛(1+𝛾) · 𝛽 1+𝛾

2 = 𝜌0, 𝑆 [ℎ] is the scalar curvature of ℎ, and 𝑛 is the dimension of the fiber ℎ
(refer to [2]). We are assuming the unknowns 𝛼 and 𝛽 are positive functions of 𝑟 defined on (𝑟∗, 𝑟0]

where 0 ≤ 𝑟∗ < 𝑟0,

0 < 𝛼(𝑟0) < ∞,

0 < 𝛽(𝑟0) < ∞,

and 𝑟∗ is the first singularity; that is,

𝑟∗ = inf{𝑟 ≥ 0| There exist solutions 𝛼, 𝛽 which are continuous over (𝑟, 𝑟0]

and differentiable over (𝑟, 𝑟0)}.

𝑟∗ is well defined by the Peano Existence Theorem. We also assume

0 < 𝛾 < 1

𝜌0 > 0

𝑛 ≥ 2.

In a future paper, we will establish the mathematical definition for Big Bang singularites and classify

all physically meaningful cosmological models with explicit asymptotes toward the Big Bang time.

8

In particular, the Big Bang time may be zero (𝑟∗ = 0) or nonzero (𝑟∗ > 0). We investigate the

nonhomogeneous instability of the Big Bang for 𝑟∗ > 0 in Chapter 3. This is also the main topic

this thesis intends to focus on.

1.2.2 Recovering the FLRW spacetime - 𝑟∗ = 0

In a future paper, we do the asymptote analysis for 𝑟∗ = 0 case. In this section, we only emphasize

that a special case for 𝑟∗ = 0 recovers the Friedmann-Lemaître-Robertson-Walker spacetime. Recall

that our warped product spacetime has the metric

˜𝑔 = −𝛼𝑑𝑟 2 + 𝛽𝑑𝑠2 + 𝑟 2ℎ.

√

Setting

𝛼𝑑𝑟 = 𝑑𝑡, 𝛽 = 𝐶2𝑟 2, ℎ = 𝐶2(𝑑𝑦2 + 𝑑𝑧2), 𝑅 = 𝐶𝑟, we recover the Friedmann-Lemaître-

Robertson-Walker spacetime for Λ = 0, 𝜖 = 0:

˜𝑔 = −𝑑𝑡2 + 𝑅2(𝑑𝑠2 + 𝑑𝑦2 + 𝑑𝑧2).

Notice that when

𝛼 =

27𝐶2
4𝑀

· (1 + 𝛾)2 · 𝑟 1+3𝛾,

𝛽 = 𝐶2𝑟 2,

𝜌 =

𝑀
𝐶3(1+𝛾)𝑟 3(1+𝛾)

,

where 𝐶1+3𝛾 = 9

4 (1 + 𝛾)2, 𝑀 = 𝜌 · 𝑅3(1+𝛾), 𝑅 = 𝐶𝑟, one can verify that 𝛼, 𝛽, 𝜌 defined above
satisfy the equations (1.2), (1.3) , the reduced Einstein field equations for homogeneous solutions.

In other words, the warped product spacetimes can be regarded as an extension of the classical

cosmological models.

1.2.3 Classification of singularities - 𝑟∗ > 0 case

Our goal is to classify all the possibilities when 𝑟∗ > 0 (Proposition 1.2.1). We begin with an

observation.

Lemma 1.2.1. Suppose there exists a pair of solution (𝛼, 𝛽) to the equations (1.2) and (1.3) having

a singularity at 𝑟 = 𝑟∗ > 0. Then

𝑟 𝑛−1
𝛼(𝑟)

lim
𝑟→𝑟 +
∗

= ∞.

9

Remark. Actually in this case, we would have

(𝛼(𝑟) 𝛽(𝑟)) = 0,

lim
𝑟→𝑟 +
∗

as we will see later.

Proof. Firstly, we claim that

𝑟 𝑛−1
𝛼(𝑟)

lim sup
𝑟→𝑟 +
∗

= lim inf
𝑟→𝑟 +
∗

𝑟 𝑛−1
𝛼(𝑟)

.

This is because (1.2) implies that the quantity in the parenthesis on the left hand side

𝑟 𝑛−1
𝛼

+

𝑆 [ℎ]
𝑛(𝑛 − 1)

𝑟 𝑛−1 −

2Λ
𝑛(𝑛 + 1)

𝑟 𝑛+1

is monotonic, and hence it has a limit (may be infinity) when 𝑟 → 𝑟 +

∗ . Since both 𝑆 [ℎ]
𝑛(𝑛−1)

𝑟 𝑛−1 and

𝑟 𝑛+1 have a finite limit as 𝑟 → 𝑟 +

∗ , the above claim follows. Here we use the assumption

2Λ
𝑛(𝑛+1)
𝑛 ≥ 2.

Secondly, we show that

𝑟 𝑛−1
𝛼(𝑟)

lim
𝑟→𝑟 +
∗

≠ 0.

Suppose it is zero. Since we can rewrite (1.2) as

𝜕𝑟

(cid:17)

(cid:16) 𝑟 𝑛−1
𝛼

= −

2𝛾𝜌0
𝑛𝑟 𝑛𝛾 𝛽 1+𝛾

2

𝑆 [ℎ]
𝑛

−

𝑟 𝑛−2 +

2Λ
𝑛

𝑟 𝑛

= −

2𝛾𝜌0

𝑛𝑟 𝑛𝛾𝜎 · (cid:0) 𝑟 𝑛−1
𝛼

(cid:1) 1+𝛾

2

· 𝑟 (𝑛−1)· 1+𝛾

2 −

𝑆 [ℎ]
𝑛

𝑟 𝑛−2 +

2Λ
𝑛

𝑟 𝑛

(1.4)

2 , and 𝜎 cannot go to infinity as 𝑟 → 𝑟 +

where 𝜎 = (cid:0)𝛼𝛽(cid:1) 1+𝛾
find that the first term on the right hand side of (1.4) will go to negative infinity as 𝑟 → 𝑟 +
both 𝜎 and 𝑟 𝑛−1

𝛼 have a limit). This is impossible if we require

∗ by (1.3) (because 𝜎(𝑟0) is finite), we

∗ (because

𝑟 𝑛−1
𝛼(𝑟)

lim
𝑟→𝑟 +
∗

= 0

and 𝑟 𝑛−1
𝛼(𝑟)

> 0 for 𝑟∗ < 𝑟 ≤ 𝑟0, a contradiction.

10

Lastly, we show that it is impossible to have

0 < lim
𝑟→𝑟 +
∗

𝑟 𝑛−1
𝛼(𝑟)

< ∞.

This will require the following lemma.

Lemma 1.2.2. Suppose the solution (𝛼, 𝛽) has a singularity at 𝑟 = 𝑟∗ > 0. It is impossible that

lim𝑟→𝑟 +
∗

𝜎(𝑟) = 0 but 0 < lim𝑟→𝑟 +

∗

𝛼(𝑟) < ∞, where

𝜎 = (𝛼𝛽)

1+𝛾
2

as in the proof of Lemma 1.2.1.

We would postpone the proof of Lemma 1.2.2 .

□

This lemma implies that 𝑟 𝑛−1

𝛼(𝑟) will stay away from 0 when 𝑟 is close to 𝑟∗ > 0. It gives a hint to

do the following change of variable

𝜏 =

𝛼
𝑟 𝑛−1

𝜎 = (𝛼𝛽)

1+𝛾
2 .

The two equations (1.2) and (1.3) become

𝜕𝑟 𝜏 = −𝜏2 (cid:16)

−

2𝛾𝜌0
𝑛

· 𝑟 1

2

𝑛− 1

2 −

𝛾
2

𝑛−

𝛾
2 ·

𝜏 1+𝛾
2
𝜎

𝑆 [ℎ]
𝑛

−

𝑟 𝑛−2 +

𝑟 𝑛(cid:17)

2Λ
𝑛

2𝛾𝜌0
𝑛

=

· 𝑟 1

2

𝑛− 1

2 −

𝛾
2

𝑛−

𝛾
2 ·

𝜏 5+𝛾
2
𝜎

𝑆 [ℎ]
𝑛

+

𝑟 𝑛−2 · 𝜏2 −

2Λ
𝑛

𝑟 𝑛 · 𝜏2

𝜕𝑟 𝜎 =

(1 + 𝛾)2𝜌0
𝑛

· 𝑟 1

2

𝑛− 1

2 −

𝛾
2

𝑛−

𝛾

2 · 𝜏 3+𝛾
2 ,

(1.5)

(1.6)

which imply

𝜕𝑟 (𝜏𝑎𝜎) = (cid:0)𝑎 · 2𝛾 + (1 + 𝛾)2(cid:1) ·

𝑟 1

2

𝑛− 1

2 −

𝛾
2

𝑛−

𝜌0
𝑛

𝛾

2 · 𝜏 3+𝛾

2 +𝑎 + 𝑎

(cid:16) 𝑆 [ℎ]
𝑛

𝑟 𝑛−2 −

𝑟 𝑛(cid:17)

2Λ
𝑛

· 𝜏𝑎+1𝜎

for any real number 𝑎 ∈ R. In order to eliminate the first term on the right hand side, we calculate

(cid:16) 𝜏

𝜕𝑟

(1+𝛾)2
2𝛾
𝜎

(cid:17)

=

(1 + 𝛾)2
2𝛾

(cid:16) 𝑆 [ℎ]
𝑛

𝑟 𝑛−2 −

𝑟 𝑛(cid:17)

𝜏 ·

2Λ
𝑛

(cid:16) 𝜏

(1+𝛾)2
2𝛾
𝜎

(cid:17)

11

=

(1 + 𝛾)2
2𝛾

𝐺 (𝑟)𝜏 ·

(cid:16) 𝜏

(1+𝛾)2
2𝛾
𝜎

(cid:17)

,

where we denote (cid:0) 𝑆 [ℎ]

𝑛 𝑟 𝑛−2 − 2Λ

𝑛 𝑟 𝑛(cid:1) by 𝐺 (𝑟). This implies

Lemma 1.2.3. Suppose there exists a pair of solution (𝛼, 𝛽) to the equations (1.2) and (1.3) having

a singularity at 𝑟 = 𝑟∗ > 0 and 𝜏, 𝜎 are defined as above. If 𝜏 satisfies

𝜏(𝑟) ≤ 𝑀,

𝑟∗ < 𝑟 ≤ 𝑟0

for some constant 𝑀 < ∞, then the following limit

𝜏

(1+𝛾)2
2𝛾
𝜎

< ∞

0 < lim
𝑟→𝑟 +
1

exists.

Note that this lemma proves Lemma 1.2.2 and the remark after Lemma 1.2.1. The above lemma

suggests doing the following change of variable

(cid:16) 𝜏

(1+𝛾)2
2𝛾
𝜎

(cid:17)

𝜂 =

and then we have the new system of equations

𝜕𝑟 𝜎 =

(1 + 𝛾)2𝜌0
𝑛

· 𝑟 𝑏 ·

(cid:16) 𝜏

(1+𝛾)2
2𝛾
𝜎

(cid:17)

𝛾 (3+𝛾)
(1+𝛾)2 · 𝜎

𝛾 (3+𝛾)
(1+𝛾)2

=

(1 + 𝛾)2𝜌0
𝑛

· 𝑟 𝑏 · 𝜂

𝛾 (3+𝛾)
(1+𝛾)2 · 𝜎

𝛾 (3+𝛾)
(1+𝛾)2

𝜕𝑟𝜂 =

(1 + 𝛾)2
2𝛾

𝐺 (𝑟) ·

(cid:16) 𝜏

(1+𝛾)2
2𝛾
𝜎

(cid:17)

2𝛾
(1+𝛾)2 · 𝜎

2𝛾
(1+𝛾)2 ·

(cid:16) 𝜏

(1+𝛾)2
2𝛾
𝜎

(cid:17)

=

(1 + 𝛾)2
2𝛾

𝐺 (𝑟) · 𝜂

2𝛾

(1+𝛾)2 +1 · 𝜎

2𝛾
(1+𝛾)2 ,

(1.7)

(1.8)

where 𝑏 = 1
2

𝑛 − 1

2 −

𝛾
2

𝑛 −

𝛾

2 . If we try to separate 𝜎 and 𝜂, we find

𝜕𝑟 (cid:0)𝜎

1−𝛾
(1+𝛾)2 (cid:1) =

1 − 𝛾
(1 + 𝛾)2

(1 + 𝛾)2𝜌0
𝑛

·

· 𝑟 𝑏 · 𝜂

𝛾 (3+𝛾)
(1+𝛾)2

12

=

1 − 𝛾
(1 + 𝛾)2

(1 + 𝛾)2𝜌0
𝑛

·

· 𝑟 𝑏 · (cid:0)𝜂− 2𝛾

(1+𝛾)2 (cid:1) − 3+𝛾

2

𝜕𝑟 (cid:0)𝜂− 2𝛾

(1+𝛾)2 (cid:1) = −

= −

2𝛾
(1 + 𝛾)2

2𝛾
(1 + 𝛾)2

·

·

(1 + 𝛾)2
2𝛾

(1 + 𝛾)2
2𝛾

𝐺 (𝑟) · 𝜎

2𝛾
(1+𝛾)2

𝐺 (𝑟) · (cid:0)𝜎

1−𝛾

(1+𝛾)2 (cid:1) 2𝛾
1−𝛾 .

This means if we do another change of variable

1−𝛾
(1+𝛾)2

𝑢 = 𝜎
𝑣 = 𝜂− 2𝛾

(1+𝛾)2 ,

the two evolution equations become

𝜕𝑟𝑢 =

(1 − 𝛾) 𝜌0
𝑛

· 𝑟 𝑏 · 𝑣− 3+𝛾

2

𝜕𝑟 𝑣 = −𝐺 (𝑟) · 𝑢

2𝛾
1−𝛾 .

Since the initial data of 𝑢 is going to be zero in our application and it is the only difficulty

to extend the differential equation over 𝑟 = 𝑟∗, we could consider the following better evolution

equations instead.

𝜕𝑟𝑢 =

(1 − 𝛾) 𝜌0
𝑛

· |𝑟 |𝑏 · 𝑣− 3+𝛾

2

𝜕𝑟 𝑣 = −𝐺 (𝑟) · |𝑢|

2𝛾
1−𝛾 .

(1.9)

(1.10)

This does not change what (𝑢, 𝑣) is because 𝑢(𝑟∗) = 0 and the right hand side of (1.9) is positive.

By the Peano existence theorem, for 𝜖 small enough, there exists a solution (𝑢, 𝑣) solving (1.9) and

(1.10) on [𝑟∗ − 𝜖, 𝑟∗ + 𝜖], with initial data 𝑢(𝑟∗) = 0, 𝑣(𝑟∗) > 0, where we are assuming 𝑟1 > 0.

13

Lemma 1.2.4. Let 𝑟∗ > 0, 𝐴 > 0, and 𝑣∗ > 0 be given. There exists an 𝜖 = 𝜖 (𝑟∗, 𝐴, 𝑣∗) > 𝑟∗ so

that there exists a solution (𝑢(𝑟), 𝑣(𝑟)) to the system of equations (1.9) and (1.10) with the initial

data

and the bounds

𝑢(𝑟∗) = 0

𝑣(𝑟∗) = 𝑣∗

0 < 𝑢(𝑟) ≤ 𝐴
1
2

𝑣∗ ≤ 𝑣(𝑟) ≤ 2𝑣∗

for 𝑟∗ − 𝜖 ≤ 𝑟 ≤ 𝑟∗ + 𝜖. Notice that with these initial data, lim𝑟→𝑟 +

∗

𝛼(𝑟) = 0, so that corresponds

to a singularity of 𝛼.

Proposition 1.2.1. Let 𝑟∗ > 0. We have the following.

1. If there exists a solution (𝛼, 𝛽) to the equations (1.2) and (1.3), having a singularity at

𝑟 = 𝑟∗ > 0, then

𝑟 𝑛−1
𝛼(𝑟)

= ∞

(𝛼(𝑟) · 𝛽(𝑟)) = 0.

lim
𝑟→𝑟 +
∗

lim
𝑟→𝑟 +
∗

2. Conversely, given any 𝐴 > 0 and 𝑣∗ > 0, there exists an 𝑟0 = 𝑟0(𝑟∗, 𝐴, 𝑣∗) > 𝑟∗ so that there

exists a solution (𝛼, 𝛽) (on (𝑟∗, 𝑟0]) to the system of equations (1.2) and (1.3) with

(cid:0)𝛼(𝑟) 𝛽(𝑟)(cid:1) 1−𝛾

2(1+𝛾) = 0

𝑟 𝑛−1 ·

1
𝛼

· (cid:0)𝛼𝛽(cid:1)

𝛾

1+𝛾 = 𝑣∗,

lim
𝑟→𝑟 +
∗

lim
𝑟→𝑟 +
∗

which implies lim𝑟→𝑟 +
∗

𝛼(𝑟) = 0, and with the bounds

2(1+𝛾) ≤ 𝐴

0 ≤ (cid:0)𝛼(𝑟) 𝛽(𝑟)(cid:1) 1−𝛾
1
2

𝑣∗ ≤ 𝑟 𝑛−1 ·

1
𝛼

· (cid:0)𝛼𝛽(cid:1)

𝛾

1+𝛾 ≤ 2𝑣∗.

14

Remark. The relation between (𝑢, 𝑣) and the original unknowns (𝛼, 𝛽) is

2(1+𝛾)

𝑢 = (cid:0)𝛼𝛽(cid:1) 1−𝛾
1
𝛼

𝑣 = 𝑟 𝑛−1 ·

· (cid:0)𝛼𝛽(cid:1)

𝛾
1+𝛾 ,

or

𝛼 = 𝑢

2𝛾
1−𝛾 ·

1
𝑣

· 𝑟 𝑛−1

𝛽 = 𝑢 2

1−𝛾 · 𝑣 ·

1
𝑟 𝑛−1

.

Since lim𝑟→𝑟 +
∗

𝑢 = 0 and 0 < lim𝑟→𝑟 +

∗

𝑣 < ∞, we have

𝛼 ≈ (𝑟 − 𝑟∗)

2𝛾
1−𝛾

𝛽 ≈ (𝑟 − 𝑟∗)

2
1−𝛾 ,

where the implicit constant depends on 0 < 𝛾 < 1, 𝜌0, 𝑛, 𝐺 (𝑟∗), and 𝑟∗ > 0.

15

CHAPTER 2

TECHNIQUES FOR SHOCKS

This chapter aims to introduce the dynamics of compressible fluids (Section 2.1 and 2.2) and the

techniques to deal with shocks (Section 2.3 and 2.4).

2.1 Newtonian fluids

This section aims to derive the conservation laws for classical Newtonian fluids. We will

basically follow Chapter 1 of Toro’s book [26]. We use 𝜌 : R × R3 → R as the fluid mass density,

𝑣 : R × R3 → R3 as the fluid velocity, 𝑝 : R × R3 → R as the fluid pressure, 𝑒 as the specific

internal energy, 𝑠 as the specific entropy.

Let 𝑈 (𝑡) ⊂ R3 (called control volume in the Physics context) be a family of open, bounded,

connected regions, bounded by the smooth boundary 𝜕𝑈 (𝑡) moving with the fluid. For any quantity

Ψ(𝑡) =

∫

𝑈 (𝑡)

𝜓(𝑡, 𝑥)𝑑𝑥

with 𝜓 : R × R3 → R, we have the material derivative is

𝑑Ψ
𝑑𝑡

=

∫

𝑈 (𝑡)

𝜕𝑡𝜓(𝑡, 𝑥)𝑑𝑥 +

∫

𝜕𝑈 (𝑡)

𝜓(𝑡, 𝑥) (𝑣 · 𝑛)𝑑𝑆

where 𝑣 : R × R3 → R3 is the fluid velocity, the velocity of the boundary 𝜕𝑈, and 𝑛 is the outward

normal vector on 𝜕𝑈. Intuitively, the second surface integral says that if the boundary tends to

expand (𝑣 · 𝑛 > 0), 𝜓 should contribute to Ψ at the points 𝜕𝑈 is expanding. Applying Divergence

theorem to this surface integral, we derive

𝑑Ψ
𝑑𝑡

=

∫

𝑈 (𝑡)

𝜕𝑡𝜓(𝑡, 𝑥)𝑑𝑥 +

∫

𝑈 (𝑡)

𝑑𝑖𝑣(𝜓𝑣)𝑑𝑥.

One can take 𝜓(𝑡, 𝑥) to be the density 𝜌, the momentum 𝜌𝑣, or the energy 𝐸 to derive the following

conservation laws:

(𝜕𝑡 𝜌) + 𝑑𝑖𝑣(𝜌𝑣) = 0

𝜕𝑡 (𝜌𝑣𝑖) + 𝑑𝑖𝑣(𝜌𝑣𝑖𝑣) + (𝜕𝑖 𝑝) = 0,

1 ≤ 𝑖 ≤ 3

(2.1)

(2.2)

16

(𝜕𝑡 𝐸) + 𝑑𝑖𝑣(𝐸𝑣) + 𝑑𝑖𝑣( 𝑝𝑣) = 0

(2.3)

where 𝑝 is the pressure coming from the stress tensor, 𝐸 = 1
2

𝜌|𝑣|2 + 𝜌𝑒, and 𝑒 is the specific internal

energy. The first equation (2.2) comes from the conservation of mass. For (2.2), we are assuming

that the stress tensor is diagonal and a multiple of the identity matrix 𝐼 ∈ 𝑀3×3(R). Therefore, the

source term for the rate of change of the momentum, coming from the stress tensor acting on the

boundary 𝜕𝑈, is

∫

−

𝜕𝑈

( 𝑝𝐼)𝑛𝑑𝑆 = −

∫

𝑈

(∇𝑝)𝑑𝑥,

which appears as the last term (𝜕𝑖 𝑝) in (2.2). In (2.3), we are assuming the stress tensor is 𝑝𝐼 again

and moreover there is no net heat flowing across the boundary. The stress energy will provide the

source term for the rate of change of energy:

∫

−

𝜕𝑈

( 𝑝𝐼)𝑛 · 𝑣𝑑𝑆 = −

∫

𝑈

𝑑𝑖𝑣( 𝑝𝑣)𝑑𝑥.

The term ( 𝑝𝐼)𝑛 · 𝑣 comes from the fact that power = force · velocity. This explains the last term

𝑑𝑖𝑣( 𝑝𝑣) in (2.3). These three conservation laws combined with equation of state give the following

observation:

Observation 1. In the isentropic case, the conservation law of energy (2.3) is redundant.

Proof. To simplify the notation, we assume the fluid is in R rather than R3, but the result still holds

in R3. The first two conservation laws now become

(𝜕𝑡 𝜌) + 𝜕𝑥 (𝜌𝑣) = 0

𝜕𝑡 (𝜌𝑣) + 𝜕𝑥 (𝜌𝑣2) + (𝜕𝑥 𝑝) = 0.

Using the Leibniz rule, we can convert them to the following equations

(𝜕𝑡 𝜌) + 𝜕𝑥 (𝜌𝑣) = 0

(𝜕𝑡𝑣) + 𝑣(𝜕𝑥𝑣) +

1
𝜌

(𝜕𝑥 𝑝) = 0.

17

(2.4)

(2.5)

On the other hand, the isentropic condition means 𝑑𝑠 = 0 along the fluid line (the 𝜕𝑡 + 𝑣𝜕𝑥 direction),

where 𝑠 denotes the specific entropy. This implies that

0 = 𝑇 𝑑𝑠 = 𝑑𝑒 + 𝑝𝑑

(cid:17)

(cid:16) 1
𝜌

= 𝑑𝑒 −

1
𝜌2

𝑝𝑑𝜌

(2.6)

from the first law of Thermodynamics. Here 𝑒 denotes the specific internal energy, 𝑝 is the pressure,

and 1

𝜌 is the specific volume. In order to show that the energy conservation law is redundant, we

calculate

(𝜕𝑡 𝐸) + 𝜕𝑥 (𝐸𝑣) + 𝜕𝑥 ( 𝑝𝑣) = 𝜕𝑡

𝜌𝑣2 + 𝜌𝑒

(cid:17)

+ 𝜕𝑥

(cid:16) 1
2

(cid:16) 1
2

𝜌𝑣3 + 𝜌𝑣𝑒

(cid:17)

+ 𝜕𝑥 ( 𝑝𝑣)

= 𝜌𝜕𝑡

𝑣2(cid:17)

(cid:16) 1
2

+ 𝜌(𝜕𝑡𝑒) + 𝜌𝑣𝜕𝑥

𝑣2(cid:17)

(cid:16) 1
2

+ 𝜌𝑣(𝜕𝑥𝑒) + 𝜕𝑥 ( 𝑝𝑣)

= 𝜌𝑣(𝜕𝑡 + 𝑣𝜕𝑥)𝑣 + 𝜌(𝜕𝑡 + 𝑣𝜕𝑥)𝑒 + 𝜕𝑥 ( 𝑝𝑣)

= 𝜌𝑣

(cid:16)

−

(𝜕𝑥 𝑝)

(cid:17)

+

1
𝜌

𝑝
𝜌

(𝜕𝑡 + 𝑣𝜕𝑥) 𝜌 + 𝜕𝑥 ( 𝑝𝑣)

=

𝑝
𝜌

(𝜕𝑡 + 𝑣𝜕𝑥) 𝜌 + 𝑝(𝜕𝑥𝑣) = 0,

where we make use of the definition of 𝐸, (2.4), (2.5), and (2.6).

□

Indeed, the isentropic condition and the energy conservation law are equivalent by the same

computation. A natural question arises:

What if the entropy is no longer constant in time?

Historically ([8]), when Riemann considered one dimensional fluids, he discovered that even starting

from smooth initial data, shocks can appear in finite time. Before the formation of shocks, the

solution is smooth, and therefore the energy conservation law is equivalent to the adiabatic condition

Riemann was considering. However, after the shock formation, the two are no longer equivalent to

each other. Thus, if one wants to continue the solution after the shock formation, one must choose

between energy equations and the adiabatic condition. Riemann made the wrong choice before the

18

concept of entropy was introduced. After Clausius introduced the concept of entropy, it is clear

that one should let the entropy increase across the shock boundary while maintaining the energy

conservation law. The correct jump condition across the shock boundary is the Rankine–Hugoniot

conditions.

Example 2.1.1. (RH condition) This example follows Section 3.4 in Evans’ book [10]. Consider

the partial differential equation

𝜕𝑡𝑢 + 𝜕𝑥 (𝐹 (𝑢)) = 0

in [0, ∞) × R.

(2.7)

Let 𝐶 be a regular curve cutting through [0, ∞) × R described by

𝐶 = {(𝑡, 𝑥(𝑡)) | 𝑡 ≥ 0}.

Set the left part of [0, ∞) × R to be Ω𝑙 and the right part to be Ω𝑟. In other words,

[0, ∞) × R = Ω𝑙 ∪ 𝐶 ∪ Ω𝑟 .

The formulation for weak solutions to (2.7) is as follows. Given a test function 𝑣 ∈ 𝐶1

𝑐 ( [0, ∞) × R),

we should have

−

∫ ∞

∫

(cid:16)

0

R

𝑢(𝜕𝑡𝑣) + 𝐹 (𝑢) (𝜕𝑥𝑣)

(cid:17)

𝑑𝑥𝑑𝑡 −

∫

R

𝑔𝑣𝑑𝑥 = 0,

(2.8)

where 𝑔(𝑥) = 𝑢(0, 𝑥) for all 𝑥 ∈ R. Chossing the test function 𝑣 to be compactly supported in Ω𝑙

and Ω𝑟 respectively, we conclude that

and

𝜕𝑡𝑢 + 𝜕𝑥 (𝐹 (𝑢)) = 0

in Ω𝑙

𝜕𝑡𝑢 + 𝜕𝑥 (𝐹 (𝑢)) = 0

in Ω𝑟 .

Chossing the test function 𝑣 that does not vanish on 𝐶, (2.8) gives

∬

(cid:16)

−

Ω𝑙

𝑢(𝜕𝑡𝑣) + 𝐹 (𝑢)(𝜕𝑥𝑣)

(cid:17)

𝑑𝑥𝑑𝑡 −

∬

(cid:16)

Ω𝑟

𝑢(𝜕𝑡𝑣) + 𝐹 (𝑢) (𝜕𝑥𝑣)

(cid:17)

𝑑𝑥𝑑𝑡 −

∫

R

𝑔𝑣𝑑𝑥 = 0.

19

Using the divergence theorem, we have

∫

−

𝐶

(𝑢𝑙𝑣 (cid:174)𝑛𝑡 + 𝐹 (𝑢𝑙)𝑣 (cid:174)𝑛𝑥)𝑑𝑠 +

∫

𝐶

(𝑢𝑟 𝑣 (cid:174)𝑛𝑡 + 𝐹 (𝑢𝑟)𝑣 (cid:174)𝑛𝑥)𝑑𝑠 = 0,

where (cid:174)𝑛 is the normal vector on 𝐶 pointing from Ω𝑙 to Ω𝑟, 𝑑𝑠 = √︁(𝑑𝑡)2 + (𝑑𝑥)2, 𝑢𝑙 denotes the

limit of 𝑢 from the left of 𝐶, and 𝑢𝑟 denotes the limit of 𝑢 from the right of 𝐶. This implies that the

velocity of 𝐶

is parallel to

(1, (cid:164)𝑥(𝑡))

(𝑢𝑙 − 𝑢𝑟, 𝐹 (𝑢𝑙) − 𝐹 (𝑢𝑟)).

In other words, the velocity of 𝐶 is the jump of 𝐹 (𝑢) divided by the jump of 𝑢:

(cid:164)𝑥(𝑡) =

𝐹 (𝑢𝑙) − 𝐹 (𝑢𝑟)
𝑢𝑙 − 𝑢𝑟

.

This is called Rankine-Hugoniot condition.

2.2 Relativisitic fluids

Our goal in this section is to explain the connection between relativsitc fluids and Newtonian

fluids. This section basically follows Chapter 4 of [21]. We use R1,3 as the Minkowski spacetime,

𝜌 : R1,3 → R as the fluid proper mass density, 𝜉 as the fluid velocity, a (4, 0)-tensor on the

Minkowski spacetime R1,3 with ⟨𝜉, 𝜉⟩𝜂 = −𝑐2, 𝑝 : R1,3 → R as the fluid pressure, 𝜖 as the internal
energy density (different from the specific internal energy 𝑒 = 𝜖

𝜌 from the previous section), 𝑠 as
the specific entropy, 𝑐 as the light speed, 𝜇 = 𝜌𝑐2 + 𝜖 as the energy density. The energy momentum

tensor for a perfect fluid in this section is

𝑇 =

𝜇 + 𝑝
𝑐2

𝜉 ⊗ 𝜉 + 𝑝𝜂,

where 𝜂 = −𝑑 (𝑥0)2 + 𝑑 (𝑥1)2 + 𝑑 (𝑥2)2 + 𝑑 (𝑥3)2 with 𝑥0 = 𝑐𝑡. We write 𝜉 = 𝜉 𝑎𝜕𝑎 as

𝜉 𝑎 =

√︃

1
1 − |𝑣|2
𝑐2

(𝑐, 𝑣)

20

(2.9)

where 𝑣 = (𝑣1, 𝑣2, 𝑣3). The idea is that when |𝑣| ≪ 𝑐, the relativistic conservation law for

fluids should reduce to Newtonian conservation law. The results in this section for the Minkowski

spacetime can be generalized to a general curved spacetime.

The conservation laws for relativistic fluids are

𝑑𝑖𝑣(𝜌𝜉) (cid:66) ∇𝑎 (𝜌𝜉 𝑎) = 0

𝑑𝑖𝑣(𝑇) (cid:66) ∇𝑎𝑇 𝑎𝑏 = 0.

(2.10)

(2.11)

Using (2.9), one can show that the first conservation law (2.10) reduces to (2.1) when |𝑣| ≪ 𝑐. The

second conservation law actually implies (2.2) and (2.3) as we show below. Expanding (2.11) gives

1
(∇𝜉 𝜇)𝜉 +
𝑐2
(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:124)

𝜇 + 𝑝
(∇𝑎𝜉 𝑎)𝜉
𝑐2
(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)

+

𝜇 + 𝑝
𝑐2
(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)

∇𝜉 𝜉 + Π · ∇𝑝
(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)

(cid:124)

(cid:123)(cid:122)
𝜉⊥ direction

(cid:125)

(cid:123)(cid:122)
𝜉 direction

where Π = 1

𝑐2 (𝜉 ⊗ 𝜉) + 𝜂 is the orthogonal projection to the spatial hyperplane. The first two terms
are in the 𝜉 direction, and the latter two terms are orthogonal to the 𝜉 direction. Therefore, we

arrive at

The first equation (2.12) gives

(∇𝜉 𝜇) + (𝜇 + 𝑝) (∇𝑎𝜉 𝑎) = 0

𝜇 + 𝑝
𝑐2

∇𝜉 𝜉 + Π · ∇𝑝 = 0.

∇𝜉𝜖 −

𝜖 + 𝑝
𝜌

∇𝜉 𝜌 = 0,

(2.12)

(2.13)

(2.14)

where we use 𝜇 = 𝜌𝑐2 + 𝜖 and ∇𝑎𝜉 𝑎 = − 1

𝜌 ∇𝜉 𝜌 from (2.10). This equation (2.14) is precisely the

isentropic condition along the fluid velocity 𝜉: one can rewrite (2.14) as

∇𝜉

(cid:17)

(cid:16) 𝜖
𝜌

+ 𝑝∇𝜉

(cid:17)

(cid:16) 1
𝜌

= 0,

which implies ∇𝜉 𝑠 = 0 by the first law of Thermodynamics, identifying 𝜖

𝜌 as the specific internal

energy and 1

𝜌 as the specific volume. The second equation (2.13) can be reduced to (2.3) using

𝜇 = 𝜌𝑐2 + 𝜖 and |𝑣| ≪ 𝑐.

21

2.3 Total variation: John’s technique

This section aims to prove Proposition 2.3.1. This is a more transparent version of John’s

technique (refer to [15]).

In John’s original paper, he incorporated a lot of ingredients to the

bootstrap mechanism. The equation considered there does not have source terms. We will describe

the crucial technique for shock formation in a simpler setting, and our equation involves a source

term.

2.3.1 Main equation

The main equation (analogous to (2.1), (2.2), (2.3)) for this section is

𝜕𝑡𝑢 + 𝜕𝑥 ( 𝐴(𝑢)) = ˜𝐹 (𝑢)

where 𝑢 : R × R → R2 is the unknown, and 𝐴 : R2 → R2 and ˜𝐹 : R2 → R2 are functions with 𝑢

as the input. We assume the equation satisfies the following assumption.

Assumption 1. The main equation can be reduced to the evolution equations for Riemann invari-

ants:

(𝜕𝑡 + 𝜆1𝜕𝑥)𝑣1 = 𝐹1

(𝜕𝑡 + 𝜆2𝜕𝑥)𝑣2 = 𝐹2,

where 𝑆 𝑝𝑎𝑛R{𝑢1, 𝑢2} = 𝑆 𝑝𝑎𝑛R{𝑣1, 𝑣2}, and 𝜆1 ≠ 𝜆2, 𝐹1, 𝐹2 are scalar functions of 𝑣1 and 𝑣2.
When 𝐹1 = 0 = 𝐹2, 𝑣1 and 𝑣2 are the classical Riemann invariants.

Remark. We provide a special case where the above assumption is satisfied. When the derivative

𝑑𝐴 can be decomposed to

𝑑𝐴 = 𝑃𝐷𝑃−1

with 𝐷 a diagonal matrix, and 𝑃−1 satisfying that (𝑃−1)11 = (𝑃−1)21 are functions of 𝑢1, and

(𝑃−1)12, (𝑃−1)22 are constants, one can show that the assumption holds.

Since 𝜆1 ≠ 𝜆2, we can use the characteristics to foliate the spacetime R × R, assuming the

solutions 𝑣1, 𝑣2 exist in a certain spacetime region, and therefore the functions 𝜆1, 𝜆2 of 𝑣1, 𝑣2 can

22

be regarded as functions of (𝑡, 𝑥). Regarding {0} × R as the initial slice, we define the coordinates

for characteristics as follows.

{(𝑡, 𝑋𝑖 (𝑡; 𝑧)) | 𝑡 > 0}

is the characteristic curve for (𝜕𝑡 + 𝜆𝑖𝜕𝑥) passing through the point (0, 𝑧), where 𝑖 = 1, 2 and 𝑧 ∈ R.

In other words, 𝑋𝑖 is 𝑥-coordinate of the characteristic curve, and we are using 𝑡 as the parameter

of the curve emanating from (0, 𝑧).

The crucial property is as follows. Using the characteristic viewpoints, we actually have a linear

control (not quadratic!) on the total variation of the Riemann invariants 𝑣1, 𝑣2. More specifically,

on the one hand, the evolution equations for Riemann invariants implies

(𝜕𝑡 + 𝜆1𝜕𝑥)(𝜕𝑥𝑣1) = (𝜕𝑥 𝐹1) − (𝜕𝑥𝜆1) (𝜕𝑥𝑣1).

On the other hand, the definition of 𝑋1 gives an evolution equation for

𝜕 𝑋1
𝜕𝑧

(𝜕𝑡 + 𝜆1𝜕𝑥)

(cid:17)

(cid:16) 𝜕 𝑋1
𝜕𝑧

=

𝑑
𝑑𝑡

(cid:16) 𝜕 𝑋1
𝜕𝑧

(cid:17)

=

𝜕
𝜕𝑧

(cid:16) 𝑑𝑋1
𝑑𝑡

(cid:17)

=

𝜕
𝜕𝑧

𝜆1 = (𝜕𝑥𝜆1)

(cid:16) 𝜕 𝑋1
𝜕𝑧

(cid:17)

,

where 𝑑

𝑑𝑡 = (𝜕𝑡 + 𝜆1𝜕𝑥) denotes the directional derivative along the characteristic. Combining these

two, we see that we have an evolution for the integrand of the total variation.

(𝜕𝑡 + 𝜆1𝜕𝑥)

(cid:16)

(𝜕𝑥𝑣1) ·

(cid:17)

𝜕 𝑋1
𝜕𝑧

= (𝜕𝑥 𝐹1) ·

𝜕 𝑋1
𝜕𝑧

.

It is important that the (𝜕𝑥𝜆1) terms cancel. The term −(𝜕𝑥𝜆1) (𝜕𝑥𝑣1) is expected to be the source

for shock formation since it gives a term like −(𝐷𝑣1

𝜆1) (𝜕𝑥𝑣1)2 and the equation becomes a Riccati

equation with finite blowup time if the sign is correct. What John found is that, despite this blowup

tendancy, one can still get a control on the total variation, as we explain in the following important

calculation. For 𝑡 > 0,

∫

(Γ𝑡 )

|𝜕𝑥𝑣1|𝑑𝑥 =

=

∫ 𝑧𝑅

𝑧𝐿𝐿 (Γ𝑡 )

∫ 𝑧𝑅

𝑧𝐿

(Γ0)

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

𝜕𝑥𝑣1 ·

𝜕 𝑋1
𝜕𝑧

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

𝑑𝑧

|𝜕𝑥𝑣1|𝑑𝑧 +

∫ 𝑡

∫ 𝑧𝑅

0

𝑧𝐿𝐿 (Ω𝑡 )

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

(𝜕𝑥 𝐹1) ·

𝜕 𝑋1
𝜕𝑧

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

𝑑𝑧𝑑𝑡

23

∫ 𝑧𝑅

≤

𝑧𝐿

(Γ0)

|𝜕𝑥𝑣1|𝑑𝑧 +

∬

(Ω𝑡 )

|𝐷𝑣1

𝐹1||𝜕𝑥𝑣1| + |𝐷𝑣2

𝐹1||𝜕𝑥𝑣2|𝑑𝑥𝑑𝑡.

(2.15)

This means that, if we have control on the initial total variation of 𝑣1, 𝑣2, and if |𝐷𝑣1
are pointwise uniformly bounded, we will have a closed feedback for the total variation of 𝑣1 and

𝐹1|, |𝐷𝑣2

𝐹1|

𝑣2, and then we can do the Gronwall’s inequality and run the bootstrap mechanism accordingly.

Figure 2.1 The strategy to control the total variation.

Figure 2.2 The picture for Γ𝑡, Ω𝑡.

Here we are using the notations (assuming 𝜆1 < 𝜆2)

𝑧𝐿 < 𝑧𝑅

24

Γ𝑡 = {(𝑡, 𝑥) | 𝑋2(𝑡; 𝑧𝐿) ≤ 𝑥 ≤ 𝑋1(𝑡; 𝑧𝑅)}

Ω𝑡 = {(𝜏, 𝑥) | 0 ≤ 𝜏 ≤ 𝑡, 𝑋2(𝜏; 𝑧𝐿) ≤ 𝑥 ≤ 𝑋1(𝜏; 𝑧𝑅)}.

We are also assuming the solutions exhibit homogeneous behavior outside the wave propagation

cone so that we only have to focus on the region Ω𝑡.

Assumption 2. The solution (𝑣1, 𝑣2), or equivalently (𝑢1, 𝑢2), satisfies

(𝜕𝑥𝑣1) = 0 = (𝜕𝑥𝑣2)

on (R+ × R) − Ω∞.

Proposition 2.3.1. Assumption 1 and 2 imply the control of total variation (2.15).

Remark. If our equation allows homogeneous solutions, then Assumption 2 holds if the initial

perturbation is 0 outside a compact subset Γ0, according to the finite speed of propagation property.

2.4 Piontwise blowup: Riccati equation

This section aims to establish Proposition 2.4.1. Continuing from the Assumption 1, we see a

Riccati type structure for the evolution of 𝜕𝑠𝑣1:

(𝜕𝑡 + 𝜆1𝜕𝑥)(𝜕𝑥𝑣1) + (𝜕𝑥𝜆1)(𝜕𝑥𝑣1) = (𝜕𝑥 𝐹1)

(𝜕𝑡 + 𝜆1𝜕𝑥)(𝜕𝑥𝑣1) + (𝐷𝑣1

𝜆1)(𝜕𝑥𝑣1)2 + (𝐷𝑣2

𝜆1) (𝜕𝑥𝑣1) (𝜕𝑥𝑣2) = (𝐷𝑣1

𝐹1) (𝜕𝑥𝑣1) + (𝐷𝑣2

𝐹1) (𝜕𝑥𝑣2).

Using the integral factor method for ordinary differential equations, we derive

(𝜕𝑡 + 𝜆1𝜕𝑥)(𝑒 𝑓 · 𝜕𝑥𝑣1) = −𝑒− 𝑓 (𝐷𝑣1

𝜆1) (𝑒 𝑓 · 𝜕𝑥𝑣1)2 + 𝑒 𝑓 (𝐷𝑣2

𝐹1) (𝜕𝑥𝑣2),

(2.16)

where

∫ 𝑡

𝑓 =

0 (𝑋1)

(𝐷𝑣2

𝜆1) (𝜕𝑥𝑣2) − (𝐷𝑣1

𝐹1)𝑑𝑡

is the integral over the curve (𝑡, 𝑋1(𝑡; 𝑧)) and thus depends on the initial position 𝑧 ∈ R. Regarding
the weighted derivative as a new variable 𝑦 = 𝑒 𝑓 · 𝜕𝑥𝑣1, one can see that there is a chance for 𝑦 to

25

blow up at finite time if the sign of −(𝐷𝑣1

𝜆1) is correct, based on the finite blowup time property

for a Riccati equation. This induces the following assumption.

Assumption 3. Fix 𝑧 ∈ R and let 𝑋1 denote the characteristic for (𝜕𝑡 + 𝜆1𝜕𝑥) issuing from

(𝑡, 𝑥) = (0, 𝑧). We assume there exist constants 𝑐, 𝐶, 𝜖, which may depend on 𝑧, so that along 𝑋1,

• (𝐷𝑣1

𝜆1) ≥ 𝑐 > 0

• 𝑐 ≤ 𝑒 𝑓 ≤ 𝐶

• |𝜕𝑠𝑣2| ≤ 𝜖

• |𝐷𝑣2

𝐹1| ≤ 𝐶.

Remark. In Chapter 3, the lower bounds for (𝐷𝑣1

𝜆1) and 𝑒− 𝑓 (the coefficient for (𝑒 𝑓 · 𝜕𝑥𝑣1)2) are

stronger. We also have to worry about whether the shock happens before the Big Bang blowup time

there.

Remark. We only require |𝜕𝑠𝑣2| ≤ 𝜖 along 𝑋1, not necessarily globally in space.

Therefore, if Assumption 3 holds, the weighted derivative 𝑒 𝑓 · 𝜕𝑥𝑣1, with sufficiently large initial

data 𝜕𝑥𝑣1(0, 𝑧), follows a Riccati equation (2.16) similar to

𝑑𝑦
𝑑𝑡

≥ 𝑦2,

which blows up at finite time. Since 𝑒 𝑓 has a positive lower bound, 𝜕𝑠𝑣1 also goes to infinity and

thus concludes the shock formation.

Proposition 2.4.1. Assumption 1, 2, 3 imply the shock formation within finite time.

26

CHAPTER 3

INSTABILITY OF THE BIG BANG

3.1

Introduction

Starting with the homogeneous solutions to Einstein-Euler equations derived from Chapter

1, we prove the instability of the 𝑟∗ > 0 Big Bang models under non-homogeneous, compactly

supported perturbations. For notations, see 3.3.4.

Theorem 3.1.1. Fix a background homogeneous fluid over (𝑟∗, 𝑟0] having the Big Bang sin-

gularity at 𝑟 = 𝑟∗ > 0. The asymptote of ( ˚𝜌 ˚𝛼) will determine 𝑟𝑚𝑖𝑑 ∈ (𝑟∗, 𝑟0). We re-

gard 𝑟∗, 𝑟𝑚𝑖𝑑, 𝑟0 as parameters (defined in Section 3.3.4). Fix 𝑟𝑐𝑢𝑡 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑) which is not

a parameter. There exist 𝐿𝐵 = 𝐿𝐵(𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠), 𝜖𝑎,0 = 𝜖𝑎,0(𝐿𝐵, 𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠),

𝜖0 = 𝜖0(𝐿𝐵, 𝜖𝑎,0, 𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠) so that for every compactly supported data, specified on

{𝑟 = 𝑟0}, satisfying

• 𝐸 (𝑟0) = ∫

Γ𝑟
0

|𝜕𝑠𝑣1| + |𝜕𝑠𝑣2|𝑑𝑠 < 𝜖0

(small total variation)

• ∫
Γ𝑟
0

|𝜕𝑠 ln 𝛼| + |𝜕𝑠 ln 𝛽|𝑑𝑠 < 𝜖𝑎,0 (small metric total variation)

• |𝜕𝑠𝑣2(𝑟0, 𝑠)| < 𝜖𝑎,0 for 𝑠 ∈ [−1, 0]

(small opponent)

• 𝜕𝑠𝑣1(𝑟0, − 1

2) ≥ 𝐿𝐵,

𝜕𝑠𝑣1 goes to infinity along the (𝜆1) characteristic before 𝑟 = 𝑟𝑐𝑢𝑡 (meaning somewhere in [𝑟𝑐𝑢𝑡, 𝑟0]).

Moreover

lim
𝑟𝑐𝑢𝑡 →𝑟 +
∗

𝐿𝐵(𝑟𝑐𝑢𝑡) = 0.

We construct a sequence of initial data that satisfies the above constraint in Section 3.6. Section

3.3.4 includes all the notations.

During the time 𝑟𝑐𝑢𝑡 ≤ 𝑟 ≤ 𝑟0, it is guaranteed that the total variations 𝐸 (𝑟), ∫

Γ𝑟

|𝜕𝑠 ln 𝛼| +

|𝜕𝑠 ln 𝛽|𝑑𝑠 remain small. Note that this control is valid only up to 𝑟𝑐𝑢𝑡.

27

Proof. The proof is in Section 3.5.3.

□

Remark. The theorem says that, for a family of background solutions that satisfy certain condition

involving 𝐶2, we can find a sequence of initial data that goes to background in 𝑊 1,∞ so that shock
forms before 𝑟 = 𝑟∗. In other words, this family of background solutions (modelling the Big Bang)

are unstable.

Remark. We can of course arrange things so that 𝜖0 = 𝜖𝑎,0. The reason we introduce another

notation 𝜖𝑎,0 (where 𝑎 for auxiliary) is to keep track of their contribution throughout the bootstrap

framework.

Figure 3.1 The picture for the big bang model considered in this work.

3.2 Main Equations

3.2.1 Geometry

As in [2], we consider warped product spacetimes defined as follows.

Definition 3.2.1. A warped product spacetime is a spacetime 𝑄 ×𝑟 𝐹 with the metric

˜𝑔 = 𝑔 + 𝑟 2ℎ,

28

where 𝑄 is a simply-connected, 2-dimensional Lorentzian manifold with the metric 𝑔 and 𝐹 is an

𝑛-dimensional Riemannian manifold with the metric ℎ. Here 𝑟 : 𝑄 → (0, ∞) is a positive function

on 𝑄. We further assume that 𝑟 serves as the time function for the spacetime satisfying

⟨𝑑𝑟, 𝑑𝑟⟩𝑔 < 0.

Notice that in this paper, we assume all the dynamics for the warped product spacetime are

exhibited in 𝑄, and we regard 𝐹 as a fixed fiber. Since 𝑄 is assumed to be simply-connected, we

are able to assume the metric 𝑔 has the following form

𝑔 = −𝛼𝑑𝑟 2 + 𝛽𝑑𝑠2,

where 𝛼, 𝛽 are positive functions on 𝑄. The reason is as follows. By the simply-connectedness,

we can construct the integral curves along ∇𝑟. We define another function 𝑠 on 𝑄 by setting these

integral curves to be 𝑠 = 𝑐𝑜𝑛𝑠𝑡. Thus, 𝜕𝑟 is timelike since ⟨∇𝑟, ∇𝑟⟩𝑔 = ⟨𝑑𝑟, 𝑑𝑟⟩𝑔 < 0. 𝜕𝑠 is spacelike

since 𝑄 is Lorentzian. 𝜕𝑟 is orthogonal to 𝜕𝑠 since ∇𝑟 is orthogonal to 𝑟 = 𝑐𝑜𝑛𝑠𝑡 slice.

3.2.2 Einstein-Euler equations

We consider the Einstein-Euler equations in this paper:

𝑅𝑖𝑐[ ˜𝑔] = 𝑇 −

1
𝑛

𝑡𝑟 (𝑇) ˜𝑔 +

2
𝑛

Λ ˜𝑔,

where 𝑇 = ( 𝑝 + 𝜌)𝜉 ⊗ 𝜉 + 𝑝 ˜𝑔 is the energy momentum tensor for a perfect fluid. Here 𝑝, 𝜌, 𝜉

represent the pressure, density, and fluid velocity of the fluid respectively. We further impose the

equation of state

𝑝 = 𝛾𝜌

for ultrarelativistic fluids. Here 0 < √𝛾 < 1 is the sound speed. Since the fluid veloctiy has unit
length ⟨𝜉, 𝜉⟩ = −1, we use 𝜃 to parametrize 𝜉 = 𝜉𝑟 𝜕𝑟 + 𝜉 𝑠𝜕𝑠:

√

𝛼𝜉𝑟 =

√︁

1 + 𝜃2

√︁𝛽𝜉 𝑠 = 𝜃,

29

where 𝜃 ∈ R is a scalar indicating how much the fluid velocity deviates from the time direction

𝜕𝑟. In summary, we have four unknowns in this paper: two metric components 𝛼, 𝛽 and two fluid

variables 𝜃, 𝜌.

3.2.3 Reduced Einstein Field Equations

After expanding the definition of Ricci curvature, we get the following reduced Einstein field

equations

(cid:16)√︁𝛽 · 𝑟 𝑛 𝜌 1

1+𝛾 ·

𝜕𝑟

1 + 𝜃2(cid:17)
√︁

+ 𝜕𝑠

(cid:16)√

𝛼 · 𝑟 𝑛 𝜌 1

1+𝛾 · 𝜃

(cid:17)

= 0

(cid:16)√︁𝛽 · 𝜌

𝜕𝑟

𝛾

1+𝛾 · 𝜃

(cid:17)

+ 𝜕𝑠

(cid:16)√

𝛼 · 𝜌

𝛾

1+𝛾 √︁

1 + 𝜃2(cid:17)

= 0

(𝜕𝑟 ln 𝛼) =

(cid:16)

2(1 + 𝛾)𝜃2 + 2𝛾

(cid:17)

(𝜕𝑟 ln 𝛽) =

(cid:16)

2(1 + 𝛾)𝜃2 + 2

(cid:17)

·

𝑟
𝑛

(𝜌𝛼) +

𝑛 − 1
𝑟

− 𝛼

(cid:16) 2Λ
𝑛

𝑟 −

(cid:17)

𝑆 [ℎ]
𝑛𝑟

(𝜌𝛼) −

𝑛 − 1
𝑟

+ 𝛼

(cid:16) 2Λ
𝑛

𝑟 −

(cid:17)

𝑆 [ℎ]
𝑛𝑟

·

𝑟
𝑛

with the constraint

(𝜕𝑠 ln 𝛼) = −2(1 + 𝛾) ·

𝑟
𝑛

(𝜌𝛼) ·

√
√

𝛽
𝛼

· 𝜃√︁

1 + 𝜃2.

The main unknowns are 𝜃, 𝜌, 𝛼, 𝛽 (and become 𝑣1, 𝑣2, 𝛼, 𝛽 later). The first two equations form a

hyperbolic system and describe the evolution of the fluid variables 𝜃 and 𝜌. The third and the fourth

equations describe the evolution of the metric components 𝛼 and 𝛽. The last constraint equation is

a condition for initial data and is compatible with the third equation. Once the initial data satisfies

the constraint, the constraint holds forever.

Lemma 3.2.1. The Einstein field equations are equivalent to the reduced Einstein field equations

for smooth solutions.

Proof. Assume the solution (𝜃, 𝜌, 𝛼, 𝛽) satisfies the Einstein field equations. We have

1. Conservation of energy momentum tensor, ∇𝑎𝑇𝑎𝑏 = 0.

2. Einstein equation restricted on 𝑄, 𝑅𝑖𝑐 [𝑔] − 𝑛

𝑟 ∇2𝑟 = 𝑇𝑄 − 1

𝑛 (𝑡𝑟 ˜𝑔𝑇) + 2

𝑛 Λ𝑔.

30

3. Einstein equation restricted on 𝐹, 𝑅𝑖𝑐[ℎ] −

(cid:16)

𝑟Δ𝑔𝑟 + (𝑛 − 1)⟨𝑑𝑟, 𝑑𝑟⟩𝑔

(cid:17)

ℎ = 𝑇𝐹 − 1

𝑛 (𝑡𝑟 ˜𝑔𝑇) (𝑟 2ℎ) +

𝑛 Λ(𝑟 2ℎ).
2

Here 𝑇𝑄, 𝑇𝐹 denote the restriction of the energy momentum tensor 𝑇 on the base 𝑄 and the fiber 𝐹,

respectively. Expanding the conservation law gives the first two reduced Einstein field equations.

Applying 𝑡𝑟𝑔 to the restrction on 𝑄 (simplified by using (2) on the 𝜕𝑠 ⊗ 𝜕𝑠 direction) and expanding

the restriction on 𝐹 give the third and the fourth reduced Einstein field equations. Applying the

striction on 𝑄 to the 𝜕𝑟 ⊗ 𝜕𝑠 direction gives the constraint equation for ln 𝛼.

Conversely, assume the solution (𝜃, 𝜌, 𝛼, 𝛽) satisfies the reduced Einstein field equations. De-

note the tensor 𝑅𝑖𝑐[ ˜𝑔] − 𝑇 + 1

𝑛 Λ ˜𝑔 by 𝐸𝑖𝑛𝑠𝑡𝑒𝑖𝑛. From the above argument, we know that
the third and the fourth reduced Einstein equations imply (𝐸𝑖𝑛𝑠𝑡𝑒𝑖𝑛)𝐹 = 0, where (𝐸𝑖𝑛𝑠𝑡𝑒𝑖𝑛)𝐹 is

𝑛 𝑡𝑟 (𝑇) ˜𝑔 − 2

the Einstein tensor restricted on 𝐹. To prove (𝐸𝑖𝑛𝑠𝑡𝑒𝑖𝑛)𝑄 = 0, we observe that

• The constraint reduced Einstein equation implies (𝐸𝑖𝑛𝑠𝑡𝑒𝑖𝑛)𝑄 (𝜕𝑟, 𝜕𝑠) = 0.

• The term (𝐸𝑖𝑛𝑠𝑡𝑒𝑖𝑛)𝑄 (𝜕𝑟, 𝜕𝑟) involves (𝜕2

𝑠 𝛼) and (𝜕2

𝑟 𝛽). Therefore, we can start from

the reduced Einstein equations for (𝜕𝑠 ln 𝛼) and (𝜕𝑟 ln 𝛽), take one more derivative, and

algebraically prove that (𝐸𝑖𝑛𝑠𝑡𝑒𝑖𝑛)𝑄 (𝜕𝑟, 𝜕𝑟) = 0 holds. During the process, we also need the

first two reduced Einstein equations.

• The way to prove (𝐸𝑖𝑛𝑠𝑡𝑒𝑖𝑛)𝑄 (𝜕𝑠, 𝜕𝑠) = 0 is similar.

Indeed, since both the reduced Einstein equations (with a compatible constraint equation) and

the Einstein equations are locally well-posed, they should be equivalent if one implies the other

under enough regularity conditions.

□

3.2.4 Riemann invariants

If we perform a matrix diagonalization on the hyperbolic system, the first two equations in the

reduced Einstein field equations, we are able to get the following evolution equations for Riemann

invariants 𝑣1 and 𝑣2 (refer to [15]). Notice that they are two evolution equations for two quantities

31

𝑣1 and 𝑣2 along two different directions (𝜕𝑟 + 𝜆1𝜕𝑠 and 𝜕𝑟 + 𝜆2𝜕𝑠 directions).

(𝜕𝑟 + 𝜆1𝜕𝑠)

(𝜕𝑟 + 𝜆2𝜕𝑠)

1 + 𝜃2 + 𝜃(cid:1) +

√𝛾 ln (cid:0)√︁

(cid:16) 1
2
(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: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:124)

ln 𝜌 +

ln 𝛼
(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: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)

1
2(1 + 𝛾)

1 + 𝜃2 + 𝜃(cid:1) −

√𝛾 ln (cid:0)√︁

(cid:16) 1
2
(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: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:124)

ln 𝜌 −

ln 𝛼
(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: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)

1
2(1 + 𝛾)

1
2(1 + 𝛾)
(cid:123)(cid:122)
𝑣1

1
2(1 + 𝛾)
(cid:123)(cid:122)
𝑣2

(cid:17)

= 𝐹1

(cid:17)

= 𝐹2.

(cid:125)

(cid:125)

Here the eigenvalues, or the speeds of propagation, are

√
√

𝛼
𝛽

·

𝜆1 =

and the source terms are

√

(1 − 𝛾)𝜃

1 + 𝜃2 +
1 + (1 − 𝛾)𝜃2

√𝛾

,

𝜆2 =

√
√

𝛼
𝛽

·

√

(1 − 𝛾)𝜃

1 + 𝜃2 −
1 + (1 − 𝛾)𝜃2

√𝛾

,

(cid:17)

(cid:17)

𝐹1 =

(cid:16)

−

1
2

+

𝛾
1 + 𝛾

(cid:17)

·

𝑟
𝑛

(𝜌𝛼)

(cid:16)

−

+

𝑛 − 1
𝑟

+ 𝛼 (cid:0) 2
𝑛

Λ𝑟 −

𝑆 [ℎ]
𝑛𝑟

(cid:16)

(cid:1)(cid:17)

·

−

√

√𝛾

(1 − 𝛾)𝜃
4

1 + 𝜃2 +
√𝛾 (cid:0)1 + (1 − 𝛾)𝜃2(cid:1) −

(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)

1
2(1 + 𝛾)
(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)
2(1+𝛾) ± 1
√𝛾

4

(cid:123)(cid:122)

bounded between− 1

(cid:124)

+

𝑛
2𝑟

·

√
1 + 𝜃2 (cid:0)√𝛾𝜃 −

√

1 + 𝜃2(cid:1)

1 + (1 − 𝛾)𝜃2

𝐹2 =

(cid:16) 1
2

−

𝛾
1 + 𝛾

(cid:17)

·

𝑟
𝑛

(𝜌𝛼)

(cid:16)

−

+

𝑛 − 1
𝑟

+ 𝛼 (cid:0) 2
𝑛

Λ𝑟 −

𝑆 [ℎ]
𝑛𝑟

(cid:16)

(cid:1)(cid:17)

·

−

√

√𝛾

(1 − 𝛾)𝜃
1 + 𝜃2 −
√𝛾 (cid:0)1 + (1 − 𝛾)𝜃2(cid:1) +
4

(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:124)

(cid:123)(cid:122)
bounded between

1
2(1 + 𝛾)
(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)
2(1+𝛾) ± 1
√𝛾

1

4

+

𝑛
2𝑟

·

√
1 + 𝜃2 (cid:0)√𝛾𝜃 +

√

1 + 𝜃2(cid:1)

1 + (1 − 𝛾)𝜃2

.

For convenience, we record these Riemann invariants as a definition.

Definition 3.2.2. We use

𝑣1 =

2

1

√𝛾 ln (cid:0)√︁

1 + 𝜃2 + 𝜃(cid:1) +

1
2(1 + 𝛾)

ln 𝜌 +

1
2(1 + 𝛾)

ln 𝛼

32

𝑣2 =

1

√𝛾 ln (cid:0)√︁

2

1 + 𝜃2 + 𝜃(cid:1) −

1
2(1 + 𝛾)

ln 𝜌 −

1
2(1 + 𝛾)

ln 𝛼

to denote the Riemann invariants. They encode the information about the fluid variables 𝜃 and 𝜌.

The ln 𝛼 term is only for convenience and does not play a big role.

Notice that for homogeneous fluids, 𝜃 = 0, which implies 𝜆1 =

√𝛾.
The metric components 𝛼, 𝛽 for the homogeneous fluids have the asymptotes (refer to Section 1.2)

√𝛾 and 𝜆2 = −

𝛼
𝛽 ·

𝛼
𝛽 ·

√
√

√
√

𝛼 ≈ (𝑟 − 𝑟∗)

2𝛾
1−𝛾 ,

𝛽 ≈ (𝑟 − 𝑟∗)

2
1−𝛾

as 𝑟 → 𝑟 +

∗ . Therefore, since 0 < 𝛾 < 1, the speeds 𝜆1, 𝜆2 go to infinity when 𝑟 approaches 𝑟∗

(Figure 3.2).

Figure 3.2 The speeds of propagation go to infinity close to 𝑟∗ for homogeneous fluids.

3.3 Strategy

The fluid density 𝜌 and the metric components 𝛼, 𝛽 for the background homogeneous fluid

exhibit singular behavior as 𝑟 → 𝑟 +
∗ :

𝛼 ≈ (𝑟 − 𝑟∗)

2𝛾
1−𝛾 ,

𝛽 ≈ (𝑟 − 𝑟∗)

2
1−𝛾 ,

𝜌 ≈ (𝑟 − 𝑟∗)− 1+𝛾
1−𝛾 .

Thus, after initial nonhomogeneous perturbation, we may expect these variables exhibit a similar,

if not worse, singular behavior. The main difficulty is how to control quantities when the time 𝑟 is

33

close to 𝑟∗. The solution is that we avoid this issue by setting an 𝑟𝑐𝑢𝑡 with

𝑟∗ < 𝑟𝑐𝑢𝑡 < 𝑟0.

The main philosophy is that, the equations may have unbounded coefficients in the time interval

(𝑟∗, 𝑟0], but for each fix 𝑟𝑐𝑢𝑡 ∈ (𝑟∗, 𝑟0), the equations are expected to have large but bounded
coefficients in [𝑟𝑐𝑢𝑡, 𝑟0]. Our hope is to argue that the solution exists in 𝑊 1,1 during the time
[𝑟𝑐𝑢𝑡, 𝑟0] but the derivative blows up pointwise before 𝑟𝑐𝑢𝑡 (meaning at some time in [𝑟𝑐𝑢𝑡, 𝑟0]) for

sufficiently large initial spatial derivative ≥ 𝐿𝐵, where 𝐿𝐵 is a lower bound depending on 𝑟𝑐𝑢𝑡. We

next claim that

lim
𝑟𝑐𝑢𝑡 →𝑟∗

𝐿𝐵(𝑟𝑐𝑢𝑡) = 0

to establish the instability and conclude our main theorem. Note that this 𝑟𝑐𝑢𝑡 trick only makes

sense when we aim to prove the instability: it is enough to construct a sequence of initial data that

form shocks in (𝑟∗, 𝑟0], where we label the sequence by 𝑟𝑐𝑢𝑡, and the above 𝐿𝐵 → 0 fact implies

the instability for small initial non-homogeneous perturbation.

It turns out the above hope is true, and we actually use the total variation to claim the solu-
tion exists in 𝑊 1,1 during [𝑟𝑐𝑢𝑡, 𝑟0]. Heuristically, we are trying to argue that ∥𝜕𝑠𝑣1∥ 𝐿1 ({𝑟=𝑟}) +
∥𝜕𝑠𝑣2∥ 𝐿1 ({𝑟=𝑟}) remains small during [𝑟𝑐𝑢𝑡, 𝑟0], while 𝜕𝑠𝑣1 goes to infinity pointwise somewhere in
[𝑟𝑐𝑢𝑡, 𝑟0]. It is a basic fact in analysis that a function can go to infinity at one point while remaining

small 𝐿1 norm, and it is the main idea in John’s work [15]. Our proof thus consists of two parts:

total variation (for 𝑣1, 𝑣2) control and pointwise derivative (𝜕𝑠𝑣1, 𝜕𝑠𝑣2) control. We describe the

strategy for each part separately. Recall that our unknowns are 𝑣1, 𝑣2 (Riemann invariants, involving

fluid variables) and 𝛼, 𝛽 (metric components).

3.3.1 Total variation

Our total variation is defined as

𝐸 (𝑟) =

∫

Γ𝑟

(cid:0)|𝜕𝑠𝑣1| + |𝜕𝑠𝑣2|(cid:1) 𝑑𝑠

34

for 𝑟∗ < 𝑟 ≤ 𝑟0 (Figure 3.7). The idea is to use the trick in John’s work [15] to argue that, if

the initial total variation is small, it remains small up to 𝑟 = 𝑟𝑐𝑢𝑡. The idea of John’s trick is to

consider the evolution of the integrand of the total variation (Figure 3.3). It turns out that we can

use Gronwall’s inequality to argue that, if the initial perturbation has small total variation less than

𝜖0, it remains small in [𝑟𝑐𝑢𝑡, 𝑟0]. Note that 𝜖0 depends on 𝑟𝑐𝑢𝑡.

Figure 3.3 John’s trick to control the total variation.

3.3.2 Pointwise 𝜕𝑠𝑣 behavior

In order to get the control on the pointwise behavior of 𝜕𝑠𝑣1, a natural way is to take a spatial

derivative on (𝜕𝑟 + 𝜆1𝜕𝑠)𝑣1 = 𝐹1. This yields

(𝜕𝑟 + 𝜆1𝜕𝑠) (𝜕𝑠𝑣1) + (𝜕𝑠𝜆1) (𝜕𝑠𝑣1) = (𝜕𝑠𝐹1).

Note that 𝜆1 = 𝜆1(𝜃, 𝛼, 𝛽) = 𝜆1(𝑣1, 𝑣2, 𝛼, 𝛽) and 𝐹1 = 𝐹1(𝜃, 𝜌, 𝛼) = 𝐹1(𝑣1, 𝑣2, 𝛼).

It is worth

noting that the source terms 𝐹1 and 𝐹2 do not depend on 𝛽. One can expect that (𝜕𝑠𝜆1) will

generate (𝜕𝑠𝑣1), (𝜕𝑠𝑣2), (𝜕𝑠𝛼), and (𝜕𝑠 𝛽). Thus, this can be regarded as a complicated Riccati

equation since it involves (𝜕𝑠𝑣1)2 term (refer to Lemma 3.5.1). This Riccati structure is the main

mechanism to generate shocks in John’s work [15], but his equations differ from our system by

the following features: his Riccati equation does not have linear terms, and his coefficents are all

bounded. Fortunately, our goal is also different from his. John proved the shock formation for all

35

𝐶2-small data, while our goal is to prove shock formation for some sequence of initial data that

converges to 0 in 𝑊 1,1.

Going back to our evolution for (𝜕𝑠𝑣1), there are several difficulties from (𝜕𝑠𝜆1) (and also from

(𝜕𝑠𝐹1)):

• how to deal with linear (𝜕𝑠𝑣1) term,

• how to deal with linear (𝜕𝑠𝑣2) term, and

• how to deal with (𝜕𝑠 𝛽) term.

Our solution for the first issue is to use the integral factor method as in the ordinary differential

equation context (refer to Lemma 3.5.1). The way to deal with (𝜕𝑠𝑣2) term is to use a bootstrap

argument to claim it remains small in a certain region (refer to Lemma 3.5.9). The way to resolve

the (𝜕𝑠 𝛽) issue is our main contribution (refer to Lemma 3.5.2), and we describe our method as

follows.

Recall that 𝛼, 𝛽 are our metric components. The reason why we only identify the issue for (𝜕𝑠 𝛽)

but not for (𝜕𝑠𝛼) is because the feature of Einstein field equations under symmetry assumption

naturally lacks an equation (constraint equation in our context) for one metric component. One can

see this in our reduced Einstein field equations, where we have an equation for (𝜕𝑠 ln 𝛼) but not

for (𝜕𝑠 ln 𝛽). This is a difficulty in our argument since we only have control for quantities without

derivative, so if there is no equation for (𝜕𝑠 ln 𝛽), there will be no control on (𝜕𝑠 ln 𝛽).

Our innovative way is to use the divergence structure of the hyperbolic system to control

∫

(𝜆1)

𝜆1(𝜕𝑠 ln 𝛽)𝑑𝑟.

Note that this is the term that (𝜕𝑠 ln 𝛽) appears in the integral factor (refer to Lemma 3.5.1). Recall

that our first equation in the reduced Einstein field equations is

(cid:16) √

𝛼 · 𝑟 𝑛 𝜌 1
(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)
𝐵

1+𝛾 · 𝜃
(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)

(cid:124)

(cid:17)

= 0.

𝜕𝑟

(cid:16) √︁𝛽 · 𝑟 𝑛 𝜌 1
1+𝛾 ·
(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:124)
𝐴

√︁

1 + 𝜃2
(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)

(cid:17)

+ 𝜕𝑠

36

If we denote the first parenthesis to be 𝐴 and the second parenthesis to be 𝐵, we have the following

calculation.

[ln 𝐴 − ln 𝐴(𝑟0)] (𝜆1) =

∫ 𝑟

𝑟0 (𝜆1)

(𝜕𝑟 𝐴) + 𝜆1(𝜕𝑠 𝐴)
𝐴

𝑑𝑟 =

∫ 𝑟

𝑟0 (𝜆1)

−(𝜕𝑠𝐵) + 𝜆1(𝜕𝑠 𝐴)
𝐴

𝑑𝑟,

where [·] (𝜆1) and ∫ 𝑟
from (𝜕𝑟 + 𝜆1𝜕𝑠). Note that (𝜕𝑠 𝐴) involves the term we aim to estimate since 𝐴 includes

𝑑𝑟 denote the corresponding computation along the characteristic generated

√

𝛽, while

𝑟0 (𝜆1)

(𝜕𝑠𝐵) does not include 𝛽. It turns out the integral on the above right hand side can be simplified
to ∫

𝜆1(𝜕𝑠 ln 𝛽) plus the integral of a function of (𝜕𝑠 ln 𝛼) and (𝜕𝑠𝑣2) (but no (𝜕𝑠𝑣1)). We have

(𝜆1)

an equation for (𝜕𝑠 ln 𝛼), and we can actually integrate the (𝜕𝑠𝑣2) term. As a remark, one can
alternatively apply John’s trick to control ∫

|𝜕𝑠𝑣2|𝑑𝑟 although we did not choose this way. We

(𝜆1)

can also estimate [ln 𝐴 − ln 𝐴(𝑟0)] (𝜆1) using the total variation and 𝜕𝑟 integral curves. For more

details, refer to Lemma 3.5.2.

After estimating the integral factor and other error terms, we have a precise behavior (up to

a constant) of the integral factor. More specifically, we know how the integral factor depends on

𝑟 ∈ (𝑟∗, 𝑟𝑐𝑢𝑡) (refer to Proposition 3.5.1). We thus aruge that lim𝑟𝑐𝑢𝑡 →𝑟 +

∗

𝐿𝐵(𝑟𝑐𝑢𝑡) = 0 and prove our

main theorem in Section 3.5.3.

3.3.3 Method of characteristics

Our proof is based on the method of characteristics. We define the notation for characteristics

and relevant regions in this section.

Let 𝑋1(𝑟; 𝑧) be the function so that 𝑠 = 𝑋1(𝑟; 𝑧) is the characteristic tangent to (𝜕𝑟 + 𝜆1𝜕𝑠) and

starting from the point (𝑟, 𝑠) = (𝑟0, 𝑧). Similarly 𝑋2(𝑟; 𝑧) is the function so that 𝑠 = 𝑋2(𝑟; 𝑧) is

the characteristic tangent to (𝜕𝑟 + 𝜆2𝜕𝑠) and starting from the point (𝑟, 𝑠) = (𝑟0, 𝑧). Therefore, for
example, the integral notation ∫ 𝑟

𝑓 𝑑𝑟 in the previous section means ∫ 𝑟
𝑟0

𝑓 (𝑟, 𝑋1(𝑟; 𝑧))𝑑𝑟. We

𝑟0 (𝜆1)

may denote the characteristic 𝑠 = 𝑋1(𝑟; 𝑧) itself by 𝑋1 or (𝜆1) interchangeably.

Since we will estimate quantities along different characteristics, for a given point 𝑃 in the

spacetime 𝑄, we define 𝑃1 to be the point on the initial slice that is connected to 𝑃 by the

characteristic 𝑋1. Similarly we define 𝑃2 to be the point on the initial slice that is connected to 𝑃

37

by the characteristic 𝑋2 (see Figure 3.4). We also define 𝑃0 to be the point on the initial slice that

is connected to 𝑃 by the vertical straight line tangent to 𝜕𝑟 (see Figure 3.5).

In addition to the curves capturing the evolution of the unknowns, we define another notation

for total variation control. Given a point 𝑃 in the spacetime 𝑄, we define ˚𝑃 to be the point on the

same time slice as 𝑃 but lying in the background homogeneous fluid (HF) region (see Figure 3.6).

For any quantity 𝑞, we denote 𝑞( ˚𝑃) by ˚𝑞.

Figure 3.4 Characteristics along (𝜕𝑟 + 𝜆1𝜕𝑠), (𝜕𝑟 + 𝜆2𝜕𝑠).

Figure 3.5 The grey vertical line is the integral curve for 𝜕𝑟.

38

Figure 3.6 The green horizontal line denotes a constant time slice.

3.3.4 Notation

We give the definition and physical meaning of all the variables we will use in this paper.

• (𝑟, 𝑠) = (𝑡𝑖𝑚𝑒, 𝑠𝑝𝑎𝑐𝑒) are independent variables.

◦ 𝑟0 is the initial time for the perturbation.

◦ 𝑟∗ > 0 is the blowup time for background homogeneous metric. See Figure 3.1.

◦ 𝑟𝑚𝑖𝑑 ∈ (𝑟∗, 𝑟0) is to capture the asymptote for ( ˚𝜌 ˚𝛼). See Figure 3.10.

◦ 𝑟𝑐𝑢𝑡 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑) is the auxiliary time introduced in this paper.

• (𝜃, 𝜌, 𝛼, 𝛽) are the main unknowns, which are functions of 𝑟 and 𝑠.

◦ (𝜃, 𝜌) are fluid variables, where 𝜃 is the angle between 𝜉 and 𝜕𝑟 (measuring how 𝜉

deviates from 𝜕𝑟), and 𝜌 > 0 is the density of the fluid.

◦ (𝛼, 𝛽) are metric components, where 𝛼 > 0 and 𝛽 > 0.

◦ On the background homogeneous fluid, these unknowns satisfy

˚𝛼(𝑟) = 0,

lim
𝑟→𝑟 +
∗

˚𝛽(𝑟) = 0

lim
𝑟→𝑟 +
∗

39

(cid:17)

(cid:16) ˚𝛼
˚𝛽

lim
𝑟→𝑟 +
∗

= ∞

˚𝜃 = 0

over (𝑟∗, 𝑟0]

˚𝜌(𝑟) = ∞.

lim
𝑟→𝑟 +
∗

• 0 < √𝛾 < 1 is a parameter representing the sound speed.

• 𝜉 is the fluid velocity satisfying ⟨𝜉, 𝜉⟩𝑔 = −1.

• 𝑔 = −𝛼𝑑𝑟 2 + 𝛽𝑑𝑠2 is the metric for the 2-dimensional Lorentzian manifold 𝑄.

• parameters = parameters(background profile, 𝑟𝑚𝑖𝑑, 𝑟0, 𝛾, 𝑛, Λ, 𝑆 [ℎ]) where background pro-

file includes 𝑟∗, 𝜇∗, 𝜏∗, which are determined by a Big Bang background solution.

• We can also regard (𝑣1, 𝑣2, 𝛼, 𝛽) as our main unknowns, where

𝑣1 =

𝑣2 =

2

2

1
√𝛾 ln |

√︁

1 + 𝜃2 + 𝜃| +

1
2(1 + 𝛾)

ln(𝜌𝛼),

1
√𝛾 ln |

√︁

1 + 𝜃2 + 𝜃| −

1
2(1 + 𝛾)

ln(𝜌𝛼)

are Riemann invariatnts.

• For 𝜆1, 𝜆2, 𝐹1, 𝐹2, refer to Section 3.2.4.

• 𝑠 = 𝑋1(𝑟; 𝑧) denotes the characteristic starting from (𝑟, 𝑠) = (𝑟0, 𝑧) along the 𝜆1 direction.

Similarly, 𝑠 = 𝑋2(𝑟; 𝑦) denotes the characteristic starting from (𝑟, 𝑠) = (𝑟0, 𝑦) along the 𝜆2

direction. See Figure 3.4 and Figure 3.9.

• 𝜖, 𝜖𝑎 control the size of total variation over [𝑟𝑐𝑢𝑡, 𝑟0].

• 𝜖0, 𝜖𝑎,0 control the size of total variation on the initial slice.

• 𝑀 generally means a constant that depends on 𝑟𝑐𝑢𝑡. Usually 𝑀 = 𝑀 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠).

• 𝐶 generally means a constant that does not depend on 𝑟𝑐𝑢𝑡. Usually 𝐶 = 𝐶 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠).

40

• 𝐸 (𝑟) denotes the total variation on the {𝑟 = 𝑟} time slice and is defined by

𝐸 (𝑟) =

∫

{𝑟=𝑟 }

|𝜕𝑠𝑣1| + |𝜕𝑠𝑣2|𝑑𝑠.

• For a 𝑃 in the spacetime, define 𝑃1, 𝑃2 in Figure 3.4, 𝑃0 in Figure 3.5, ˚𝑃 in Figure 3.6.

• ˚𝑞 denotes 𝑞( ˚𝑃).

• Γ𝑟 and Ω𝑟 are defined in Figure 3.7. 𝑧𝐿, 𝑧𝐿𝐿, 𝑧𝑅, 𝑧𝑅𝑅 are defined in Figure 3.8. Ω𝑟𝑐𝑢𝑡 ,𝑧 𝑀 is

defined in Lemma 3.5.9.

• 𝛿1, 𝛿2 depend on parameters, including 𝑟∗ and 𝑟𝑚𝑖𝑑

𝛿1 (cid:66) 1
1 + 𝛾
𝛿2 (cid:66) 1
1 + 𝛾

−

−

(1.1) (1 − 𝛾)
2(1 + 𝛾)
(1 − 𝛾)
(2.2) (1 + 𝛾)

·

𝑟𝑚𝑖𝑑
𝑟∗

> 0,

> 0

serving as exponents of the (cid:0) 1
𝑟−𝑟∗

(cid:1) term in the integral factor. See Proposition 3.5.1.

3.4 Control of total variations

Proposition 3.4.1. Fix 𝑟𝑐𝑢𝑡 ∈ (𝑟∗, 𝑟0] and fix 0 < 𝜖 < 1. There exists an

𝜖0 = 𝜖0(𝑟𝑐𝑢𝑡, 𝜖, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

so that the total variation always satisfies

𝐸 (𝑟) =

∫

Γ𝑟

|𝜕𝑠𝑣1| + |𝜕𝑠𝑣2|𝑑𝑠 ≤ 𝜖

for 𝑟 ∈ [𝑟𝑐𝑢𝑡, 𝑟0]

as long as the initial data satisfies the Initial Data Assumption 1. Moreover,

√

|𝜃| ≤ 2

𝛾𝜖

| ln(𝜌𝛼) − ln( ˚𝜌 ˚𝛼)| ≤ (1 + 𝛾)𝜖

sup | ln 𝛼 − ln ˚𝛼| + sup | ln 𝛽 − ln ˚𝛽| ≤ 𝜖 .

In particular, |𝜃|, 𝜌, | ln 𝛼|, and | ln 𝛽| stay finite over the region [𝑟𝑐𝑢𝑡, 𝑟0].

41

We postpone the proof to the end of Section 4.

The reason we choose 𝐸 to denote the total variation is that we are going to bound 𝐿∞-norm of

several quantities by this total variation 𝐸 and run a bootstrap argument. This is similar to the role

of energy in the wave equation context where we try to build 𝐿∞-𝐿2 estimates. In the following

lemma, we explain how we bound the 𝐿∞-norm of our unknowns.

Lemma 3.4.1. We estimate 𝜃,

√

1 + 𝜃2, ln(𝜌𝛼) − ln( ˚𝜌 ˚𝛼), and ln(𝛼) − ln( ˚𝛼) by the total variation

𝐸 (𝑟). For ln(𝛽) − ln( ˚𝛽), we use sup 𝐸 where sup denotes supΩ𝑟 and Ω𝑟 is defined in the Figure
3.7. We use ˚𝑞 to dentoe 𝑞( ˚𝑃).

Figure 3.7 The picture for Γ𝑟 and Ω𝑟.

Proof. For 𝜃, we have (since 𝜃 ( ˚𝑃) = 0)

|𝜃| = |𝜃 (𝑃) − 𝜃 ( ˚𝑃)|

≤

≤

∫

|𝜕𝑠𝜃|𝑑𝑠

Γ𝑟

∫

√

Γ𝑟

𝛾√︁

1 + 𝜃2 (cid:0)|𝜕𝑠𝑣1| + |𝜕𝑠𝑣2|(cid:1) 𝑑𝑠

42

√

≤

𝛾 · sup

√︁

1 + 𝜃2

∫

Γ𝑟

(cid:0)|𝜕𝑠𝑣1| + |𝜕𝑠𝑣2|(cid:1) 𝑑𝑠

√

≤

𝛾 · sup

√︁

1 + 𝜃2 · 𝐸 (𝑟)

where Γ𝑟 denotes {𝑟 = 𝑟} slice inside of the sound cone as in Figure 3.7 and sup denotes supΩ𝑟 .
Here 𝑟 is the time coordinate of 𝑃. Note that

√︁

1 + 𝜃2 ≤ 1 + sup |𝜃| ≤ 1 +

sup

√

𝛾 · sup

√︁

1 + 𝜃2 · 𝐸.

For ln(𝜌𝛼) − ln( ˚𝜌 ˚𝛼), we have

| ln(𝜌𝛼) − ln( ˚𝜌 ˚𝛼)| = | ln(𝜌𝛼) (𝑃) − ln(𝜌𝛼) ( ˚𝑃)|

∫

≤

Γ𝑟

|𝜕𝑠 ln(𝜌𝛼)|𝑑𝑠

≤ (1 + 𝛾)

∫

Γ𝑟

(cid:0)|𝜕𝑠𝑣1| + |𝜕𝑠𝑣2|(cid:1) 𝑑𝑠

≤ (1 + 𝛾)𝐸 (𝑟).

For ln 𝛼 − ln ˚𝛼, we have

| ln 𝛼 − ln ˚𝛼| = | ln 𝛼(𝑃) − ln 𝛼( ˚𝑃)|

≤

=

≤

∫

Γ𝑟

∫

Γ𝑟

|𝜕𝑠 ln 𝛼|𝑑𝑠

2(1 + 𝛾)
𝑛

· 𝑟 (𝜌𝛼) ·

√
√

𝛽
𝛼

· |𝜃|

√︁

1 + 𝜃2𝑑𝑠

2(1 + 𝛾)
𝑛

· 𝑟 sup(𝜌𝛼) · sup

√
√

𝛽
𝛼

· sup |𝜃| sup

√︁

1 + 𝜃2 · 𝑊 (𝑟)

#
≤

2(1 + 𝛾)
𝑛

· 𝑟 · ( ˚𝜌 ˚𝛼) ·

· 𝑒(1+𝛾)𝐸 · 𝑒ln

√

√

𝛽−ln

√

√

˚𝛼−ln

˚𝛽 · 𝑒ln

𝛼 · sup |𝜃| sup

√︁

1 + 𝜃2 · 𝑊 (𝑟)

≤ 𝐶𝛼 · 𝑒(1+𝛾)𝐸 · 𝑒ln

√

𝛽−ln

˚𝛽 · 𝑒ln

√

√

˚𝛼−ln

𝛼 · sup |𝜃| sup

√︁

1 + 𝜃2 · 𝑊 (𝑟)

√︃

˚𝛽

√

˚𝛼
√

where 𝑊 (𝑟) = ∫

Γ𝑟

#
≤, and

for

1𝑑𝑠 denotes the width for {𝑟 = 𝑟 }, we use the above ln(𝜌𝛼) − ln( ˚𝜌 ˚𝛼) estimate

𝐶𝛼 = 𝐶𝛼 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠).

43

Notice that the sup |𝜃| on the right hand side is the key to make sure the right hand side is small.

For ln 𝛽 − ln ˚𝛽, we have

| ln 𝛽 − ln ˚𝛽| = | ln 𝛽(𝑃) − ln 𝛽( ˚𝑃)|

|𝜕𝑠 ln 𝛽|𝑑𝑠

|𝜕𝑠 ln 𝛽|𝑑𝑠 +

∬

Ω𝑟

|𝜕𝑟 𝜕𝑠 ln 𝛽|𝑑𝑟𝑑𝑠

|𝜕𝑠 ln 𝛽|𝑑𝑠

≤

≤

≤

∫

Γ𝑟

∫

Γ𝑟
0

∫

Γ𝑟
0
∬

+

4(1 + 𝛾)
𝑛

· |𝜃||𝜕𝑠𝜃| · 𝑟 (𝜌𝛼) +

(cid:16) 2(1 + 𝛾)
𝑛

· 𝜃2 +

(cid:17)

2
𝑛

· 𝑟 |𝜕𝑠 (𝜌𝛼)|

Ω𝑟

+ 𝛼(𝜕𝑠 ln 𝛼)

(cid:16) 2
𝑛

Λ𝑟 −

(cid:17)

𝑆 [ℎ]
𝑛𝑟

𝑑𝑟𝑑𝑠

∫

#
≤

Γ𝑟
0

|𝜕𝑠 ln 𝛽|𝑑𝑠

+

4(1 + 𝛾)
𝑛

· (sup |𝜃|) · (𝑟0 − 𝑟) ·

(cid:16)

· sup

𝑟 ( ˚𝜌 ˚𝛼) · 𝑒ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼)(cid:17)
(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)

(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:124)

(cid:123)(cid:122)
𝑟 (𝜌𝛼)

√︁
𝛾 sup
(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:124)

1 + 𝜃2 · sup 𝐸
(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:125)
∫ |𝜕𝑠𝜃|𝑑𝑠

+

(cid:16) 2(1 + 𝛾)
𝑛

· (cid:0) sup 𝜃(cid:1) 2 +

(cid:17)

2
𝑛

· 𝑟0(𝑟0 − 𝑟) (1 + 𝛾) sup(𝜌𝛼) · sup 𝐸
(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)

(cid:124)

(cid:123)(cid:122)
∫ |𝜕𝑠 (𝜌𝛼)|𝑑𝑠

+ sup 𝛼 ·

· 𝑟0 sup(𝜌𝛼) · sup

(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:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)

2(1 + 𝛾)
𝑛

(cid:124)

(cid:123)(cid:122)
(𝜕𝑠 ln 𝛼)

√
1 + 𝜃2
· sup |𝜃| sup
√
(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:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)

𝛽
𝛼

√︁

(cid:125)

· sup

(cid:16) 2
𝑛

Λ𝑟 −

(cid:17)

𝑆 [ℎ]
𝑛𝑟

· (𝑟0 − 𝑟) sup 𝑊

∫

≤

(initial slice)

|𝜕𝑠 ln 𝛽|𝑑𝑠

+ 𝑀𝛽 · sup |𝜃| sup

√︁

1 + 𝜃2 · sup 𝐸 · 𝑒sup | ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼)|

44

+ 𝑀𝛽

(cid:16)

1 + (sup |𝜃|)2(cid:17)

· 𝑒sup | ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼)| · sup 𝐸

+ 𝐶𝛽 · 𝑒sup | ln 𝛼−ln ˚𝛼| · 𝑒sup | ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼)| · 𝑒sup | ln

√

√

𝛽−ln

˚𝛽| · 𝑒sup | ln

√

˚𝛼−ln

√

𝛼|

· sup |𝜃| sup

√︁

1 + 𝜃2 · sup 𝑊

where for

#
≤ we rewrite 𝜕𝑠 (𝜌𝛼) as (𝜌𝛼)(𝜕𝑠 ln(𝜌𝛼)) and apply the above estimate to argue

∫

Γ𝑟′

sup
𝑟 ≤𝑟 ′≤𝑟0

|𝜕𝑠 (ln(𝜌𝛼))|𝑑𝑠 ≤ (1 + 𝛾) sup 𝐸.

In addition, we use sup 𝑊 to denote sup𝑟 ≤𝑟 ′≤𝑟0

𝑊 (𝑟′). In the last inequality, we have that

𝑀𝛽 = 𝑀𝛽 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

goes to infinity at the rate

1
𝑟𝑐𝑢𝑡 −𝑟∗

as 𝑟𝑐𝑢𝑡 → 𝑟∗ and

𝐶𝛽 = 𝐶𝛽 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠).

We also decompose sup(𝜌𝛼) to sup( ˚𝜌 ˚𝛼) · 𝑒sup | ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼)| and absorb sup( ˚𝜌 ˚𝛼) to 𝑀𝛽 or 𝐶𝛽

constants.

□

Lemma 3.4.2. We derive a more explicit form of 𝑊 (𝑟). In particular,

sup 𝑊 = sup 𝑊 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

goes to infinity as 𝑟𝑐𝑢𝑡 → 𝑟∗.

Proof. Let

𝑧𝐿 = inf{𝑧 ∈ R | initial perturbation(𝑟0, 𝑧) ≠ 0}

𝑧𝑅 = sup{𝑧 ∈ R | initial perturbation(𝑟0, 𝑧) ≠ 0}.

We have

𝑊 = 𝑋2(𝑟; 𝑧𝑅) − 𝑋1(𝑟; 𝑧𝐿)

45

=

=

∫ 𝑟

𝑟0

∫ 𝑟

𝑟0

𝜆2(𝑟, 𝑋2(𝑟; 𝑧𝑅))𝑑𝑟 −

𝜆1(𝑟, 𝑋1(𝑟; 𝑧𝐿))𝑑𝑟

√

˚𝛼
√︃

˚𝛽

√

· (cid:0) −

𝛾(cid:1) 𝑑𝑟 −

∫ 𝑟

𝑟0

· (cid:0)√

𝛾(cid:1) 𝑑𝑟

∫ 𝑟

𝑟0
√

˚𝛼
√︃

˚𝛽

√

𝛾

= 2

∫ 𝑟0

𝑟

√

˚𝛼
√︃

˚𝛽

𝑑𝑟.

Therefore,

goes to infinity as 𝑟𝑐𝑢𝑡 → 𝑟∗.

sup 𝑊 = sup 𝑊 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

□

We introduce the backbone of our total variation control: John’s trick [15]. This trick is to

use the evolution of the integrand of total variation and the Gronwall’s inequality to get a control

on the total variation. The main philosophy is that, although the evolution of (𝜕𝑠𝑣2) is a Riccati
equation which does not give a good control, the evolution of (𝜕𝑠𝑣2) · 𝜕 𝑋2

𝜕𝑧 (the integrand of the total

variation) gives a linear equation and allows one to apply the Gronwall’s inequality.

Lemma 3.4.3. (John’s trick) We have

∫

|𝜕𝑠𝑣2|𝑑𝑠 ≤

∫

Γ𝑟

Γ𝑟
0

|𝜕𝑠𝑣2|𝑑𝑠 +

∬

Ω𝑟

|𝐷𝑣1

𝐹2||𝜕𝑠𝑣1| + |𝐷𝑣2

𝐹2||𝜕𝑠𝑣2| + |𝐷𝛼𝐹2||𝜕𝑠𝛼|𝑑𝑟𝑑𝑠.

Proof. Based on the evolution equations for 𝜕𝑠𝑣1 and 𝜕𝑠𝑣2 (using 𝜕𝑠𝑣2 as an example):

(𝜕𝑟 + 𝜆2𝜕𝑠)𝑣2 = 𝐹2

(𝜕𝑟 + 𝜆2𝜕𝑠) (𝜕𝑠𝑣2) = (𝜕𝑠𝐹2) − (𝜕𝑠𝜆2) (𝜕𝑠𝑣2)

and the evolution for the Jacobian

𝜕
𝜕𝑟

(cid:16) 𝜕 𝑋2
𝜕𝑧

(cid:17)

=

𝜕
𝜕𝑧

(cid:16) 𝜕 𝑋2
𝜕𝑟

(cid:17)

=

𝜕
𝜕𝑧

(𝜆2) = (𝜕𝑠𝜆2)

(cid:17)

(cid:16) 𝜕 𝑋2
𝜕𝑧

46

we find that the evolution equation for the integrand is

(𝜕𝑟 + 𝜆2𝜕𝑠)

(cid:16)

𝜕𝑠𝑣2 ·

(cid:17)

𝜕 𝑋2
𝜕𝑧

= (𝜕𝑠𝐹2) ·

𝜕 𝑋2
𝜕𝑧

.

Note that the (𝜕𝑠𝜆2) terms cancel, which indicates that there is no quadratic feedback if we consider
the integrand of the total variation (𝜕𝑠𝑣2)· 𝜕 𝑋2
𝜕𝑧 . This is one of the key observations in John’s paper and
also one of the essential steps in this paper. Define 𝑧𝐿 = inf{𝑧 ∈ R | initial perturbation(𝑟0, 𝑧) ≠ 0}

and 𝑧𝑅 = sup{𝑧 ∈ R | initial perturbation(𝑟0, 𝑧) ≠ 0} as before. For each 𝑟 ∈ [𝑟𝑐𝑢𝑡, 𝑟0], define

𝑧𝐿𝐿 ∈ R on the initial slice so that 𝑋1(𝑟, 𝑧𝐿) = 𝑋2(𝑟, 𝑧𝐿𝐿) and similarly 𝑧𝑅𝑅 ∈ R on the initial slice

so that 𝑋2(𝑟, 𝑧𝑅) = 𝑋1(𝑟, 𝑧𝑅𝑅) (see Figure 3.8). We have (recall Figure 3.7)
𝜕 𝑋2
𝜕𝑧

|𝜕𝑠𝑣2|𝑑𝑠 =

𝜕 𝑋2
𝜕𝑧

|𝜕𝑠𝑣2| ·

𝜕𝑠𝑣2 ·

∫ 𝑍𝑅

∫ 𝑍𝑅

𝑑𝑧 =

𝑑𝑧

∫

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

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

Γ𝑟

𝑍𝐿𝐿 (Γ𝑟 )

𝑍𝐿𝐿 (Γ𝑟 )

≤

=

≤

∫ 𝑍𝑅

𝑍𝐿𝐿 (Γ𝑟
0

)

∫ 𝑍𝑅

𝑍𝐿

(Γ𝑟
0

)

∫ 𝑍𝑅

𝑍𝐿

(Γ𝑟
0

)

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

𝜕𝑠𝑣2 ·

𝜕 𝑋2
𝜕𝑧

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

𝑑𝑧 +

∫ 𝑍𝑅

∫ 𝑟0

𝑍𝐿𝐿

𝑟

(Ω𝑟 )

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

𝜕𝑠𝐹2 ·

𝜕 𝑋2
𝜕𝑧

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

𝑑𝑟𝑑𝑧

|𝜕𝑠𝑣2|𝑑𝑠 +

|𝜕𝑠𝑣2|𝑑𝑠 +

∬

∬

(Ω𝑟 )

(Ω𝑟 )

|𝜕𝑠𝐹2|𝑑𝑟𝑑𝑠

|𝐷𝑣1

𝐹2||𝜕𝑠𝑣1| + |𝐷𝑣2

𝐹2||𝜕𝑠𝑣2| + |𝐷𝛼𝐹2||𝜕𝑠𝛼|𝑑𝑟𝑑𝑠.

Notice that

∫ 𝑍𝑅

𝑍𝐿𝐿 (Γ𝑟
0

)

|𝜕𝑠𝑣2|𝑑𝑠 =

∫ 𝑍𝑅

𝑍𝐿

(Γ𝑟
0

)

|𝜕𝑠𝑣2|𝑑𝑠

since 𝜕𝑠𝑣2 = 0 outside the interval [𝑧𝐿, 𝑧𝑅] on the initial slice. Thus, although the definition of 𝑧𝐿𝐿

depends on 𝑟, the term involving 𝑧𝐿𝐿 can be reduced to the one with 𝑧𝐿, and thus the result does

not depend on 𝑟. The only dependence on 𝑟 is through the spacetime integral over Ω𝑟.

□

Proof of Proposition 3.4.1. We use a bootstrap argument.

Bootstrap Assumption.

𝐸 ≤ 𝜖

over [𝑟, 𝑟0]

47

Figure 3.8 Definitions of 𝑧𝐿, 𝑧𝐿𝐿, 𝑧𝑅, 𝑧𝑅𝑅.

√︁

1 + 𝜃2 ≤ 2

sup

sup | ln 𝛼 − ln ˚𝛼| + sup | ln 𝛽 − ln ˚𝛽| ≤ 𝜖𝑎

𝜖 ≤ 𝜖𝑎

with 𝑟 ∈ (𝑟∗, 𝑟0], 0 < 𝜖 < 1, sup = supΩ𝑟 .

The 𝑎 in 𝜖𝑎 refers to auxiliary. The last assumption implies that we are trying to use the

smallness of 𝐸 to improve the estimate for sup | ln 𝛼 − ln ˚𝛼| and sup | ln 𝛽 − ln ˚𝛽|.

Step 1. Gronwall’s inequality

Looking at the inequality from Lemma 3.4.3

∫

|𝜕𝑠𝑣2|𝑑𝑠 ≤

∫

Γ𝑟

Γ𝑟
0

|𝜕𝑠𝑣2|𝑑𝑠 +

∬

Ω𝑟

|𝐷𝑣1

𝐹2||𝜕𝑠𝑣1| + |𝐷𝑣2

𝐹2||𝜕𝑠𝑣2| + |𝐷𝛼𝐹2||𝜕𝑠𝛼|𝑑𝑟𝑑𝑠,

we aim to estimate the right hand side in terms of 𝐸 and apply the Gronwall’s inequality. For

𝐷𝑣1

𝐹2, we have

𝐷𝑣1

𝐹2 =

(cid:16) 1
2

−

𝛾
1 + 𝛾

(cid:17)

·

𝑟
𝑛

(𝜌𝛼) · (1 + 𝛾)

(cid:16)

−

+

𝑛 − 1
𝑟

+ 𝛼 (cid:0) 2
𝑛

Λ𝑟 −

(cid:1)(cid:17)

𝑆 [ℎ]
𝑟

(cid:32)

· 𝐷𝜃

−

48

√

√𝛾
(1 − 𝛾)𝜃
√𝛾 (cid:0)1 + (1 − 𝛾)𝜃2(cid:1)
4

1 + 𝜃2 −

(cid:33)

√

·

𝛾√︁

1 + 𝜃2

+

𝑛
2𝑟

· 𝐷𝜃

(cid:32) √

1 + 𝜃2 (cid:0)√𝛾𝜃 +

√

1 + 𝜃2(cid:1)

(cid:33)

1 + (1 − 𝛾)𝜃2

√

·

𝛾√︁

1 + 𝜃2.

Therefore,

∬

Ω𝑟

|𝐷𝑣1

𝐹2||𝜕𝑠𝑣1|𝑑𝑟 𝑑𝑠 ≤

(cid:16)

𝑀 · 𝑒sup | ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼)| + 𝐶 · 𝑒sup | ln 𝛼−ln ˚𝛼| + 𝐶

(cid:17) ∫ 𝑟0
𝑟

𝐸 (𝑟)𝑑𝑟

(cid:16)

≤

𝑀 · 𝑒(1+𝛾) sup 𝐸 + 𝐶 · 𝑒𝑥 𝑝

(cid:16)

𝐶𝛼 · 𝑒(1+𝛾) sup 𝐸 · 𝑒 1

2

𝜖𝑎 · 𝑒 1

2

𝜖𝑎 · (cid:0) √
(cid:124)

𝛾 · 2 · sup 𝐸
(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:125)
sup |𝜃|

(cid:1)

∫ 𝑟0

𝐸 (𝑟)𝑑𝑟

𝑟
(cid:17) ∫ 𝑟0
𝑟

𝐸 (𝑟)𝑑𝑟

· 2 · sup 𝑊

(cid:17)

+ 𝐶

(cid:17)

·

(cid:16)

≤

𝑀 · 𝑒(1+𝛾)𝜖 + 𝑀 + 𝐶

≤ 𝑀1

∫ 𝑟0

𝑟

𝐸 (𝑟)𝑑𝑟.

In the first ≤ we decompose (𝜌𝛼) to ( ˚𝜌 ˚𝛼) · 𝑒ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼) and absorb ( ˚𝜌 ˚𝛼) to 𝑀. In the second ≤

we apply Lemma 3.4.1 and use the bootstrap assumption to estimate

√

1 + 𝜃2 by 2. Here we have

𝑀 = 𝑀 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠), 𝑀1 = 𝑀1(𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

blow up when 𝑟𝑐𝑢𝑡 → 𝑟 +

∗ and

𝐶 = 𝐶 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

remains bounded since 0 < 𝛾 < 1. Similarly,

∬

Ω𝑟

|𝐷𝑣2

𝐹2||𝜕𝑠𝑣2|𝑑𝑟𝑑𝑠 ≤ 𝑀1

∫ 𝑟0

𝑟

𝐸 (𝑟)𝑑𝑟

For |𝐷𝛼𝐹2||𝜕𝑠𝛼|, we have

∬

Ω𝑟

|𝐷𝛼𝐹2||𝜕𝑠𝛼|𝑑𝑟 𝑑𝑠

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

≤

∬

2
𝑛

Λ𝑟 −

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
≤ 𝐶 · 𝑒sup | ln 𝛼−ln ˚𝛼| (cid:16) 2(1 + 𝛾)

𝑆 [ℎ]
𝑛𝑟

(blue region)

𝑛

√

−

(1 − 𝛾)𝜃
4

1 + 𝜃2 −
√𝛾 (cid:0)1 + (1 − 𝛾)𝜃2(cid:1) +

√𝛾

· 𝛼|𝜕𝑠 ln 𝛼|𝑑𝑟𝑑𝑠

(cid:12)
(cid:12)
1
(cid:12)
(cid:12)
2(1 + 𝛾)
(cid:12)
√

√

𝛽−ln

· 𝑟 · 𝑒sup | ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼)| · 𝑒sup | ln

˚𝛽| · 𝑒sup | ln

√

˚𝛼−ln

√

𝛼|

49

· sup

1 + 𝜃2(cid:17)
√︁

·

∬

|𝜃|𝑑𝑟 𝑑𝑠

Ω𝑟

≤ 𝐶 · 𝑒𝑥 𝑝

(cid:16)

𝑒2𝜖𝑎 · 𝑒(1+𝛾)𝜖 · sup 𝑊

∫ 𝑟0

(cid:17)

·

𝑟

𝐸 (𝑟)𝑑𝑟

≤ 𝑀2 ·

∫ 𝑟0

𝑟

𝐸 (𝑟)𝑑𝑟.

In the second ≤, we use the constraint equation from our reduced Einstein field equations. Notice
√
˚𝛼 can be absorbed to the constant 𝐶 due to their asymptotes. In the third ≤,
√

that the terms ( ˚𝜌 ˚𝛼) ·

˚𝛽

we incorporate the bootstrap assumption and replace |𝜃| by 𝐸 (𝑟) up to a constant. Here

goes to infinity as 𝑟𝑐𝑢𝑡 → 𝑟∗, and

𝑀2 = 𝑀2(𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

𝐶 = 𝐶 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

remains bounded. Together, we have

𝐸 (𝑟) ≤ 𝐸 (𝑟0) + 2𝑀1

∫ 𝑟0

𝑟

𝐸 𝑑𝑟 + 2𝑀2

∫ 𝑟0

𝑟

𝐸 𝑑𝑟.

By Gronwall’s inequality, we have

𝐸 ≤ 𝐸 (𝑟0) · 𝑒(2𝑀1+2𝑀2)(𝑟0−𝑟) ≤ 𝐸 (𝑟0) · 𝑒(2𝑀1+2𝑀2)(𝑟0−𝑟∗).

Step 2. Improved estimate

In order to close the bootstrap argument, we have to show that our bootstrap assumptions are

improved. For the total variation 𝐸, we can choose the initial data (where 𝑀3, 𝑀4 are determined

below)

𝐸 (𝑟0) ≤

𝜖
20𝑒(2𝑀1+2𝑀2)(𝑟0−𝑟∗)

· min

(cid:110)

1
1 + 10𝑀3

,

1
1 + 10𝑀4

(cid:111)

,

which implies the improved estimate

𝐸 ≤

𝜖

20

· min

(cid:110)

1
1 + 10𝑀3

,

1
1 + 10𝑀4

(cid:111)

≤

𝜖

20

.

50

Since 0 < 𝜖 < 1 and 0 < 𝛾 < 1, we have

√︁

1 + 𝜃2 ≤ 1 + |𝜃|

sup

√

√

≤ 1 +

≤ 1 +

𝛾 · sup

√︁

1 + 𝜃2 · 𝐸

𝛾 · 2 ·

𝜖

20

≤ 1 +

𝜖

10

≤

.

3
2

Finally, for sup | ln 𝛼 − ln ˚𝛼| and sup | ln 𝛽 − ln ˚𝛽|, we have

| ln 𝛼 − ln ˚𝛼| ≤ 𝐶𝛼 · 𝑒(1+𝛾)𝐸 · 𝑒 1

2

𝜖𝑎 · 𝑒 1

2

𝜖𝑎 · (cid:0) sup |𝜃| · sup

√︁

1 + 𝜃2(cid:1) · sup 𝑊

≤ 𝐶𝛼 · 𝑒(1+𝛾) · 𝑒 · (cid:0)√

𝛾 · 4 · sup 𝐸 (cid:1) · sup 𝑊

≤ 𝑀3 · sup 𝐸

≤

𝜖

200

≤

𝜖𝑎
200

with

𝑀3 = 𝑀3(𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

going to infinity as 𝑟𝑐𝑢𝑡 → 𝑟 +

∗ , and

| ln 𝛽 − ln ˚𝛽| ≤

∫

Γ𝑟
0

|𝜕𝑠 ln 𝛽|𝑑𝑠

+ 𝑀𝛽 ·

(cid:16) √

√︁
𝛾 sup
(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:124)

1 + 𝜃2 sup 𝐸
(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)
(cid:123)(cid:122)
sup |𝜃|

(cid:17)

· sup

√︁

1 + 𝜃2 · sup 𝐸 · 𝑒(1+𝛾)𝜖

+ 𝑀𝛽

(cid:16)

1 + sup |𝜃|2(cid:17)

· 𝑒(1+𝛾)𝜖 · sup 𝐸

+ 𝐶𝛽 · 𝑒(1+𝛾)𝜖 · 𝑒2𝜖𝑎 ·

(cid:16) √

√︁
𝛾 · sup
(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:124)

1 + 𝜃2 · sup 𝐸
(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:125)
sup |𝜃|

(cid:17)

· sup

√︁

1 + 𝜃2 · sup 𝑊

∫

≤

Γ𝑟
0

|𝜕𝑠 ln 𝛽|𝑑𝑠 + 𝑀4 · sup 𝐸

51

≤

≤

≤

∫

Γ𝑟
0

∫

Γ𝑟
0

𝜖𝑎
200

|𝜕𝑠 ln 𝛽|𝑑𝑠 +

|𝜕𝑠 ln 𝛽|𝑑𝑠 +

𝜖

200

𝜖𝑎
200

+

𝜖𝑎
200

=

𝜖𝑎
100

provided that the initial perturbation ∫

Γ𝑟
0

|𝜕𝑠 ln 𝛽|𝑑𝑠 is sufficiently small. Here

𝑀4 = 𝑀4(𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

goes to infinity as 𝑟𝑐𝑢𝑡 → 𝑟 +
∗ .

Initial data Assumption 1.

𝐸 (𝑟0) ≤

∫

(initial slice)

|𝜕𝑠 ln 𝛽|𝑑𝑠 ≤

𝜖
20𝑒(2𝑀1+2𝑀2)(𝑟0−𝑟∗)
𝜖𝑎
200

· min

(cid:110)

1
1 + 10𝑀3

,

1
1 + 10𝑀4

(cid:111)

where 𝑀1 comes from |𝐷𝑣1
from | ln 𝛼 − ln ˚𝛼| in Step 2, and 𝑀4 comes from | ln 𝛽 − ln ˚𝛽| in Step 2.

𝐹2||𝜕𝑠𝑣1| in Step 1, 𝑀2 comes from |𝐷𝛼𝐹2||𝜕𝑠𝛼| in Step 1, 𝑀3 comes

□

Remark. From the computation, we see that

𝑀3 = 4𝐶𝛼 · 𝑒2+𝛾 ·

√

(cid:16)

𝛾

(cid:17)

𝑊

sup
[𝑟𝑐𝑢𝑡 ,𝑟0]

𝑀4 = 4𝑀𝛽 ·

√

𝛾 · 𝑒(1+𝛾) + 𝑀𝛽

(cid:16)

1 + 4𝛾

(cid:17)

· 𝑒1+𝛾 + 4𝐶𝛽 ·

√

𝛾𝑒3+𝛾 ·

(cid:16)

(cid:17)

.

𝑊

sup
[𝑟𝑐𝑢𝑡 ,𝑟0]

Note that sup[𝑟𝑐𝑢𝑡 ,𝑟0] 𝑊 will go to infinity as 𝑟𝑐𝑢𝑡 → 𝑟 +
𝑀3, 𝑀4 depend on 𝑟𝑐𝑢𝑡.

∗ ; this is the reason why we emphasize that

52

3.5 Riccati equation for derivatives

In this section, we derive the pointwise behavior of the derivative (𝜕𝑠𝑣1) when the total variation
is small. This is reduced to deriving the pointwise behavior of the integral factor 𝑒 𝑓 as this involves

the integral of derivatives along (𝜆1) characteristic and is hard to control. The main difficulty is to

control the integral of (𝜕𝑠 𝛽) term.

We begin with performing the integral factor method as in the ordinary differential equation

context to absorb the linear terms of (𝜕𝑠𝑣1). The result is a Riccati equation for (𝜕𝑠𝑣1).

Lemma 3.5.1. For 𝜕𝑠𝑣1, the derivative of the first Riemann invariant, we have

(𝜕𝑟 + 𝜆1𝜕𝑠)(𝑒 𝑓 · 𝜕𝑠𝑣1) = −(𝑒− 𝑓 )(𝐷𝑣1

𝜆1) (𝑒 𝑓 · 𝜕𝑠𝑣1)2 + 𝑒 𝑓 (𝐷𝑣2

𝐹1) (𝜕𝑠𝑣2) + 𝑒 𝑓 (𝐷𝛼𝐹1) (𝜕𝑠𝛼)

where

− 𝑓 =

∫ 𝑟

𝑟0 (𝜆1)

with 𝑑𝑟 < 0.

(−𝐷𝑣2

𝜆1)(𝜕𝑠𝑣2) − (𝐷𝛼𝜆1) (𝜕𝑠𝛼) − (𝐷 𝛽𝜆1) (𝜕𝑠 𝛽) + (𝐷𝑣1

𝐹1)𝑑𝑟

Proof. Starting from the evolution equation for the first Riemann invariant

(𝜕𝑟 + 𝜆1𝜕𝑠)𝑣1 = 𝐹1,

we take spatial derivative on both sides and get

(𝜕𝑟 + 𝜆1𝜕𝑠)(𝜕𝑠𝑣1) = −(𝜕𝑠𝜆1)(𝜕𝑠𝑣1) + (𝜕𝑠𝐹1)

(𝜕𝑟 + 𝜆1𝜕𝑠)(𝜕𝑠𝑣1) = −(𝐷𝑣1

𝜆1)(𝜕𝑠𝑣1)2

(cid:16)

+

− (𝐷𝑣2

𝜆1) (𝜕𝑠𝑣2) − (𝐷𝛼𝜆1) (𝜕𝑠𝛼) − (𝐷 𝛽𝜆1) (𝜕𝑠 𝛽) + (𝐷𝑣1

𝐹1)

(cid:17)

(𝜕𝑠𝑣1)

+ (𝐷𝑣2

𝐹1)(𝜕𝑠𝑣2) + (𝐷𝛼𝐹1) (𝜕𝑠𝛼)

(𝜕𝑟 + 𝜆1𝜕𝑠)(𝑒 𝑓 · 𝜕𝑠𝑣1) = −(𝑒− 𝑓 )(𝐷𝑣1

𝜆1) (𝑒 𝑓 · 𝜕𝑠𝑣1)2 + 𝑒 𝑓 (𝐷𝑣2

𝐹1) (𝜕𝑠𝑣2) + 𝑒 𝑓 (𝐷𝛼𝐹1) (𝜕𝑠𝛼).

□

53

Note that the quadratic term (𝑒 𝑓 · 𝜕𝑠𝑣1)2 is the main driving force to generate a shock. We are
hoping that the coefficient 𝐷𝑣1
𝜆1 > 0
(refer to Lemma 3.5.7) and 𝑒− 𝑓 is nondegenerate (refer to Proposition 3.5.1) provided that the total

𝜆1 has a fixed sign and 𝑒− 𝑓 is nondegenerate. It turns out 𝐷𝑣1

variation is small. Regarding all the other terms as error terms, we prove our main theorem in

Section 3.5.3.

Definition 3.5.1. Our error term has different definitions in different contexts. For the terms inside

of the integral factor 𝑒− 𝑓 or 𝑒 𝑓 , an error term is defined to be a constant term independent of 𝑟𝑐𝑢𝑡.

For example,

𝐶, 𝑀𝜖

can be regarded as error terms. Although 𝑀 = 𝑀 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠) goes to infinity as 𝑟𝑐𝑢𝑡 → 𝑟 +
∗ ,

𝜖 can depend on 𝑟𝑐𝑢𝑡, so 𝑀𝜖 can be independent of 𝑟𝑐𝑢𝑡. In the Riccati equation context, an error

term is defined to be a small term that goes to 0 when 𝜖 → 0. For example,

𝑀𝜖

can be regarded as an error term. Notice that 𝜖 can depend on 𝑟𝑐𝑢𝑡. Recall that 𝜖 is the notation

for the upper bound of total variation.

It turns out that in the integral factor − 𝑓 , only (𝐷 𝛽𝜆1) (𝜕𝑠 𝛽) and (𝐷𝑣1

𝐹1) contribute non-error

terms coming from the background homogeneous fluid.

3.5.1 Pointwise behavior of the integral factor − 𝑓

In this section, we try to analyze each term in the integral factor − 𝑓 . We begin with the most

difficult term.

Lemma 3.5.2. We separate the background influence (with 𝑟𝑐𝑢𝑡) and error terms (without 𝑟𝑐𝑢𝑡) for

the term

∫ 𝑟

𝑟0 (𝜆1)

− (𝐷 𝛽𝜆1) (𝜕𝑠 𝛽)𝑑𝑟

54

in the integral factor − 𝑓 . It turns out the background influence is

−

1 − 𝛾
1 + 𝛾

·

1
𝑛

∫ 𝑟0

𝑟

𝑟 ( ˚𝜌 ˚𝛼)𝑑𝑟 +

1
1 + 𝛾 ln( ˚𝜌 ˚𝛼) −

1
1 + 𝛾 ln( ˚𝜌(𝑟0) ˚𝛼(𝑟0))

Proof. From the divergence structure of the first equation in the reduced Einstein field equations

𝜕𝑟

(cid:16) √︁𝛽 · 𝑟 𝑛 𝜌 1
1+𝛾 ·
(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:124)
𝐴

√︁

1 + 𝜃2
(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)

(cid:17)

+ 𝜕𝑠

(cid:16) √

𝛼 · 𝑟 𝑛 𝜌 1
(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)
𝐵

1+𝛾 · 𝜃
(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)

(cid:124)

(cid:17)

= 0

and letting the first parenthesis be 𝐴, the second parenthesis be 𝐵, we have

[ln( 𝐴)− ln( 𝐴(𝑟0))]

=

=

=

∫ 𝑟

𝑟0 (𝜆1)

∫ 𝑟

𝑟0 (𝜆1)

∫ 𝑟

𝑟0 (𝜆1)

(cid:12)
(cid:12)
(cid:12)𝜆1
(𝜕𝑟 + 𝜆1𝜕𝑠) 𝐴
𝐴

𝑑𝑟

−(𝜕𝑠𝐵) + 𝜆1(𝜕𝑠 𝐴)
𝐴

𝑑𝑟

(cid:16) 1
2
√
√

+ (𝜕𝑠𝜃) ·

𝛼
𝛽

·

√

(cid:17)

1
1 + 𝜃2

−

(𝜕𝑠 ln 𝛼) ·

√
√

𝛼
𝛽

·

√

𝜃

1 + 𝜃2

+

1
1 + 𝛾

(𝜕𝑠 ln 𝜌) ·

√
√

𝛼
𝛽

·

𝜃

√

1 + 𝜃2

+ 𝜆1 ·

(cid:16) 1
2

(𝜕𝑠 ln 𝛽) +

1
1 + 𝛾

(𝜕𝑠 ln 𝜌) + (𝜕𝑠 ln

√︁

1 + 𝜃2)

(cid:17)

𝑑𝑟

=

∫ 𝑟

𝑟0 (𝜆1)

1
2

· 𝜆1(𝜕𝑠 ln 𝛽) −

(cid:16)

−

+

1
1 + 𝛾

·

√
√

𝛼
𝛽

·

1
2

𝜃

· (𝜕𝑠 ln 𝛼) ·

√
√

𝛼
𝛽

·

√

𝜃

1 + 𝜃2

√

1 + 𝜃2

+

1
1 + 𝛾

(cid:17) (cid:16)

· 𝜆1

(1 + 𝛾) (𝜕𝑠𝑣1 − 𝜕𝑠𝑣2) − (𝜕𝑠 ln 𝛼)
(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: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)
(cid:123)(cid:122)
(cid:124)
(𝜕𝑠 ln 𝜌)

(cid:17)

(cid:16)

−

+

√
√

𝛼
𝛽

√

·

𝛾 + 𝜆1 ·

√𝛾𝜃
√
1 + 𝜃2

(cid:17)

(𝜕𝑠𝑣1 + 𝜕𝑠𝑣2)𝑑𝑟

∫ 𝑟

=

1
2

· 𝜆1(𝜕𝑠 ln 𝛽)

𝑟0 (𝜆1)
√
√

+

𝛼
𝛽

·

1
1 + (1 − 𝛾)𝜃2

(cid:16) (cid:0) −

·

1
2

+

𝛾
1 + 𝛾

(cid:1) ·

√

𝜃

1 + 𝜃2

55

−

1 − 𝛾
2

·

√

𝜃3

1 + 𝜃2

−

√

(cid:17)

𝛾

·

(𝜕𝑠 ln 𝛼)

√
√

𝛼
𝛽

−

√

· 2

𝛾 ·

1 −

1 + (1 − 𝛾)𝜃2

· (𝜕𝑠𝑣2)𝑑𝑟.

1
1 + 𝛾
√𝛾𝜃
√
1+𝜃2

Notice that the first term 1

In order to derive the background influence, we have a closer look at [ln( 𝐴) − ln( 𝐴(𝑟0))]

2 · 𝜆1(𝜕𝑠 ln 𝛽) in the integrand is precisely the −(𝐷 𝛽𝜆1) (𝜕𝑠 𝛽) term in − 𝑓 .
. First

(cid:12)
(cid:12)
(cid:12)𝜆1

we expand the definition of ln( 𝐴).

ln( 𝐴) = ln

(cid:16)

√

𝛽
𝛼 1
1+𝛾

· 𝑟 𝑛 (𝜌𝛼)

1
1+𝛾 ·

√︁

1 + 𝜃2(cid:17)

.

• Since the evolution equation we have for

√

𝛽 is only along 𝜕𝑟 direction, we estimate ln

(cid:17)

(cid:16) √

𝛽
1
1+𝛾

𝛼

along this direction:

(cid:16)

(cid:104)

ln

(cid:17)

√

𝛽
𝛼 1
1+𝛾

(cid:16)

(cid:16)

− ln

= ln

(cid:17)(cid:105)(cid:12)
(cid:12)
(cid:12)𝜆1

1
1+𝛾

√︁𝛽(𝑟0)
𝛼(𝑟0)
√
𝛽
𝛼 1
1+𝛾

(cid:17)

(𝑃) − ln

(cid:17)

(cid:16)

√

𝛽
𝛼 1
1+𝛾

(𝑃0) + ln

(cid:17)

(cid:16)

√

𝛽
𝛼 1
1+𝛾

(𝑃0) − ln

(cid:16)

√

𝛽
𝛼 1
1+𝛾

(cid:17)

(𝑃1)

=

∫ 𝑟

𝑟0

𝜕𝑟 ln

(cid:16)

√

𝛽
𝛼 1
1+𝛾

(cid:17)

𝑑𝑟 +

(cid:16)

ln √︁𝛽(𝑃0) − ln √︁𝛽(𝑃1)
(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)
(cid:124)
(cid:123)(cid:122)
√
𝛽
2 initial perturbation for ln

(cid:17)

(cid:16)

+

ln(𝛼 1
(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:124)

1+𝛾 ) (𝑃1) − ln(𝛼 1
(cid:123)(cid:122)

1+𝛾 ) (𝑃0)
(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)

(cid:17)

.

2 initial perturbation for ln(𝛼

Note that

1
1+𝛾 )

𝜕𝑟 ln

(cid:16)

√

𝛽
𝛼 1
1+𝛾

(cid:17)

=

(cid:16) 1 − 𝛾
1 + 𝛾

− (1 − 𝛾) · 𝜃2(cid:17)

·

𝑟
𝑛

(𝜌𝛼) −

(cid:16) 1
2

+

(cid:17)

·

1
1 + 𝛾

𝑛 − 1
𝑟

+

+

(cid:16) 1
2
(cid:16) 1 − 𝛾
1 + 𝛾

=

(cid:17)

· 𝛼

(cid:16) 2Λ
𝑛

1
1 + 𝛾
− (1 − 𝛾) · 𝜃2(cid:17)

(cid:17)

𝑆 [ℎ]
𝑛𝑟

( ˚𝜌 ˚𝛼) · 𝑒ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼)

𝑟 −

𝑟
𝑛

·

−

(cid:16) 1
2

+

(cid:17)

·

1
1 + 𝛾

𝑛 − 1
𝑟

+

(cid:16) 1
2

+

(cid:17)

1
1 + 𝛾

· 𝛼

(cid:16) 2Λ
𝑛

𝑟 −

𝑆 [ℎ]
𝑛𝑟

(cid:17)

.

56

Therefore, the contribution from the background to ln

1 − 𝛾
1 + 𝛾

·

1
𝑛

∫ 𝑟

𝑟0

𝑟 ( ˚𝜌 ˚𝛼)𝑑𝑟 = −

1 − 𝛾
1 + 𝛾

·

The remaining terms are bounded by

(cid:16) √

𝛽
1
1+𝛾

(cid:17)

is

𝛼

1
𝑛

∫ 𝑟0

𝑟

𝑟 ( ˚𝜌 ˚𝛼)𝑑𝑟.

(3.1)

∫ 𝑟0

𝑟

1 − 𝛾
1 + 𝛾

𝑟
𝑛

·

( ˚𝜌 ˚𝛼) · (cid:12)

(cid:12)𝑒ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼) − 1(cid:12)

(cid:12) + (1 − 𝛾) · 𝜃2 ·

𝑟
𝑛

( ˚𝜌 ˚𝛼) · 𝑒ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼)

+ 2 ·

𝑛 − 1
𝑟

+ 2 · ˚𝛼 · 𝑒ln 𝛼−ln ˚𝛼 (cid:16) 2Λ
𝑛

𝑟 −

(cid:17)

𝑆 [ℎ]
𝑛𝑟

≤ 𝑀𝜖 + 𝑀𝜖 2 + 𝐶 + 𝐶,

where we use Proposition 3.4.1 to estimate |𝑒ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼) − 1| ≤ 𝑒| ln(𝜌𝛼) − ln( ˚𝜌 ˚𝛼)| and

ln 𝛼 − ln ˚𝛼, and we use the fact that ˚𝛼 is bounded over the entire time interval [𝑟∗, 𝑟0] (recall

that lim𝑟→𝑟∗ ˚𝛼 = 0). Here

goes to infinity as 𝑟𝑐𝑢𝑡 → 𝑟 +

∗ and

𝑀 = 𝑀 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

𝐶 = 𝐶 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

remains bounded (recall that 𝑟∗ > 0). Since 𝜖 is allowed to depend on 𝑟𝑐𝑢𝑡, these terms can

be regarded as error terms.

• Next, we consider the ln

(cid:16)

(𝜌𝛼)

1

1+𝛾 (cid:17)

term. Since we have control on the total variation of 𝑣1

and 𝑣2, we write this term as

1
1 + 𝛾 ln(𝜌𝛼) =

1
1 + 𝛾 ln( ˚𝜌 ˚𝛼) +

∫

Γ𝑟

(𝜕𝑠𝑣1 − 𝜕𝑠𝑣2)𝑑𝑠.

Therefore, we have

(cid:104) 1
1 + 𝛾 ln(𝜌𝛼) −

1
1 + 𝛾 ln(𝜌(𝑟0)𝛼(𝑟0))

(cid:105)(cid:12)
(cid:12)
(cid:12)𝜆1

=

1
1 + 𝛾 ln( ˚𝜌 ˚𝛼) −
∫

1
1 + 𝛾 ln( ˚𝜌(𝑟0) ˚𝛼(𝑟0))
∫

+

(𝜕𝑠𝑣1 − 𝜕𝑠𝑣2)𝑑𝑠 −

(𝜕𝑠𝑣1 − 𝜕𝑠𝑣2)𝑑𝑠.

(3.2)

Γ𝑟

Γ𝑟

57

Notice that the last two integrals are bounded by 2𝜖 (since 𝐸 ≤ 𝜖 by Proposition 3.4.1) and

hence can be taken as error terms.

• We consider the ln(𝑟 𝑛

√

1 + 𝜃2) term. Since we know that

√

1 + 𝜃2 ≤ 2 (by Proposition 3.4.1),

we have

where

(cid:104)(cid:12)
(cid:12) ln (cid:0)𝑟 𝑛√︁
(cid:12)

1 + 𝜃2(cid:1) − ln (cid:0)𝑟 𝑛
0

√︃

1 + 𝜃2
0

(cid:105)

(cid:1)(cid:12)
(cid:12)
(cid:12)

𝜆1

≤ 2𝑛| ln 𝑟 | + 2𝑛| ln 𝑟0| ≤ 𝐶

𝐶 = 𝐶 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠).

We still have to control the two error terms on the right hand side of the equation for [ln 𝐴 −

ln 𝐴(𝑟0)]

(cid:12)
(cid:12)
(cid:12)(𝜆1)

.

• The first error term is

∫ 𝑟0

𝑟

√
√

𝛼
𝛽

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

·

1
1 + (1 − 𝛾)𝜃2

(cid:16)(cid:0) −

·

1
2

+

𝛾
1 + 𝛾

·

√

𝜃

1 + 𝜃2

−

1 − 𝛾
2

·

√

𝜃3

1 + 𝜃2

−

1
1 + 𝛾

√

(cid:17)

𝛾

·

(𝜕𝑠 ln 𝛼)

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

𝑑𝑟 ≤ 𝑀𝜖

where 𝜖 comes from the 𝜃 in (𝜕𝑠 ln 𝛼) equation (refer to reduced Einstein field equations)

and 𝑀 comes from

√
√

𝛼
𝛽 . Here

𝑀 = 𝑀 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

goes to infinity when 𝑟𝑐𝑢𝑡 → 𝑟 +
∗ .

• The second error term is

∫ 𝑟

𝑟0 (𝜆1)

√
√

𝛼
𝛽

−

√

· 2

𝛾 ·

1 −

√𝛾𝜃
√
1+𝜃2

1 + (1 − 𝛾)𝜃2

· (𝜕𝑠𝑣2)𝑑𝑟.

One can apply John’s trick to this term, but here we choose to use a more precise method

to control this term. Since the integrand is similar to the evolution equation of 𝜃 along 𝜆1

58

direction, we argue that a major part of this is actually integrable along 𝜆1 direction. From

the evolution of Riemann invariants (refer to Section 3.2.4), we have

(cid:17)

= (𝜆1 − 𝜆2) (𝜕𝑠𝑣2) + (𝐹1 + 𝐹2)

(𝜕𝑟 + 𝜆1𝜕𝑠)

(cid:16) 1

√𝛾 ln (cid:0)√︁

1 + 𝜃2 + 𝜃(cid:1)
(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)

(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:123)(cid:122)
(𝑣1+𝑣2)
√𝛾
2
1 + (1 − 𝛾)𝜃2
(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)

(cid:123)(cid:122)
(𝜆1−𝜆2)

√
𝛼
·
√
𝛽
(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:124)

·(𝜕𝑠𝑣2) = (𝜕𝑟 + 𝜆1𝜕𝑠)

(cid:16) 1

√𝛾 ln (cid:0)√︁

1 + 𝜃2 + 𝜃(cid:1)(cid:17)

− (𝐹1 + 𝐹2)

√
√

𝛼
𝛽

−

√

𝛾 ·

· 2

1 −

√𝛾𝜃
√
1+𝜃2

1 + (1 − 𝛾)𝜃2

· (𝜕𝑠𝑣2) = −

(cid:16)

1 −

(cid:16)

+

1 −

(cid:17)

·

√𝛾

1
√
1 + 𝜃2

(𝜕𝑟 + 𝜆1𝜕𝑠)𝜃

(cid:17)

(𝐹1 + 𝐹2)

√𝛾𝜃
√
1 + 𝜃2
√𝛾𝜃
√
1 + 𝜃2
(cid:16)

= (𝜕𝑟 + 𝜆1𝜕𝑠)

1

√𝛾 ln (cid:0)√︁

1 + 𝜃2 + 𝜃(cid:1) +

(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)

1
2

ln(1 + 𝜃2)
(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:17)

(cid:125)

−

(cid:124)

(cid:123)(cid:122)
𝑄(𝜃)

(cid:16)

+

1 −

√𝛾𝜃
√
1 + 𝜃2

(cid:17)

(𝐹1 + 𝐹2)

Integrating both sides along 𝜆1 direction, we have

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

∫ 𝑟

𝑟0 (𝜆1)

√
√

𝛼
𝛽

−

√

𝛾 ·

· 2

1 −

√𝛾𝜃
√
1+𝜃2

1 + (1 − 𝛾)𝜃2

· (𝜕𝑠𝑣2)𝑑𝑟

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

≤ |𝑄(𝜃 (𝑃))| + |𝑄(𝜃 (𝑃1))|

∫ 𝑟0

+

𝑟

(𝜆1)

|𝐹1 + 𝐹2|𝑑𝑟

≤ 𝐶.

Note that |𝜃| remains small so the 𝑄 terms have no problem. There is no (𝜌𝛼) term in 𝐹1 + 𝐹2

so this term is also bounded. Here

𝐶 = 𝐶 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠).

59

Lemma 3.5.3. We separate the background influence and error terms for

∫ 𝑟

𝑟0 (𝜆1)

(𝐷𝑣1

𝐹1)𝑑𝑟

in the integral factor − 𝑓 . It turns out the background influence is

(cid:16) 1 + 𝛾
2

(cid:17)

·

− 𝛾

1
𝑛

∫ 𝑟0

𝑟

𝑟 ( ˚𝜌 ˚𝛼)𝑑𝑟.

Proof. By the definition of 𝐹1, we have

𝐷𝑣1

𝐹1 = (𝐷 (𝜌𝛼) 𝐹1)(𝐷𝑣1 (𝜌𝛼)) + (𝐷𝜃 𝐹1) (𝐷𝑣1

𝜃)

·

(cid:16)

(cid:17)

+

=

−

𝑟
𝑛

𝛾
1 + 𝛾

1
2
1 + 𝛾
2
Therefore, the contribution of the background to the ∫ 𝑟
∫ 𝑟0

+ 𝛾

𝑟
𝑛

−

=

(cid:17)

(cid:16)

·

(𝜌𝛼) · (1 + 𝛾) + (𝐷𝜃 𝐹1) ·

√

𝛾√︁

1 + 𝜃2

( ˚𝜌 ˚𝛼) · 𝑒ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼) + (𝐷𝜃 𝐹1) ·

√

𝛾√︁

1 + 𝜃2.

(𝐷𝑣1

𝐹1)𝑑𝑟 term is

𝑟0 (𝜆1)

(cid:16) 1 + 𝛾
2

(cid:17)

− 𝛾

·

1
𝑛

𝑟

𝑟 ( ˚𝜌 ˚𝛼)𝑑𝑟.

□

(3.3)

Notice that the error terms are

∫ 𝑟0

𝑟

(cid:16) 1 + 𝛾
2

(cid:17)

·

− 𝛾

𝑟
𝑛

( ˚𝜌 ˚𝛼) ·

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

𝑒ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼) − 1

+ |𝐷𝜃 𝐹1| ·

√

𝛾√︁

1 + 𝜃2𝑑𝑟

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

≤ 𝑀𝜖 + 𝐶,

where we use Proposition 3.4.1 to control |𝑒ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼) − 1| ≤ 𝑒| ln(𝜌𝛼) − ln( ˚𝜌 ˚𝛼)| and use the fact

√

that |𝐷𝜃 𝐹1| ·

1 + 𝜃2 is bounded by a fraction of 𝜃, of which the numerator and the denominator

having the same degree. Here

goes to infinity as 𝑟𝑐𝑢𝑡 → 𝑟 +

∗ and

𝑀 = 𝑀 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

𝐶 = 𝐶 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠).

60

□

Lemma 3.5.4. We prove that the term

∫ 𝑟

𝑟0 (𝜆1)

− (𝐷𝑣2

𝜆1) (𝜕𝑠𝑣2)𝑑𝑟

in the integral factor − 𝑓 can be regarded as an error term.

Proof. Recall that we have the evolution for Riemann invariants

(𝜕𝑟 + 𝜆1𝜕𝑠)𝑣1 = 𝐹1

(𝜕𝑟 + 𝜆2𝜕𝑠)𝑣2 = 𝐹2.

Adding these two equations, we get

(𝜕𝑟 + 𝜆1𝜕𝑠)(𝑣1 + 𝑣2) = (𝐹1 + 𝐹2) + (𝜆1 − 𝜆2) (𝜕𝑠𝑣2)

(𝜆1 − 𝜆2)(𝜕𝑠𝑣2) = (𝜕𝑟 + 𝜆1𝜕𝑠) (𝑣1 + 𝑣2) − (𝐹1 + 𝐹2)
√𝛾
1 + (1 − 𝛾)𝜃2

· (𝜕𝑠𝑣2) = (𝜕𝑟 + 𝜆1𝜕𝑠)

√𝛾 ln (cid:0)√︁

1 + 𝜃2 + 𝜃(cid:1)(cid:17)

(cid:16) 1

2

√
√

𝛼
𝛽

·

− (𝐹1 + 𝐹2).

On the other hand, from the definition of 𝜆1 (refer to Section 3.2.4),

−(𝐷𝑣2

𝜆1)(𝜕𝑠𝑣2) = −(𝐷𝜃𝜆1) ·

√

𝛾√︁

1 + 𝜃2(𝜕𝑠𝑣2)

√

·

𝛾√︁

1 + 𝜃2(𝜕𝑠𝑣2)

= −

√
√

(cid:124)

·

√

𝛼
𝛽
1 + 𝜃2 (cid:0)
(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)

1 − 𝛾
√
√𝛾𝜃(cid:1) 2
1 + 𝜃2 +
(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)

√
√

𝛼
𝛽

·

(cid:0)

= −

√

(cid:123)(cid:122)
𝐷 𝜃 𝜆1
√𝛾(1 − 𝛾)
1 + 𝜃2 +

√𝛾𝜃(cid:1) 2

· (𝜕𝑠𝑣2).

Putting these two equations together, we find that

−(𝐷𝑣2

𝜆1)(𝜕𝑠𝑣2) = −

1 − 𝛾
2

·

+

1 − 𝛾
2

·

√

√

1 + 𝜃2 −
1 + 𝜃2 +
√

√

1 + 𝜃2 −
1 + 𝜃2 +

√𝛾𝜃
√𝛾𝜃
√𝛾𝜃
√𝛾𝜃

· (𝜕𝑟 + 𝜆1𝜕𝑠)

(cid:16) 1

√𝛾 ln (cid:0)√︁

1 + 𝜃2 + 𝜃(cid:1)(cid:17)

(𝐹1 + 𝐹2)

61

= −

1 − 𝛾
√𝛾
2

·

+

1 − 𝛾
2

·

√

√

1 + 𝜃2 −
1 + 𝜃2 +
√

√

1 + 𝜃2 −
1 + 𝜃2 +

√𝛾𝜃
√𝛾𝜃
√𝛾𝜃
√𝛾𝜃

· (𝜕𝑟 + 𝜆1𝜕𝑠)𝜃

·

√

1
1 + 𝜃2

(𝐹1 + 𝐹2)

= −

1 − 𝛾
√𝛾
2

· (𝜕𝑟 + 𝜆1𝜕𝑠)

(cid:110)

−

1 −
1 +

√𝛾
√𝛾 ln (cid:0)1 −

√

𝜃

(cid:1)

1 + 𝜃2 + 1

𝜃

(cid:1)

√

1 + 𝜃2 + 1
𝜃

√𝛾
1 +
√𝛾 ln (cid:0)1 +
1 −
√𝛾
1 − 𝛾 ln (cid:0)(cid:0)

√

2

+

−

√

√

1 + 𝜃2 + 1
√𝛾𝜃
√𝛾𝜃

1 + 𝜃2 −
1 + 𝜃2 +

+

1 − 𝛾
2

·

(𝐹1 + 𝐹2)

√

(cid:1) 2 + 2

𝛾 (cid:0)

√

𝜃

(cid:1) + 1(cid:1)(cid:111)

1 + 𝜃2 + 1

= −

1 − 𝛾
√𝛾
2

· (𝜕𝑟 + 𝜆1𝜕𝑠)𝑄(𝜃) +

1 − 𝛾
2

·

√

√

1 + 𝜃2 −
1 + 𝜃2 +

√𝛾𝜃
√𝛾𝜃

(𝐹1 + 𝐹2).

where 𝑄 denotes the function inside of the big parenthesis. Since 𝑄(0) = 0 and |𝜃| remains small,

we conclude that

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

∫

(𝜆1)

−(𝐷𝑣2

𝜆1)(𝜕𝑠𝑣2)𝑑𝑟

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

≤

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

1 − 𝛾
√𝛾
2

· 𝑄(𝜃 (𝑃))

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

+

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

1 − 𝛾
√𝛾
2
√

· sup

√

· 𝑄(𝜃 (𝑃1))

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

1 + 𝜃2 −
1 + 𝜃2 +

√𝛾𝜃
√𝛾𝜃

(𝐹1 + 𝐹2)

+ (𝑟0 − 𝑟∗) ·

1 − 𝛾
2

≤ 𝐶

where we use the fact that |𝐹1 + 𝐹2| remains bounded since there is no (𝜌𝛼) term. Here

𝐶 = 𝐶 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠).

□

Lemma 3.5.5. We show that the −(𝐷𝛼𝜆1) (𝜕𝑠𝛼) term in the integral factor − 𝑓 can be regarded as

an erorr term (without 𝑟𝑐𝑢𝑡).

62

Proof. Notice that

−(𝐷𝛼𝜆1)(𝜕𝑠𝛼) = −

√
√

𝛼
𝛽

·

1
2

·

√

(1 − 𝛾)𝜃

1 + 𝜃2 +
1 + (1 − 𝛾)𝜃2

√𝛾

· (𝜕𝑠 ln 𝛼)

√
√

𝛼
𝛽

·

1
2

·

(1 + 𝛾)
𝑛

(1 + 𝛾)
𝑛

=

=

=

√

(1 − 𝛾)𝜃

1 + 𝜃2 +
1 + (1 − 𝛾)𝜃2
√

√𝛾

·

· 𝑟 (𝜌𝛼) ·

(1 − 𝛾)𝜃

1 + 𝜃2 +
1 + (1 − 𝛾)𝜃2

2(1 + 𝛾)
𝑛
√𝛾

· 𝜃√︁

1 + 𝜃2

· 𝑟 (𝜌𝛼) ·

√
√

𝛽
𝛼

· 𝜃√︁

1 + 𝜃2

· 𝑟 ( ˚𝜌 ˚𝛼) · 𝑒ln(𝜌𝛼)−ln( ˚𝜌 ˚𝛼) ·

√

(1 − 𝛾)𝜃

1 + 𝜃2 +
1 + (1 − 𝛾)𝜃2

√𝛾

· 𝜃√︁

1 + 𝜃2.

Therefore, we have

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

∫ 𝑟

𝑟0

−(𝐷𝛼𝜆1)(𝜕𝑠𝛼)𝑑𝑟

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

≤

∫ 𝑟0

𝑟

(cid:12)
(cid:12)𝐷𝛼𝜆1

(cid:12)
(cid:12)

(cid:12)𝜕𝑠𝛼(cid:12)
(cid:12)

(cid:12)𝑑𝑟 ≤ 𝑀𝜖,

where 𝑀 comes from ( ˚𝜌 ˚𝛼) and 𝜖 comes from |𝜃|. Here

𝑀 = 𝑀 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

goes to infinity as 𝑟𝑐𝑢𝑡 → 𝑟 +
∗ .

For later convenience, we compute the asymptote for the background ( ˚𝜌 ˚𝛼).

Lemma 3.5.6. (Asymptote for ( ˚𝜌 ˚𝛼).) There are constants

𝑟𝑚𝑖𝑑 = 𝑟𝑚𝑖𝑑 (𝑟∗, 𝛾, Λ, 𝑆 [ℎ]) > 𝑟∗, 𝐶𝑚𝑖𝑑 = 𝐶𝑚𝑖𝑑 (𝑟0, 𝑟𝑚𝑖𝑑, 𝑟∗, 𝛾, Λ, 𝑆 [ℎ])

□

so that

where

( ˚𝜌 ˚𝛼)𝐿 ≤ ( ˚𝜌 ˚𝛼) ≤ ( ˚𝜌 ˚𝛼)𝑅,

𝑟 ∈ (𝑟∗, 𝑟0]

( ˚𝜌 ˚𝛼)𝐿 =






𝑛
(1−𝛾)(𝑟−𝑟∗) ·

1
(1.1)𝑟∗

,

𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑]

1
𝐶𝑚𝑖𝑑

,

𝑟 ∈ (𝑟𝑚𝑖𝑑, 𝑟0]

63

( ˚𝜌 ˚𝛼)𝑅 =






𝑛

(1−𝛾)(𝑟−𝑟∗) · (1.1)
𝑟∗

,

𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑]

𝐶𝑚𝑖𝑑,

𝑟 ∈ [𝑟𝑚𝑖𝑑, 𝑟0]

< (1.1), 𝑟𝑚𝑖𝑑 − 𝑟∗ < 1 and 𝑟0 > 𝑟𝑚𝑖𝑑.

with

𝑟𝑚𝑖𝑑
𝑟∗

Remark. Note that 𝑟𝑚𝑖𝑑 does not depend on 𝑟0 nor 𝑟𝑐𝑢𝑡.

Proof. We have

𝜇 = 𝜇∗ + 𝑂 ((𝑟 − 𝑟∗)

1+𝛾
1−𝛾 )

𝜏 =

1 − 𝛾
𝑛

1+𝛾
2𝛾
∗

𝜇

(𝑛+1) (1−𝛾)
2

∗

· 𝑟

−1

(𝑟 − 𝑟∗) + 𝑂 ((𝑟 − 𝑟∗)2).

Here the notation is

˚𝜌 =

1+𝛾
2𝛾

𝜇

𝑟 (𝑛+1) (1+𝛾)

2

1+𝛾
1−𝛾

· 𝜏

˚𝛼 = 𝑟 𝑛−1 · 𝜏

2𝛾
1−𝛾 .

Therefore, we have

( ˚𝜌 ˚𝛼) =

𝜇

1+𝛾
2𝛾
𝜏

· 𝑟 𝑛−1− (𝑛+1) (1+𝛾)

2

=

1+𝛾

2𝛾 · 𝑟 𝑛−1− (𝑛+1) (1+𝛾)

2

𝜇

1−𝛾
𝑛 𝜇

1+𝛾
2𝛾
∗

· 𝑟

(𝑛+1) (1−𝛾)
2

∗

−1

(𝑟 − 𝑟∗) + 𝑂 ((𝑟 − 𝑟∗)2)

=

1
𝑟 − 𝑟∗

·

𝑛
1 − 𝛾

·

𝜇

1+𝛾

2𝛾 · 𝑟 𝑛−1− (𝑛+1) (1+𝛾)
−1

(𝑛+1) (1−𝛾)
2

2

1+𝛾
2𝛾
∗

𝜇

· 𝑟

∗

+ 𝑂 (𝑟 − 𝑟∗)

Since

lim
𝑟→𝑟 +
∗

there is a time

𝜇

1+𝛾

2𝛾 · 𝑟 𝑛−1− (𝑛+1) (1+𝛾)
−1

(𝑛+1) (1−𝛾)
2

2

1+𝛾
2𝛾
∗

𝜇

· 𝑟

∗

+ 𝑂 (𝑟 − 𝑟∗)

=

1+𝛾
2𝛾
∗

𝜇

· 𝑟 𝑛−1− (𝑛+1) (1+𝛾)

2

∗

1+𝛾
2𝛾
∗

𝜇

· 𝑟

(𝑛+1) (1−𝛾)
2

∗

−1

.

=

,

1
𝑟∗

𝑟𝑚𝑖𝑑 = 𝑟𝑚𝑖𝑑 (𝑟∗, 𝛾, Λ, 𝑆 [ℎ]) > 𝑟∗

64

so that

𝑛
(1 − 𝛾)(𝑟 − 𝑟∗)

·

1
(1.1)𝑟∗

≤ ( ˚𝜌 ˚𝛼) ≤

𝑛
(1 − 𝛾) (𝑟 − 𝑟∗)

·

(1.1)
𝑟∗

for 𝑟∗ < 𝑟 ≤ 𝑟𝑚𝑖𝑑 and
[𝑟𝑚𝑖𝑑, 𝑟0], there exists 𝐶𝑚𝑖𝑑 = 𝐶𝑚𝑖𝑑 (𝑟0, 𝑟𝑚𝑖𝑑, 𝑟∗, 𝛾, Λ, 𝑆 [ℎ]) so that

𝑟𝑚𝑖𝑑
𝑟∗

< (1.1). If 𝑟0 > 𝑟𝑚𝑖𝑑, then since ( ˚𝜌 ˚𝛼) does not have a singularity over

1
𝐶𝑚𝑖𝑑

≤ ( ˚𝜌 ˚𝛼) ≤ 𝐶𝑚𝑖𝑑

for 𝑟𝑚𝑖𝑑 ≤ 𝑟 ≤ 𝑟0.

Proposition 3.5.1. (Net influence on − 𝑓 from the background.) There exists a constnat

𝐶3 = 𝐶3( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

𝑒− 𝑓 ≥

(cid:16)

1
𝑟 − 𝑟∗

(cid:17) 𝛿1

·

𝑒− 𝑓 ≥

,

,

1
𝐶3
1
𝐶3

𝑟 ∈ [𝑟𝑐𝑢𝑡, 𝑟𝑚𝑖𝑑]

𝑟 ∈ (𝑟𝑚𝑖𝑑, 𝑟0]

𝑒 𝑓 ≤ 𝐶3,

𝑟 ∈ [𝑟𝑐𝑢𝑡, 𝑟0].

so that

In addition,

𝑒 𝑓 (𝑟0) = 1

from the definition of − 𝑓 (refer to Lemma 3.5.1).

□

(3.4)

(3.5)

(3.6)

Proof. Putting (3.1), (3.2), (3.3) together, we see that the net influence from the background solution

is

−

1 − 𝛾
1 + 𝛾

·

1
𝑛

= −

(1 − 𝛾)2
2(1 + 𝛾)

∫ 𝑟0

𝑟

·

1
𝑛

𝑟 ( ˚𝜌 ˚𝛼)𝑑𝑟 +

1
1 + 𝛾

(cid:0) ln( ˚𝜌 ˚𝛼) − ln( ˚𝜌(𝑟0) ˚𝛼(𝑟0)) +

(cid:16) 1 + 𝛾
2

(cid:17)

·

− 𝛾

1
𝑛

∫ 𝑟0

𝑟

𝑟 ( ˚𝜌 ˚𝛼)𝑑𝑟

∫ 𝑟0

𝑟

𝑟 ( ˚𝜌 ˚𝛼)𝑑𝑟 +

1
1 + 𝛾

(cid:0) ln( ˚𝜌 ˚𝛼) − ln( ˚𝜌(𝑟0) ˚𝛼(𝑟0))(cid:1).

65

From Lemma 3.5.6, we have

( ˚𝜌 ˚𝛼)𝐿 ≤ ( ˚𝜌 ˚𝛼) ≤ ( ˚𝜌 ˚𝛼)𝑅.

Therefore, we get the estimate for the background influence to − 𝑓

−

(1 − 𝛾)2
2(1 + 𝛾)

·

1
𝑛

∫ 𝑟0

𝑟

𝑟 ( ˚𝜌 ˚𝛼)𝑅𝑑𝑟 +

1
1 + 𝛾

(cid:0) ln( ˚𝜌 ˚𝛼)𝐿 − ln( ˚𝜌(𝑟0) ˚𝛼(𝑟0))𝑅(cid:1)

≤ (background influence to − 𝑓 )

≤ −

(1 − 𝛾)2
2(1 + 𝛾)

·

1
𝑛

∫ 𝑟0

𝑟

𝑟 ( ˚𝜌 ˚𝛼)𝐿 𝑑𝑟 +

1
1 + 𝛾

(cid:0) ln( ˚𝜌 ˚𝛼)𝑅 − ln( ˚𝜌(𝑟0) ˚𝛼(𝑟0))𝐿(cid:1).

Here we discuss two cases. For 𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑],

−

(1.1)(1 − 𝛾)
2(1 + 𝛾)𝑟∗

+

(cid:16)

1
1 + 𝛾

(cid:16)

ln

𝑟

∫ 𝑟𝑚𝑖𝑑

𝑑𝑟 −

𝑟𝑚𝑖𝑑
𝑟 − 𝑟∗
𝑛
(1.1)𝑟∗(1 − 𝛾) (𝑟 − 𝑟∗)

(1 − 𝛾)2
2𝑛(1 + 𝛾)

∫ 𝑟0

𝑟𝑚𝑖𝑑

(𝑟0𝐶𝑚𝑖𝑑)𝑑𝑟

(cid:17)

− ln (cid:0)𝐶𝑚𝑖𝑑(cid:1)(cid:17)

≤ (background influence to − 𝑓 )

≤ −

(1 − 𝛾)
(2.2)𝑟∗(1 + 𝛾)

∫ 𝑟𝑚𝑖𝑑

𝑟

𝑟∗
𝑟 − 𝑟∗

𝑑𝑟 −

(1 − 𝛾)2
2𝑛(1 + 𝛾)

∫ 𝑟0

𝑟𝑚𝑖𝑑

𝑟∗
𝐶𝑚𝑖𝑑

𝑑𝑟

+

(cid:16)

1
1 + 𝛾

ln

(cid:16)

(1.1)𝑛
𝑟∗(1 − 𝛾) (𝑟 − 𝑟∗)

(cid:17)

− ln

(cid:17)(cid:17)

,

(cid:16) 1
𝐶𝑚𝑖𝑑

where we use the definition of ( ˚𝜌 ˚𝛼)𝐿 and ( ˚𝜌 ˚𝛼)𝑅 in Lemma 3.5.6 and estimate 𝑟 in the integrand.

Applying exp to them, we have

(cid:16)

1
𝑟 − 𝑟∗

1+𝛾 − (1.1) (1−𝛾)
(cid:17) 1
2(1+𝛾)

·

𝑟𝑚𝑖𝑑
𝑟∗

·

1
𝐶

for 𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑]. We know that

≤ (background influence to 𝑒− 𝑓 ) ≤

1+𝛾 − (1−𝛾)
(cid:17) 1

(2.2) (1+𝛾) · 𝐶

(cid:16)

1
𝑟 − 𝑟∗

𝛿1 (cid:66) 1
1 + 𝛾
𝛿2 (cid:66) 1
1 + 𝛾

−

−

(1.1) (1 − 𝛾)
2(1 + 𝛾)
(1 − 𝛾)
(2.2) (1 + 𝛾)

·

𝑟𝑚𝑖𝑑
𝑟∗

> 0

> 0

since

𝑟𝑚𝑖𝑑
𝑟∗

< (1.1) by Lemma 3.5.6. Here

𝐶 = 𝐶 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠).

66

By Lemma 3.5.2, Lemma 3.5.3, Lemma 3.5.4, Lemma 3.5.5, we have

(cid:17) 𝛿1

(cid:16)

1
𝑟 − 𝑟∗

·

1
𝐶3

≤ 𝑒− 𝑓 ≤

(cid:17) 𝛿2

(cid:16)

1
𝑟 − 𝑟∗

· 𝐶3,

𝑟 ∈ [𝑟𝑐𝑢𝑡, 𝑟𝑚𝑖𝑑]

for some

We can rearrange 𝐶3 so that

𝐶3 = 𝐶3( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠).

𝑒 𝑓 ≤ 𝐶3,

𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑]

since 𝐶3 can depend on 𝑟𝑚𝑖𝑑. For 𝑟 ∈ (𝑟𝑚𝑖𝑑, 𝑟0], we have

−𝐶 ≤ (background influence to − 𝑓 ) ≤ 𝐶

and therefore

after possibly making 𝐶3 larger.

1
𝐶3

≤ 𝑒− 𝑓 ≤ 𝐶3

3.5.2 Pointwise behavior of the other terms

Lemma 3.5.7. We compute the (𝐷𝑣1

𝜆1) term.

Proof.

(𝐷𝑣1

𝜆1) = (𝐷𝜃𝜆1)(𝐷𝑣1

𝜃) =

√
√

𝛼
𝛽

·

√

1 + 𝜃2 (cid:0)

1 − 𝛾
√
1 + 𝜃2 +

√𝛾𝜃(cid:1) 2

√

·

𝛾√︁

1 + 𝜃2 ≥ 𝐶

(cid:16)

√

˚𝛼
√︃

˚𝛽

(cid:17)

𝐿

since |𝜃|, | ln 𝛼 − ln ˚𝛼|, | ln 𝛽 − ln ˚𝛽| remain small. Furthermore, since

√
˚𝛼√
˚𝛽

≈ 1
𝑟−𝑟∗

, we have

(cid:16)

√

˚𝛼
√︃

˚𝛽

(cid:17)

=

𝐿






𝐶
𝑟−𝑟∗

,

𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑]

𝐶,

𝑟 ∈ (𝑟𝑚𝑖𝑑, 𝑟0].

67

□

□

Lemma 3.5.8. We show that (𝐷𝛼𝐹1)(𝜕𝑠𝛼) can be regarded as an error term, meaning its upper

bound contains 𝜖. Notice that here we are regarding 𝐹1 as 𝐹1(𝑣1, 𝑣2, 𝛼).

Proof. Notice that

𝛼(𝐷𝛼𝐹1) = 𝛼 (cid:0) 2
𝑛

Λ𝑟 −

𝑆 [ℎ]
𝑛𝑟

(cid:1) ·

(cid:16)

−

(cid:124)

√

√𝛾

(1 − 𝛾)𝜃
4

1 + 𝜃2 +
√𝛾 (cid:0)1 + (1 − 𝛾)𝜃2(cid:1) −

(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:17)

1
2(1 + 𝛾)
(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)
2(1+𝛾) ± 1
√𝛾

4

(cid:123)(cid:122)

bounded between− 1

(𝜕𝑠 ln 𝛼) = −2(1 + 𝛾) ·

𝑟
𝑛

(𝜌𝛼) ·

√
√

𝛽
𝛼

· 𝜃√︁

1 + 𝜃2.

Therefore, thanks to the 𝜃 inside of (𝜕𝑠 ln 𝛼), we conclude that

|(𝐷𝛼𝐹1) (𝜕𝑠𝛼)| ≤ 𝐶𝜖

where 𝐶 = 𝐶 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠).

□

Lemma 3.5.9. We show that |𝜕𝑠𝑣2| is pointwise small over Ω𝑟𝑐𝑢𝑡 ,𝑧 𝑀 , as long as the initial data

sup𝑠∈[𝑧𝐿,𝑧 𝑀 ] |𝜕𝑠𝑣2(𝑟0, 𝑠)| and 𝜖 (total variation upper bound) are sufficienly small. Here we apply

John’s trick again. Ω𝑟𝑐𝑢𝑡 ,𝑧𝑀 is defined to be

Ω𝑟𝑐𝑢𝑡 ,𝑧 𝑀 = {(𝑟, 𝑋1(𝑟; 𝑧)) | 𝑟𝑐𝑢𝑡 ≤ 𝑟 ≤ 𝑟0, 𝑧𝐿 ≤ 𝑧 ≤ 𝑧𝑀 }.

See Figure 3.12.

Proof.

Step 1. The evolution equation for (𝜕𝑠𝑣2) is

(𝜕𝑟 + 𝜆2𝜕𝑠)(𝜕𝑠𝑣2) = −(𝜕𝑠𝜆2) (𝜕𝑠𝑣2) + (𝜕𝑠𝐹2)

= −(𝐷𝑣2

𝜆2) (𝜕𝑠𝑣2)2 +

(cid:16)

− (𝐷𝑣1

𝜆2) (𝜕𝑠𝑣1) − (𝐷𝛼𝜆2) (𝜕𝑠𝛼)

− (𝐷 𝛽𝜆2) (𝜕𝑠 𝛽) + (𝐷𝑣2

𝐹2)

(cid:17)

(𝜕𝑠𝑣2)

68

+ (𝐷𝑣1

𝐹2) (𝜕𝑠𝑣1) + (𝐷𝛼𝐹2) (𝜕𝑠𝛼).

Using the integral factor method, we get

(𝜕𝑟 + 𝜆2𝜕𝑠)(𝑒 𝑓2 · 𝜕𝑠𝑣2) = 𝑒 𝑓2 · (𝐷𝑣1

𝐹2) (𝜕𝑠𝑣1) + 𝑒 𝑓2 · (𝐷𝛼𝐹2) (𝜕𝑠𝛼),

where

− 𝑓2 =

∫ 𝑟

𝑟0 (𝜆2)

− (𝐷𝑣2

𝜆2)(𝜕𝑠𝑣2) − (𝐷𝑣1

𝜆2) (𝜕𝑠𝑣1) − (𝐷𝛼𝜆2) (𝜕𝑠𝛼) − (𝐷 𝛽𝜆2) (𝜕𝑠 𝛽) + (𝐷𝑣2

𝐹2)𝑑𝑟.

Note that here we incoporate the quadratic term (𝜕𝑠𝑣2)2 into the integral factor. By the fundamental

theorem of Calculus along (𝜆2),

𝑒 𝑓2 · (𝜕𝑠𝑣2)(𝑃) − (𝜕𝑠𝑣2)(𝑃2) =

∫ 𝑟

𝑟0 (𝜆2)

𝑒 𝑓2 · (𝐷𝑣1

𝐹2) (𝜕𝑠𝑣1) + 𝑒 𝑓2 · (𝐷𝛼𝐹2) (𝜕𝑠𝛼)𝑑𝑟.

Step 2. (John’s trick) Fix 𝑧 ∈ [𝑧𝐿𝐿, 𝑧𝑀). Let 𝑦 = 𝑦(𝑟) ∈ [𝑧𝐿, 𝑧𝑀] be the function so that

𝑋2(𝑟; 𝑧) = 𝑋1(𝑟; 𝑦(𝑟))

where 𝑟 ≤ 𝑟0 satisfies 𝑋2(𝑟, 𝑧) ≤ 𝑋1(𝑟, 𝑧𝑀) so that (𝑟, 𝑋2(𝑟; 𝑧)) ∈ Ω𝑟𝑐𝑢𝑡 ,𝑧 𝑀 (see Figure 3.12).

Taking derivative with respect to 𝑟 on both sides, we get

𝜆2 = 𝜆1 +

𝜕 𝑋1
𝜕𝑧

𝑑𝑦
𝑑𝑟

·

𝑑𝑟
𝑑𝑦

=

1
𝜆2 − 𝜆1

𝜕 𝑋1
𝜕𝑧

·

√
√

𝛽
𝛼

= −

·

1 + (1 − 𝛾)𝜃2
√𝛾
2

𝜕 𝑋1
𝜕𝑧

.

·

This calculation aims to use (𝑟, 𝑦) coordinate to foliate the spacetime region Ω𝑟𝑐𝑢𝑡 ,𝑧 𝑀 , as one can

see the 𝑑𝑟𝑑𝑦 in the following computation (see Figure 3.9). Fix 𝑃 ∈ Ω𝑟𝑐𝑢𝑡 ,𝑧 𝑀 and let 𝑃2 = (𝑟0, 𝑧).

We have

∫ 𝑟0

𝑟

(𝑋2 (·;𝑧))

𝑒 𝑓2 · |𝐷𝑣1

𝐹2||𝜕𝑠𝑣1|𝑑𝑟

∫ 𝑧𝐿

≤

𝑧 𝑀 (𝑋2 (·;𝑧))

𝑒 𝑓2 · |𝐷𝑣1

𝐹2||𝜕𝑠𝑣1| ·

𝑑𝑟
𝑑𝑦

𝑑𝑦

69

=

∫ 𝑧 𝑀

𝑧𝐿

(𝑋2 (·;𝑧))

𝑒 𝑓2 · |𝐷𝑣1

𝐹2||𝜕𝑠𝑣1| ·

√
√

𝛽
𝛼

·

1 + (1 − 𝛾)𝜃2
√𝛾
2

𝜕 𝑋1
𝜕𝑧

·

𝑑𝑦

≤ sup

Ω𝑟𝑐𝑢𝑡 ,𝑧𝑀
∫ 𝑧 𝑀

𝑧𝐿

(𝑋2 (·;𝑧))

≤ sup

(cid:16)

𝑒 𝑓2 ·

(cid:16)

𝑒 𝑓2 ·

√
√

𝛽
𝛼

|𝐷𝑣1

𝐹2| ·

1 + (1 − 𝛾)𝜃2
√𝛾
2

(cid:17)

|𝜕𝑠𝑣1| ·

𝜕 𝑋1
𝜕𝑧

𝑑𝑦

√
√

𝛽
𝛼

|𝐷𝑣1

𝐹2| ·

1 + (1 − 𝛾)𝜃2
√𝛾
2

(cid:17)

Ω𝑟𝑐𝑢𝑡 ,𝑧𝑀
(cid:16)∫ 𝑧 𝑀
𝑧𝐿

(Γ𝑟
0

Ω𝑟𝑐𝑢𝑡 ,𝑧𝑀
(cid:16)∫ 𝑧 𝑀
𝑧𝐿

(Γ𝑟
0

|𝜕𝑠𝑣1|𝑑𝑠 +

)

∬

Ω𝑟𝑐𝑢𝑡 ,𝑧𝑀

|𝜕𝑠𝐹1| ·

𝜕 𝑋1
𝜕𝑧

(cid:17)

𝑑𝑟𝑑𝑦

= sup

(cid:16)

𝑒 𝑓2 ·

√
√

𝛽
𝛼

|𝐷𝑣1

𝐹2| ·

1 + (1 − 𝛾)𝜃2
√𝛾

2

(cid:17)

|𝜕𝑠𝑣1|𝑑𝑠 +

)

∬

Ω𝑟𝑐𝑢𝑡 ,𝑧𝑀

|𝜕𝑠𝐹1|𝑑𝑟𝑑𝑠

(cid:17)

.

Notice that we can make the initial data ∫ 𝑧𝑅

𝑧𝐿 (Γ𝑟
0

|𝜕𝑠𝑣1|𝑑𝑠 small, and make

)

Figure 3.9 John’s trick to foliate the spacetime by (𝑟, 𝑦).

∬

Ω𝑟𝑐𝑢𝑡 ,𝑧𝑀

|𝜕𝑠𝐹1|𝑑𝑟 𝑑𝑠 ≤

∬

Ω𝑟𝑐𝑢𝑡 ,𝑧𝑀

|𝐷𝑣1

𝐹1||𝜕𝑠𝑣1| + |𝐷𝑣2

𝐹1||𝜕𝑠𝑣2| + |𝐷𝛼𝐹1||𝜕𝑠𝛼|𝑑𝑟𝑑𝑠

70

be bounded by 𝑀𝜖 where 𝑀 = 𝑀 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠). Thus we conclude that

∫ 𝑟0

𝑟

(𝑋2 (·;𝑧))

𝑒 𝑓2 · |𝐷𝑣1

𝐹2||𝜕𝑠𝑣1|𝑑𝑟 ≤ sup
Ω𝑟𝑐𝑢𝑡 ,𝑧𝑀

(𝑒 𝑓2) · 𝑀𝜖 .

Step 3. Note that

∫ 𝑟0

𝑟

(𝜆2)

𝑒 𝑓2 · |𝐷𝛼𝐹2||𝜕𝑠𝛼|𝑑𝑟 ≤ sup

(cid:16)

𝑒 𝑓2 · 𝛼|𝐷𝛼𝐹2||𝜕𝑠 ln 𝛼|

(cid:17)

· (𝑟0 − 𝑟)

≤ sup(𝑒 𝑓2) · 𝐶𝜖

where 𝐶 = 𝐶 ( 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠) where 𝜖 comes from the |𝜃| inside of (𝜕𝑠 ln 𝛼).

Step 4.

Bootstrap Assumption.

with 0 < 𝜖𝑎 < 1.

With this assumption, we have

|𝜕𝑠𝑣2| ≤ 𝜖𝑎

| 𝑓2| ≤ 𝑀𝜖𝑎 + 𝑀𝜖 + 𝑀𝜖 + 𝑀 + 𝑀 ≤ 5𝑀

where 𝑀 = 𝑀 (𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠). From the integral equation for 𝑒 𝑓2 · (𝜕𝑠𝑣2), we have

𝑒 𝑓2 · |𝜕𝑠𝑣2| ≤ |(𝜕𝑠𝑣2)(𝑃2)| +

∫ 𝑟

𝑟0 (𝜆2)

𝑒 𝑓2 · (𝐷𝑣1

𝐹2)|𝜕𝑠𝑣1|𝑑𝑟 +

∫ 𝑟0

𝑟

(𝜆2)

𝑒 𝑓2 · |𝐷𝛼𝐹2||𝜕𝑠𝛼|𝑑𝑟

≤ |(𝜕𝑠𝑣2)(𝑃2)| + 𝑒5𝑀 · 𝑀𝜖 + 𝑒5𝑀 · 𝐶𝜖,

or

|𝜕𝑠𝑣2(𝑃)| ≤ 𝑒5𝑀 |(𝜕𝑠𝑣2) (𝑃2)| + 𝑒10𝑀 · 𝑀𝜖 + 𝑒10𝑀 · 𝐶𝜖 .

Thus, if we choose the 𝜖 (total variation upper bound) small enough compared with 𝜖𝑎 and choose

the initial data sup𝑠∈[𝑧𝐿,𝑧𝑀 ] |(𝜕𝑠𝑣2)(𝑟0, 𝑠)| small enough, both depending on 𝑟𝑐𝑢𝑡, we can make

𝑒5𝑀 |(𝜕𝑠𝑣2)(𝑃2)| + 𝑒10𝑀 · 𝑀𝜖 + 𝑒10𝑀 · 𝐶𝜖 ≤

𝜖𝑎.

1
2

71

Therefore, we are able to improve the estimate for |𝜕𝑠𝑣2| and therefore close the bootstrap argument.

□

3.5.3 Proof of the Main Theorem

Proof of Theorem 3.1.1. Going back to the Riccati equation derived in Lemma 3.5.1

(𝜕𝑟 + 𝜆1𝜕𝑠)(𝑒 𝑓 · 𝜕𝑠𝑣1) = −(𝑒− 𝑓 )(𝐷𝑣1

𝜆1) (𝑒 𝑓 · 𝜕𝑠𝑣1)2 + 𝑒 𝑓 (𝐷𝑣2

𝐹1) (𝜕𝑠𝑣2) + 𝑒 𝑓 (𝐷𝛼𝐹1) (𝜕𝑠𝛼),

we have that, based on Proposition 3.5.1, Lemma 3.5.7, Lemma 3.5.8, and Lemma 3.5.9,

− (𝜕𝑟 + 𝜆1𝜕𝑠)(𝑒 𝑓 · 𝜕𝑠𝑣1)

≥






1
𝐶3 (𝑟−𝑟∗) 𝛿

1

·

1

𝐶 (𝑟−𝑟∗) · (𝑒 𝑓 · 𝜕𝑠𝑣1)2 − 𝑒 𝑓 (𝑀𝜖𝑎) − 𝑒 𝑓 (𝐶𝜖),

𝑟 ∈ [𝑟𝑐𝑢𝑡, 𝑟𝑚𝑖𝑑]

1
𝐶3

𝐶 · (𝑒 𝑓 · 𝜕𝑠𝑣1)2 − 𝑒 𝑓 (𝑀𝜖𝑎) − 𝑒 𝑓 (𝐶𝜖),
· 1

𝑟 ∈ (𝑟𝑚𝑖𝑑, 𝑟0].

We apply the change of variables

• 𝑦 = (𝑒 𝑓 · 𝜕𝑠𝑣1) and

• 𝑡 = 𝑟0 − 𝑟, −(𝜕𝑟 + 𝜆1𝜕𝑠) = 𝑑
𝑑𝑡 ,

and let

𝑎(𝑡) =

1
𝐶3 (𝑟−𝑟∗) 𝛿

1

·

1
𝐶 (𝑟−𝑟∗)

(cid:66)

𝑐0
(𝑟−𝑟∗)1+ 𝛿

1

,

𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑]

1
𝐶3

𝐶 (cid:66) 𝑐0,
· 1

𝑟 ∈ (𝑟𝑚𝑖𝑑, 𝑟0]






for 0 ≤ 𝑡 ≤ 𝑟0 − 𝑟𝑐𝑢𝑡 (see Figure 3.10). The above inequality simplifies to

𝑑𝑦
𝑑𝑡

≥ 𝑎(𝑡) · 𝑦2 − 𝑒 𝑓 (𝑀𝜖𝑎) − 𝑒 𝑓 (𝐶𝜖).

Rearranging 𝜖𝑎, 𝜖 to be even smaller so that

𝑒 𝑓 (𝑀𝜖𝑎) + 𝑒 𝑓 (𝐶𝜖) ≤ 𝐶3(𝑀𝜖𝑎) + 𝐶3(𝐶𝜖) ≤

𝑎(𝑡)𝑦2,

1
2

we have

𝑑𝑦
𝑑𝑡

≥

1
2

· 𝑎(𝑡) · 𝑦2.

72

Notice that the condition

𝐶3𝑀𝜖𝑎 + 𝐶3𝐶𝜖 ≤

· 𝑎(𝑡) · 𝑦2

1
2

always holds since both 𝑎(𝑡) and 𝑦 are increasing as 𝑡 increases (since 𝑟𝑚𝑖𝑑 − 𝑟∗ < 1). Thus we have

∫ 𝑟0−𝑟

∫ 𝑦

𝑦0

1
𝑦2

𝑑𝑦 ≥

1
𝑦0

−

1
𝑦

≥

0
𝑐0
2𝛿1

· 𝑎(𝑡)𝑑𝑡 ≥

1
2

∫ 𝑟𝑚𝑖𝑑

𝑟

𝑐0
2(𝑟 − 𝑟∗)1+𝛿1

𝑑𝑟

(cid:16)

·

1
(𝑟 − 𝑟∗)𝛿1

−

1
(𝑟𝑚𝑖𝑑 − 𝑟∗)𝛿1

(cid:17)

.

The above right hand side serving as a lower bound is a bit awkward since it is negative when

𝑟 ∈ (𝑟𝑚𝑖𝑑, 𝑟0]. One should however focus on 𝑟 ∈ [𝑟𝑐𝑢𝑡, 𝑟𝑚𝑖𝑑]. Therefore, if the initial data satisfies

1
𝑦0

≤

𝑐0
4𝛿1

(cid:16)

·

1
(𝑟𝑐𝑢𝑡 − 𝑟∗)𝛿1

−

1
(𝑟𝑚𝑖𝑑 − 𝑟∗)𝛿1

(cid:17)

,

𝑦 must go to infinity at some 𝑟 ∈ [𝑟𝑐𝑢𝑡, 𝑟0]. In other words, our lower bound

𝐿𝐵(𝑟𝑐𝑢𝑡, 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠)

for 𝑦0 to generate a shock can be

1
𝐿𝐵

=

𝑐0
4𝛿1

(cid:16)

·

1
(𝑟𝑐𝑢𝑡 − 𝑟∗)𝛿1

−

1
(𝑟𝑚𝑖𝑑 − 𝑟∗)𝛿1

(cid:17)

.

Note that

lim
𝑟𝑐𝑢𝑡 →𝑟∗

𝐿𝐵 = 0.

Since 𝑒 𝑓 (cid:12)

(cid:12)𝑟=𝑟0

= 1, 𝐿𝐵 is a lower bound for both 𝑦0 = (𝑒 𝑓 · 𝜕𝑠𝑣1)(cid:12)

(cid:12)𝑟=𝑟0

and (𝜕𝑠𝑣1)(cid:12)

(cid:12)𝑟=𝑟0

.

□

3.6

Initial data

3.6.1 Summary table for initial data

In this section, we use the following tables to summarize all the initial data assumption from

previous chapters.

73

Figure 3.10 Lower bound 𝑎(𝑡).

Where

Target

Requirement

Proposition 3.4.1 𝐸 ≤ 𝜖 𝐸 |𝑟0 ≤

Assumption 1

𝜖
1+2𝑀

2 ) (𝑟

20𝑒 (2𝑀

∫
(initial slice)

1
1+10𝑀3

0 −𝑟∗ ) · min (cid:8)
|𝜕𝑠 ln 𝛽|𝑑𝑠 ≤ 𝜖𝑎
200

,

1
1+10𝑀4

(cid:9)

0 < 𝜖 ≤ 𝜖𝑎 < 1

Where

Main term

Error term

Lemma 3.5.2

Lemma 3.5.3

− 1−𝛾

1+𝛾 · 1

𝑛

∫ 𝑟0
𝑟

𝑟 ( ˚𝜌 ˚𝛼)𝑑𝑟

𝑀𝜖 + 𝑀𝜖 2 + 2𝐶

√

ln

𝛽(𝑃0) − ln

√

𝛽(𝑃1)

ln(𝛼 1

1+𝛾 ) (𝑃1) − ln(𝛼 1

1+𝛾 ) (𝑃0)

1

1+𝛾 ln( ˚𝜌 ˚𝛼) − 1

1+𝛾 ln( ˚𝜌 ˚𝛼) (𝑟0)

2𝜖

(cid:0) 1+𝛾

2 − 𝛾(cid:1) · 1

𝑛

∫ 𝑟0
𝑟

𝑟 ( ˚𝜌 ˚𝛼)𝑑𝑟

𝑀𝜖 + 𝐶

74

Where

Target

Requirement

Lemma 3.5.9

|𝜕𝑠𝑣2|

|𝜕𝑠𝑣2| ≤ 𝑒5𝑀 |(𝜕𝑠𝑣2) (𝑃2)| + 𝑒10𝑀 · 𝑀𝜖 + 𝑒10𝑀 · 𝐶𝜖 ≤ 1
2

𝜖𝑎

Section 3.5.3

𝐶0𝑀𝜖𝑎 + 𝐶0𝐶𝜖 ≤

𝑐0

(𝑟0−𝑟∗)1+ 𝛿 · (𝑦0)2

Note that the smallness of 𝐸 (𝑟0), the total variation of 𝑣1, 𝑣2, will automatically imply the smallness

of ∥(𝜕𝑠 ln 𝛼(𝑟0)) ∥ 𝐿1 by the 𝜃 in the constraint equation for (𝜕𝑠 ln 𝛼) (refer to the reduced Einstein

field equations) and the fact that the width 𝑊 (𝑟0) = 𝑧𝑅 − 𝑧𝐿 of the initial perturbation is finite.
Therefore, as long as the 𝐸 (𝑟0) is small enough, the term | ln(𝛼 1

1+𝛾 ) (𝑃1) − ln(𝛼 1

1+𝛾 ) (𝑃0)| will be

small.

Therefore, in the later section, we try to pose the following conditions on the initial slice for

𝑠 ∈ [𝑧𝐿, 𝑧𝑅]:

• 𝑣1(𝑟0), 𝑣2(𝑟0) are perturbed but have a small total variation (small 𝐸 (𝑟0)).

• 𝜕𝑠𝑣1(𝑟0, 𝑠) has a large positive value for some 𝑠 ∈ (𝑧𝐿, 𝑧𝑀).

• 𝜕𝑠𝑣2(𝑟0, 𝑠) is small for all 𝑠 ∈ [𝑧𝐿, 𝑧𝑀].

• (𝜕𝑠𝛼(𝑟0)) is perturbed based on the constraint equation. In other words, ∥ (𝜕𝑠 ln 𝛼(𝑟0))∥ 𝐿1

will be small since 𝐸 (𝑟0) is small.

• We do not perturb 𝛽 for simplicity. In other words, 𝛽(𝑟0, 𝑠) = ˚𝛽(𝑟0, 𝑠) for all 𝑠 ∈ [𝑧𝐿, 𝑧𝑀].

3.6.2 Construction of initial data

In this section, we try to construct a sequence of initial data that satisfies the above conditions

and works for shock formation. Firstly, we rewrite the constraint equation for (𝜕𝑠 ln 𝛼) in terms of

Riemann invariants 𝑣1, 𝑣2. The original constraint is

(𝜕𝑠 ln 𝛼) = −2(1 + 𝛾) ·

𝑟
𝑛

(𝜌𝛼) ·

√
√

𝛽
𝛼

· 𝜃√︁

1 + 𝜃2.

75

Based on the definition of 𝑣1, 𝑣2, we have that

𝜃 =

√︁

1 + 𝜃2 =

1
2

1
2

(cid:16)

√𝛾(𝑣1+𝑣2) − 𝑒−

𝑒

√𝛾(𝑣1+𝑣2)(cid:17)

(cid:16)

√𝛾(𝑣1+𝑣2) + 𝑒−

𝑒

√𝛾(𝑣1+𝑣2)(cid:17)

(𝜌𝛼) = 𝑒(1+𝛾)(𝑣1−𝑣2).

Therefore, after replacement, we have

(𝜕𝑠 ln 𝛼) =

− 2(1 + 𝛾) ·

or equivalently,

√

(cid:0)𝜕𝑠

𝛼(cid:1) =

− (1 + 𝛾) ·

𝑟
𝑛

𝑟
𝑛

· 𝑒(1+𝛾)(𝑣1−𝑣2) ·

√
√

𝛽
𝛼

·

1
2

(cid:16)

√𝛾(𝑣1+𝑣2) − 𝑒−

𝑒

√𝛾(𝑣1+𝑣2)(cid:17)

(cid:16)

√𝛾(𝑣1+𝑣2) + 𝑒−

𝑒

√𝛾(𝑣1+𝑣2)(cid:17)

,

·

1
2

· 𝑒(1+𝛾)(𝑣1−𝑣2) · √︁𝛽 ·

(cid:16)

𝑒

1
4

√𝛾(𝑣1+𝑣2) − 𝑒−

√𝛾(𝑣1+𝑣2)(cid:17)

(cid:16)

√𝛾(𝑣1+𝑣2) + 𝑒−

𝑒

·

√𝛾(𝑣1+𝑣2)(cid:17)

.

This means that, if we impose the restriction of the perturbation of initial data that

• 𝑣1(𝑟0) + 𝑣2(𝑟0) is an odd function

• 𝑣1(𝑟0) − 𝑣2(𝑟0) is an even function

• 𝛽 is an even function

with respect to 𝑠, we will get the right hand side of the above equation become an odd function and

therefore it satisfies that

∫ 𝑧𝑅

𝑧𝐿

√

(𝜕𝑠

𝛼)𝑑𝑠 = 0,

which means we are allowed to solve for 𝛼 along the initial slice.

In conclusion, our steps for choosing the initial data are as follows.

• Fix 𝑟𝑐𝑢𝑡 ∈ (𝑟∗, 𝑟0).

76

• Fix 𝑦0 ≥ 𝐿𝐵 that depends on 𝑟𝑐𝑢𝑡.

• Choose sufficiently small 𝜖, 𝜖𝑎 that may depend on 𝑟𝑐𝑢𝑡 and 𝑦0 so that they satisfy all the

requirements in the above tables. Then derive 𝜖0, 𝜖𝑎,0.

• Construct the initial data for 𝑣1, 𝑣2, 𝛽 accordingly. For simplicity, we let 𝛽(𝑟0) = ˚𝛽(𝑟0).

• Construct the initial data for 𝛼 based on the above constraint equation.

As an example about how to construct the initial data for 𝑣1, 𝑣2 verifying the above restriction, we

let 𝑧𝐿 = −1, 𝑧𝑀 = 0, 𝑧𝑅 = 1, take a compactly supported smooth function 𝑃 : [−1, 0] → R of 𝑠

(where 𝑃 for Profile) so that

= 𝐿𝐵

≤ 𝐿𝐵,

∫ 𝑧𝑅

𝑧𝐿

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

𝑑
𝑑𝑠

𝑃

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

𝑑𝑠 ≤ 𝜖0,

∥𝑃∥ 𝐿∞ ≤ 𝜖0,

2

(cid:12)
(cid:12)
(cid:12)𝑠=− 1
(cid:13)
(cid:13)
(cid:13)
(cid:13)𝐿∞

𝑃

𝑑
𝑑𝑠

𝑃

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

𝑑
𝑑𝑠

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

(cid:110) 𝑑
𝑑𝑠

𝑃 ≠ 0

(cid:111)(cid:12)
(cid:12)
(cid:12)

≤

𝜖0
𝐿𝐵

and construct functions 𝜒(𝑜𝑑𝑑), 𝜒(𝑒𝑣𝑒𝑛) of 𝑠 so that

𝜒(𝑜𝑑𝑑) (𝑠) = 𝑃(𝑠) ∀𝑠 ∈ [−1, 0],

𝜒(𝑜𝑑𝑑) (−𝑠) = −𝜒(𝑜𝑑𝑑) (𝑠) ∀𝑠 ∈ [−1, 1]

𝜒(𝑒𝑣𝑒𝑛) (𝑠) = 𝑃(𝑠) ∀𝑠 ∈ [−1, 0],

𝜒(𝑒𝑣𝑒𝑛) (−𝑠) = 𝜒(𝑒𝑣𝑒𝑛) (𝑠) ∀𝑠 ∈ [−1, 1].

The construction implies that, on the initial slice, 𝜒(𝑜𝑑𝑑) and 𝜒(𝑒𝑣𝑒𝑛) have a large positive derivative

at 𝑠 = − 1

2 while maintaining small total variations throughout the slice (see Figure 3.11). Next, let

(cid:16)

𝜒(𝑜𝑑𝑑) + 𝜒(𝑒𝑣𝑒𝑛)(cid:17)

(cid:16)

𝜒(𝑜𝑑𝑑) − 𝜒(𝑒𝑣𝑒𝑛)(cid:17)

𝑣1 = ˚𝑣1 +

𝑣2 = ˚𝑣2 +

1
2

1
2

𝛽 = ˚𝛽.

Notice that the definition of 𝑣1 and 𝑣2 implies that 𝜕𝑠𝑣1

implies that

(cid:12)
(cid:12)(𝑟,𝑠)=(𝑟0,− 1

2 ) = 𝑑

𝑑𝑠 𝑃 (cid:0) − 1

2

(cid:1) ≥ 𝐿𝐵, which

𝑦0 = 𝑒 𝑓 · 𝜕𝑠𝑣1

(cid:12)
(cid:12)
(cid:12)(𝑟,𝑠)=(𝑟0,− 1
2 )

≥ 𝐿𝐵.

77

In addition, for 𝜕𝑠𝑣2, we have

𝜕𝑠𝑣2(𝑟0, 𝑠) = 0 ∀𝑠 ∈ [−1, 0],

which verifies the requirement for boostrap to guarantee the smallness of |𝜕𝑠𝑣2| on the left of the

initial slice (refer to Lemma 3.5.9). By finite speed of propagation property, we know that |𝜕𝑠𝑣2|

remains pointwise small on the left of the interesting region, but we do not know what happens to

|𝜕𝑠𝑣2| on the right part (see Figure 3.12).

Finally, observing that on the initial slice 𝑣1+𝑣2 = 𝜒(𝑜𝑑𝑑) is an odd function compactly supported
in 𝑠 ∈ [−1, 1] and 𝑣1 − 𝑣2 = ( ˚𝑣1 − ˚𝑣2) + 𝜒(𝑒𝑣𝑒𝑛) is an even function which is equal to ( ˚𝑣1 − ˚𝑣2) at
𝑠 = −1, 1, we see that it verifies the restriction of the initial perturbation stated above.

Figure 3.11 Construction of 𝜒(𝑜𝑑𝑑) and 𝜒(𝑒𝑣𝑒𝑛).

78

Figure 3.12 We apply Lemma 3.5.9 on the blue region Ω𝑟𝑐𝑢𝑡 ,𝑧𝑀 .

79

CHAPTER 4

STABILITY OF RELATIVISTIC FLUIDS ON FIXED BIG BANG SPACETIMES

This chapter aims to prove the stability of specially relativistic fluids on a fixed Big Bang spacetime,

the metric of which derived from Section 1.2. In other words, we consider only Euler equations

instead of Einstein-Euler equations.

4.1 The model problem and its homogeneous solutions

Our model problem is the relativistic Euler equations on a fixed warped product manifold 𝐵×𝑟 𝐹,

where 𝐵 is a 1+1 Lorentzian manifold endowed with the metric

𝑔 = −𝛼𝑑𝑟 2 + 𝛽𝑑𝑠2

and 𝐹 is an 𝑛-dimensional Riemannian manifold representing the symmetry and regarded as a fiber.

The fluid variables (𝜃, 𝜌) describing the underlying fluid are the primary variables of interest. In

this paper, we analyze the following main equations

(cid:16)√︁𝛽 · 𝑟 𝑛 𝜌 1

1+𝛾 √︁

1 + 𝜃2(cid:17)

𝜕𝑟

(cid:16)√

𝛼 · 𝑟 𝑛 𝜌 1

1+𝛾 𝜃

(cid:17)

+ 𝜕𝑠

= 0

(cid:16)√︁𝛽 · 𝜌

(cid:17)

𝛾
1+𝛾 𝜃

𝜕𝑟

+ 𝜕𝑠

(cid:16)√

𝛼 · 𝜌

𝛾

1+𝛾 √︁

1 + 𝜃2(cid:17)

= 0.

(4.1)

(4.2)

We proceed to establish the equivalence of Euler equations and our main equations. In the work

by Wong and An [2], they computed the curvature of the warped product spacetime 𝐵 ×𝑟 𝐹 and

derived the Euler equations

(cid:16)

𝑟 𝑛 𝜌 1

1+𝛾 𝜉

(cid:17)

𝑑𝑖𝑣𝑔

= 0

(cid:16)

𝛾

1+𝛾 𝜉♭(cid:17)

𝜌

𝑑

= 0

in the fluid context, with the ultra-relativistic assumption 𝑝 = 𝛾𝜌 as the equation of state. Here
0 < √𝛾 < 1 is a parameter representing sound speed, 𝑝 is the fluid pressure, 𝜌 > 0 is the fluid
density, and 𝜉 is a unit timelike vector field defined on 𝐵 representing the fluid velocity and therefore

80

satisfies the normalization condition −𝛼(𝜉𝑟)2 + 𝛽(𝜉 𝑠)2 = −1. Since the metric components 𝛼, 𝛽

are fixed, we parametrize 𝜉 by the scalar unknown 𝜃 ∈ R satisfying

√

𝛼𝜉𝑟 =

√︁
1 + 𝜃2, √︁𝛽𝜉 𝑠 = 𝜃.

We regard (𝜃, 𝜌) as fluid variables and they are the main unknown functions of (𝑟, 𝑠) in this paper.

Expanding the definition of divergence 𝑑𝑖𝑣𝑔 and exterior derivative 𝑑, we have an equivalent system

as our main equations.

Remark. It is interesting noting that, the fixed fiber 𝐹 does not play any role in our main equations.

Its scalar curvature 𝑆 [ℎ] however is involved in Einstein-Euler equations as considered in Chapter

3.

In a future work, we are able to classify all physical, spatially homogeneous solutions with a

big bang singularity that solve the above Euler equations. It turns out in the fluid (0 < 𝛾 < 1)

and positive blowup time (𝑟∗ > 0) case, we have the following asymptotic behavior for the metric

components

Assumption 4. We assume that the metric components 𝛼, 𝛽 satisfy

𝛼(𝑟), 𝛽(𝑟) > 0 ∀𝑟 ∈ (𝑟∗, 𝑟0]

𝛼(𝑟) = 0,

𝛽(𝑟) = 0

lim
𝑟→𝑟 +
∗

𝜕𝑟 ln 𝛽(𝑟) = ∞,

lim
𝑟→𝑟 +
∗

lim
𝑟→𝑟 +
∗

where 𝑟∗ > 0 is the big bang blowup time. Geometrically, we are assuming 𝜕𝑠 is a Killing vector

field and the geometry has a certain asymptotic behavior when approcahing the big bang.

In order to better understand the Euler equations, we impose the spatial homogenity on the

unknowns (𝜃, 𝜌) and analyze the equations. Assuming both 𝜃 and 𝜌 are independent of 𝑠, the

equations reduce to a coupled system of ordinary differential equations and read, after integraing

with respect to 𝑟,

√︁𝛽 · 𝑟 𝑛 𝜌 1

1+𝛾 √︁

1 + 𝜃2 = 𝐶1

81

√︁𝛽 · 𝜌

𝛾

1+𝛾 𝜃 = 𝐶2.

If 𝜃 is nonzero initially (𝜃 (𝑟0) ≠ 0), we have 𝐶2 ≠ 0 and thus

√︁

1 + 𝜃2 =

𝜃 =

𝐶1√
𝛽
𝐶2√
𝛽

·

1
𝑟 𝑛 · 𝜌− 1

1+𝛾

· 𝜌−

𝛾
1+𝛾 .

Solving the inequality

√

1 + 𝜃2 ≥ 𝜃, we have

𝐶1
𝐶2

·

1
𝑟 𝑛 ≥ 𝜌

1−𝛾
1+𝛾

and thus conclude that 𝜌 remains bounded above for 𝑟 ∈ (𝑟∗, 𝑟0] since 𝑟∗ > 0 and 0 < 𝛾 < 1.

This further implies that 𝜃 blows up at least in a rate 1√

𝛽 . This rate plays an essential role in the

remaining paper.

Remark. These homogeneous solutions with 𝜃 ≠ 0 have no general relativisitc counterpart. In

[28], it was observed that the spatially homogeneous Einstein-Euler system forces 𝜉 to be parallel

to 𝜕𝑟 (or 𝜃 = 0). This extra rigidity is a feature of Einstein’s equations when large number of

symmetries are present, and is what enables the full classification performed in a future paper.

4.2 Dynamical Stability of homogeneous 𝜃 ≠ 0 solutions

In this section, we will establish the stability of homogeneous 𝜃 ≠ 0 solutions with large enough

𝜃 (𝑟0). Specifically, given any metric components 𝛼, 𝛽 satisfying the Assumption 4 and given any

𝑟0 > 𝑟∗, there exists a lower bound (depending on 𝛼, 𝛽, 𝑟0) for 𝜃 (𝑟0) so that the homogeneous

solutions with 𝜃 (𝑟0) greater than the lower bound are stable. In particular, shocks do not form

before the big bang.

4.2.1 Evolution equations

We begin by transforming our main equations (4.1) and (4.2) to evolution equations. Our

equations form a hyperbolic system on the (1 + 1)-dimensional Lorentzian manifold 𝐵; therefore,

the method of characteristics is applicable. As indicated in [15], one derives the following equations

82

after performing a standard diagonalization process

(cid:0)𝜕𝑟 + 𝜆1𝜕𝑠(cid:1) (cid:16) 1
ln (cid:0)√︁
2
(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:124)

1 + 𝜃2 + 𝜃(cid:1) +

(cid:17)

1
√𝛾 ln 𝑓
2
(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)

(cid:123)(cid:122)
𝑣1

√

=

𝜃

(cid:110) (cid:16)

𝛼
√𝛾𝜃
√
1+𝜃2

1
4
1 +
(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: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)

1 + 𝜃2

√

−

·

−

(cid:17)

(𝑒0 ln 𝛽) −

√𝛾
4
(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: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)

𝑛√𝛾
2

(𝑒0 ln 𝑟)

(cid:111)

(4.3)

(cid:0)𝜕𝑟 + 𝜆2𝜕𝑠(cid:1) (cid:16) 1
2
(cid:124)

1 + 𝜃2 + 𝜃(cid:1) −

ln (cid:0)√︁
(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)

1
√𝛾 ln 𝑓
2
(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)

(cid:123)(cid:122)
𝑣2

(cid:17)

√

=

(cid:110) (cid:16)

𝛼
√𝛾𝜃
√
1+𝜃2

1
4
1 −
(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: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)

1 + 𝜃2

√

−

𝜃

·

+

(cid:17)

(𝑒0 ln 𝛽) +

√𝛾
4
(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: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)

𝑛√𝛾
2

(𝑒0 ln 𝑟)

(cid:111)

.

(4.4)

(cid:125)

(cid:125)

(cid:124)

(cid:124)

(cid:123)(cid:122)
𝐹1

(cid:123)(cid:122)
𝐹2
√

√𝛾
√
1+𝜃2−

𝛾

Here 𝑓 = 𝜌

𝛽 · −
to denote an orthonormal frame on 𝐵. This system of equations describe the evolution of the two

1+𝜃2+𝜃
√𝛾𝜃 . We use 𝑒0 = 1√

𝜕𝑟 and 𝑒1 = 1√
𝛽

√𝛾𝜃 , 𝜆2 =

1+𝛾 , 𝜆1 =

1+𝜃2+𝜃

𝛼
𝛽 ·

𝜕𝑠

𝛼

𝛼

√
√

√

√𝛾
√
1+𝜃2+

√
√

Riemann invariants 𝑣1, 𝑣2 in two different directions (𝜕𝑟 + 𝜆1𝜕𝑠), (𝜕𝑟 + 𝜆2𝜕𝑠). Since the coefficients

in 𝐿1, 𝐿2 involve the unknown 𝜃, one may ask whether this system satisfies the analogous genuinely

nonlinear condition defined in [15]. The answer is no. Indeed, as 𝜃 → ∞, the derivative of the

eigenvalues vanishes hence breaking the genuinely nonlinear requirement.

4.2.2 Strategy

Our strategy divides the time interval (𝑟∗, 𝑟0] into two subintervals (𝑟∗, 𝑟𝑚𝑖𝑑], [𝑟𝑚𝑖𝑑, 𝑟0] and per-

forms two different tasks in each interval. 𝑟𝑚𝑖𝑑 is a parameter depending on the metric components

𝛼, 𝛽 that will be chosen later but is expected to be close to 𝑟∗. In the first subinterval (𝑟∗, 𝑟𝑚𝑖𝑑], we

argue that the largeness of 𝜃 (𝑟𝑚𝑖𝑑) enables a bootstrap argument and thus stabilizes the equation

and prevents shock formation. In the second subinterval [𝑟𝑚𝑖𝑑, 𝑟0], we apply the standard Cauchy

stability to find the appropriate lower bound for 𝜃 (𝑟0) to ensure the largeness of 𝜃 (𝑟𝑚𝑖𝑑).

In order to execute our strategy, we have to estimate two quantities: the spatial derivative of

Riemann invariants (as indicated in [15]), and 𝜃 (related to the heuristic above about how to break

83

the genuinely nonlinear condition). For convenience, we denote the spatial derivative of Riemann
(cid:17)

√

√

(cid:17)

1 + 𝜃2 + 𝜃(cid:1) + 1
√𝛾 ln 𝑓
2

and 𝑏 = 𝑒1

(cid:16) 1
2 ln (cid:0)

1 + 𝜃2 + 𝜃(cid:1) − 1
2

√𝛾 ln 𝑓

.

invariants by 𝑎 = 𝑒1

(cid:16) 1
2 ln (cid:0)

4.2.3 Evolution equation for spatial derivatives 𝑎 and 𝑏

We start by taking a spatial derivative 𝑒1 on both Riemann invariant equations and get the

evolution equations for the spatial derivatives 𝑎 and 𝑏

−𝜕𝑟

(cid:12)
(cid:12)𝑋1

𝑎 =

1 − 𝛾
√𝛾𝜃
√
1+𝜃2

(cid:0)1 +

)2

√

·

𝛼 ·

1
1 + 𝜃2

(𝑎 + 𝑏)𝑎 +

1
4

·

1 − 𝛾
√𝛾𝜃
√
1+𝜃2

(cid:0)1 +

·

1
1 + 𝜃2

(cid:1) 2

(𝑎 + 𝑏) (𝜕𝑟 ln 𝛽)

−

𝑛𝛾

2

·

·

1
1 + 𝜃2

(cid:1) 2

1
√𝛾𝜃
√
1+𝜃2

(cid:0)1 +

(𝑎 + 𝑏) (𝜕𝑟 ln 𝑟) +

1
2

· 𝑎(𝜕𝑟 ln 𝛽)

−𝜕𝑟

(cid:12)
(cid:12)𝑋2

𝑏 =

1 − 𝛾
√𝛾𝜃
√
1+𝜃2

(cid:0)1 −

)2

√

·

𝛼 ·

1
1 + 𝜃2

(𝑎 + 𝑏)𝑏 +

1
4

·

1 − 𝛾
√𝛾𝜃
√
1+𝜃2

(cid:0)1 −

·

1
1 + 𝜃2

(cid:1) 2

(𝑎 + 𝑏) (𝜕𝑟 ln 𝛽)

−

𝑛𝛾

2

·

·

1
1 + 𝜃2

(cid:1) 2

1
√𝛾𝜃
√
1+𝜃2

(cid:0)1 −

(𝑎 + 𝑏) (𝜕𝑟 ln 𝑟) +

1
2

· 𝑏(𝜕𝑟 ln 𝛽)

where −𝜕𝑟

(cid:12)
(cid:12)𝑋1

= −(𝜕𝑟 + 𝜆1𝜕𝑠) is the first characteristic direction and −𝜕𝑟

(cid:12)
(cid:12)𝑋2

= −(𝜕𝑟 + 𝜆2𝜕𝑠) is the

second characteristic direction. Observe that these two equations are coupled mainly through the

term 1

coefficients

1+𝜃2 (𝑎 + 𝑏). If we assume the uniform bound for

1+𝜃2 (|𝑎| + |𝑏|) and use the fact that the
√𝛾(cid:1) 2 , we can regard
these two evolutions as linear ordinary differential equations along two characteristics and derive

(cid:1) 2 are both in the range from

√𝛾(cid:1) 2 to

(cid:1) 2 and

1
√𝛾 𝜃
√

1
√𝛾 𝜃
√

(cid:0)1−

(cid:0)1−

(cid:0)1+

(cid:0)1+

1+𝜃2

1+𝜃2

1

1

1

an upper bound for them separately. This is the idea we will apply in the first subinterval (𝑟∗, 𝑟𝑚𝑖𝑑].

A crucial ingredient of this idea is that we have to ensure the largeness of 𝜃 to make 1

1+𝜃2 decay fast

enough and close the bootstrap argument.

4.2.4 Evolution equation for 𝜃

Based on the previous paragraph, we have to investigate how 𝜃 evolves in order to close the

bootstrap argument. After adding equation (3), (4), (5), we get

−𝜕𝑟

(cid:16)

ln

(cid:12)
(cid:12)
(cid:12)𝑋1

√

𝜃

1 + 𝜃2

(cid:17)

𝛾

=

1 − 𝛾
2

· (𝜕𝑟 ln 𝛽) − 𝑛𝛾 · (𝜕𝑟 ln 𝑟) − 2

√

𝛾 ·

√

1 + 𝜃2
𝜃

·

1
1 + 𝜃2

√

𝛼.

· 𝑏

84

Notice that here we recover the homogeneous non-physical solution asymptote if 𝑏 = 0:

𝜃 ≈

1
√
𝛽

as 𝑟 → 𝑟 +
∗

since 𝑟 is assumed to be bounded and away from 0 (because 𝑟∗ > 0). Therefore, if we can argue

that the last term on the right hand side (the term involving

√

1+𝜃2
𝜃

√

·

𝛼) can be regarded as an error

term, we can ensure that the behavior of 𝜃 is similar to 1√

𝛽 at least when being close to the blowup

time.

4.2.5 Bootstrap argument for (𝑟∗, 𝑟𝑚𝑖𝑑]

In this section, we present our bootstrap assumption and try to incorporate this assumption into

the evolution equations we previously have derived. Our bootstrap assumption is

(cid:16)

1
1 + 𝜃2

(cid:17)

|𝑎| + |𝑏|

≤ 𝑀.

(4.5)

We begin with deriving a uniform bound for the spatial derivatives of Riemann invariants |𝑎| and

|𝑏|.

Lemma 4.2.1. (Estimate for |𝑎| and |𝑏|) Assume the bootstrap assumption 4.5 holds with the

constant 𝑀. Then we have

|𝑎|+|𝑏| ≤ 𝑒

1−𝛾

√𝛾)2 ·𝑀 ∫ 𝑟𝑚𝑖𝑑

𝑟∗

(1−

√

𝛼𝑑𝑟

√︁𝛽(𝑟𝑚𝑖𝑑)
√
𝛽

·

(cid:16)

·

sup
𝑟=𝑟𝑚𝑖𝑑

|𝑎| + sup
𝑟=𝑟𝑚𝑖𝑑

|𝑏| +

1
2

·

1 − 𝛾

(cid:0)1 −

√𝛾(cid:1) 2

· 𝑀 ln

(cid:16) 𝛽(𝑟𝑚𝑖𝑑)
𝛽

(cid:17)

+ 𝑛𝛾 ·

1
√𝛾(cid:1) 2

(cid:0)1 −

· 𝑀 ln

(cid:17)(cid:17)

(cid:16) 𝑟𝑚𝑖𝑑
𝑟

for any 𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑], where 𝑟𝑚𝑖𝑑 is a parameter that will be chosen in the next Lemma.

Proof. As mentioned in Section 3.3, we try to incorporate the bootstrap assumption 4.5 to the

evolution equations for 𝑎 and 𝑏 and try to integrate the equations along the characteristics. Since

the computation for 𝑎 and 𝑏 are mostly the same, we only perform the calculation for 𝑎 here. From

the evolution equation of 𝑎, we have

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

− 𝜕𝑟

(cid:12)
(cid:12)𝑋1

𝑎

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

≤

1 − 𝛾

(cid:0)1 −

√𝛾(cid:1) 2

1 − 𝛾

(cid:0)1 −

√𝛾(cid:1) 2

· 𝑀 (𝜕𝑟 ln 𝛽)

√

·

𝛼 · 𝑀 |𝑎| +

1
4

·

85

+

𝑛𝛾

2

·

1
√𝛾(cid:1) 2

(cid:0)1 −

· 𝑀 (𝜕𝑟 ln 𝑟) +

1
2

· |𝑎|(𝜕𝑟 ln 𝛽).

We integrate the inequality along 𝑋1 direction, where 𝑋1 is the characteristic generated by the

vector field 𝜕𝑟 + 𝜆1𝜕𝑠

|𝑎| ≤ sup
𝑟=𝑟𝑚𝑖𝑑

|𝑎| +

1
4

·

1 − 𝛾

(cid:0)1 −

√𝛾(cid:1) 2

· 𝑀 ln

(cid:17)

(cid:16) 𝛽0
𝛽

+

𝑛𝛾

2

·

1
√𝛾(cid:1) 2

(cid:0)1 −

· 𝑀 ln

(cid:17)

(cid:16) 𝑟𝑚𝑖𝑑
𝑟

+

∫ 𝑡

(cid:16)

1 − 𝛾

0

(cid:0)1 −

√𝛾(cid:1) 2

√

·

𝛼 · 𝑀 +

(cid:17)

(𝜕𝑟 ln 𝛽)

1
2

|𝑎(𝑟𝑚𝑖𝑑 − 𝑡, 𝑋1(𝑡))|𝑑𝑡.

where 𝑡 = 𝑟𝑚𝑖𝑑 − 𝑟. By Gronwall’s inequality, we have the following estimate for |𝑎|

1−𝛾

√𝛾)2 ·𝑀 ∫ 𝑟𝑚𝑖𝑑

𝑟∗

(1−

|𝑎| ≤ 𝑒

√

𝛼𝑑𝑟

√︁𝛽(𝑟𝑚𝑖𝑑)
√
𝛽

·

·

(cid:16)

sup
𝑟=𝑟𝑚𝑖𝑑

|𝑎| +

1
4

·

1 − 𝛾

(cid:0)1 −

√𝛾(cid:1) 2

· 𝑀 ln

(cid:16) 𝛽(𝑟𝑚𝑖𝑑)
𝛽

(cid:17)

+

𝑛𝛾

2

·

1
√𝛾(cid:1) 2

(cid:0)1 −

· 𝑀 ln

(cid:16) 𝑟𝑚𝑖𝑑
𝑟

(cid:17)(cid:17)

.

A similar process along the other characteristic 𝑋2 generated by the vector field 𝜕𝑟 + 𝜆2𝜕𝑠 gives the

analogous estimate for 𝑏 and therefore the result.

□

Lemma 4.2.2. (Estimate for 𝜃.) Assume the bootstrap assumption 4.5 holds with 𝑀. There exists

𝑟𝑚𝑖𝑑 > 𝑟∗ so that 𝜃 is increasing over (𝑟∗, 𝑟𝑚𝑖𝑑] (meaning −𝜕𝑟

(cid:12)
(cid:12)𝑋1

𝜃 > 0) and satisfies

𝜃1−𝛾 ≥

𝛾

√
2

·

𝜃 (𝑟𝑚𝑖𝑑)
1 + 𝜃 (𝑟𝑚𝑖𝑑)2

√︁

𝛾 ·

(cid:17) 1−𝛾

2

(cid:16) 𝛽(𝑟𝑚𝑖𝑑)
𝛽

·

(cid:16) 𝑟∗
𝑟𝑚𝑖𝑑

(cid:17) 𝑛𝛾

√𝛾𝑅𝑀 ∫ 𝑟𝑚𝑖𝑑

𝑟∗

√

𝛼𝑑𝑟 .

· 𝑒−2

as long as 𝜃 (𝑟𝑚𝑖𝑑) ≥ 1. Here

𝛽(𝑟𝑚𝑖𝑑)
𝛽

ranges in [1, ∞) and 𝑅 =

(cid:16)

sup𝜃∈[1,∞)

√

1+𝜃2
𝜃

(cid:17)

.

Proof. Similarly to the previous lemma, we replace those terms in Section 3.4 involving 1

1+𝜃2 · 𝑏 by

𝑀 based on the bootstrap assumption 4.5

−𝜕𝑟

(cid:16)

ln

(cid:12)
(cid:12)
(cid:12)𝑋1

√

𝜃

1 + 𝜃2

(cid:17)

𝛾

≥

1 − 𝛾
2

· (𝜕𝑟 ln 𝛽) − 𝑛𝛾 · (𝜕𝑟 ln 𝑟) − 2

√

(cid:16)

𝛾 ·

sup
𝜃∈[1,∞)

√

(cid:17)

1 + 𝜃2
𝜃

√

𝛼

· 𝑀

provided that 𝜃 is always greater than 1.

In order to preserve this 𝜃 ≥ 1 condition, we are

restricted to a region that is close to 𝑟∗. Notice that since 𝛼, 𝛽 satisfy the Assumption 4 (meaning

86

lim𝑟→𝑟 +

∗ (𝜕𝑟 ln 𝛽) = ∞,

√

𝛼 is bounded) and 𝑟∗ > 0 (meaning (𝜕𝑟 ln 𝑟) is bounded), there exists an

𝑟𝑚𝑖𝑑 so that restricted to the time interval (𝑟∗, 𝑟𝑚𝑖𝑑], 𝜃 keeps increasing, assuming 𝜃 begins with a
𝛽(𝑟𝑚𝑖𝑑)
𝛽

value greater than 1. In addition, we will make sure that 𝑟𝑚𝑖𝑑 is close enough to 𝑟∗ so that

ranges in [1, ∞). This range will be used in the subsequent proposition. Returning to the inequality,

we have

−𝜕𝑟

(cid:16)

ln

(cid:12)
(cid:12)
(cid:12)𝑋1

𝜃

√

1 + 𝜃2

(cid:17)

𝛾

≥

1 − 𝛾
2

· (𝜕𝑟 ln 𝛽) − 𝑛𝛾 · (𝜕𝑟 ln 𝑟) − 2

√

𝛾 · 𝑅 · 𝑀

√

𝛼.

Integrating this inequality along 𝑋1, we arrive at

𝜃

√

1 + 𝜃2

𝛾 ≥

𝜃 (𝑟𝑚𝑖𝑑)
1 + 𝜃 (𝑟𝑚𝑖𝑑)2

√︁

𝛾 ·

(cid:17) 1−𝛾

2

(cid:16) 𝛽(𝑟𝑚𝑖𝑑)
𝛽

(cid:17) 𝑛𝛾

·

(cid:16) 𝑟∗
𝑟𝑚𝑖𝑑

√𝛾𝑅𝑀 ∫ 𝑟𝑚𝑖𝑑

𝑟∗

· 𝑒−2

√

𝛼𝑑𝑟 .

In order to isolate the desired quantity 𝜃, we make use of the fact that 𝜃 ≥ 1 and get a lower bound

for 𝜃:

𝜃1−𝛾 ≥

𝛾

√
2

·

𝜃 (𝑟𝑚𝑖𝑑)
1 + 𝜃 (𝑟𝑚𝑖𝑑)2

√︁

𝛾 ·

(cid:17) 1−𝛾

2

(cid:16) 𝛽(𝑟𝑚𝑖𝑑)
𝛽

·

(cid:16) 𝑟∗
𝑟𝑚𝑖𝑑

(cid:17) 𝑛𝛾

√𝛾𝑅𝑀 ∫ 𝑟𝑚𝑖𝑑

𝑟∗

√

𝛼𝑑𝑟 .

· 𝑒−2

□

With all the ingredients in this section, we now present the crucial argument in this paper.

Proposition 4.2.1. Fix a pair of metric components (𝛼, 𝛽) satisfying Assumption 4 with positive

blowup time 𝑟∗ > 0, and fix any bootstrap constant 𝑀 > 0. Then there exist

𝑟𝑚𝑖𝑑 = 𝑟𝑚𝑖𝑑 (𝑀, 𝛾) > 𝑟∗,

𝑚𝑖𝑑 = 𝜃 𝐿𝐵
𝜃 𝐿𝐵

𝑚𝑖𝑑 (𝑀, 𝛾, 𝑟∗, 𝑟𝑚𝑖𝑑) ≥ 1

(where 𝐿𝐵 is for Lower Bound) so that as long as the initial data of the solution to the main

equations (4.1), (4.2) satisfy

• inf𝑟=𝑟𝑚𝑖𝑑 𝜃 ≥ 𝜃 𝐿𝐵
𝑚𝑖𝑑

• sup𝑟=𝑟𝑚𝑖𝑑 |𝑎| + sup𝑟=𝑟𝑚𝑖𝑑 |𝑏| ≤ 𝑀

we have that

always holds. In particular, when 𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑], since 𝜃 remains finite, shock will not form.

(cid:16)

1
1 + 𝜃2

|𝑎| + |𝑏|

(cid:17)

≤ 𝑀

87

Proof. Assume the bootstrap condition (4.5) holds with constant 𝑀. By Lemma 4.2.1 and Lemma

4.2.2, we have

(cid:16)

1
1 + 𝜃2

|𝑎| + |𝑏|

(cid:17)

≤ 𝑒

(cid:16)

1−𝛾

√𝛾)2 ·𝑀 ∫ 𝑟𝑚𝑖𝑑

𝑟∗

(1−

√

𝛼𝑑𝑡

√

𝑥·

·

sup𝑟=𝑟𝑚𝑖𝑑 (|𝑎| + |𝑏|) + 1
2 ·
𝜃 𝐿𝐵
𝑚𝑖𝑑
1+(𝜃 𝐿𝐵
𝑚𝑖𝑑)2

1−𝛾 · (cid:0)

1 + 2

√

𝛾

1−𝛾
√𝛾)2 · 𝑀 ln(𝑥) + 𝑛𝛾 ·
(cid:1) 2𝑛𝛾
1−𝛾 · 𝑥 · (cid:0) 𝑟∗
1−𝛾 · 𝑒−4
𝑟𝑚𝑖𝑑

(1−
𝛾 (cid:1) 2

1

√𝛾)2 · 𝑀 ln (cid:0) 𝑟𝑚𝑖𝑑

𝑟

(1−
√𝛾𝑅𝑀 ∫ 𝑟𝑚𝑖𝑑

𝑟∗

√

𝛼𝑑𝑟

(cid:1)(cid:17)

.

where 𝑥 =

𝛽(𝑟𝑚𝑖𝑑)
𝛽

ranges in [1, ∞) from Lemma 4.2.2. Since

√

𝑥 +

√

𝑥 ln(𝑥)
𝑥

is a bounded function for 𝑥 ∈ [1, ∞), we know that when 𝜃 𝐿𝐵

𝑚𝑖𝑑 is large enough, the term

√

𝜃 𝐿𝐵
𝑚𝑖𝑑
1+(𝜃 𝐿𝐵
𝑚𝑖𝑑)2

𝛾

in the denominator is large, so the right hand side of the above inequality will be strictly less than

𝑀, which closes the bootstrap argument. The only remaining unproven thing is that 𝜃 < ∞ for

𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑]. This can be done using the evolution equation for 𝜃 from Section 3.4

−𝜕𝑟 ln

(cid:16)

√

𝜃

1 + 𝜃2

(cid:17)

𝛾

=

1 − 𝛾
2

· (𝜕𝑟 ln 𝛽) − 𝑛𝛾 · (𝜕𝑟 ln 𝑟) − 2

√

√

1 + 𝜃2
𝜃

𝛾 ·

·

1
1 + 𝜃2

√

𝛼.

· 𝑏

Since the bootstrap assumption 4.5 holds, we have an upper bound

(cid:16)

−𝜕𝑟 ln

√

𝜃

1 + 𝜃2

(cid:17)

𝛾

≤

1 − 𝛾
2

· (𝜕𝑟 ln 𝛽) − 𝑛𝛾 · (𝜕𝑟 ln 𝑟) + 2

√

𝛾𝑅𝑀

√

𝛼

with 𝑅 defined in Lemma 4.2.2. Since the right hand side is finite (but not bounded) for 𝑟 ∈

(𝑟∗, 𝑟𝑚𝑖𝑑], 𝜃 remains finite, and therefore

|𝑎| + |𝑏| ≤ 𝑀 (1 + 𝜃2)

implies the boundedness of the spatial derivatives.

□

4.2.6 Bootstrap argument for [𝑟𝑚𝑖𝑑, 𝑟0]

In this section, we apply the standard Cauchy stability for hyperbolic system in the time interval

[𝑟𝑚𝑖𝑑, 𝑟0]. Specifically, we have

88

Proposition 4.2.2. Given any 𝑟0 > 𝑟𝑚𝑖𝑑 > 𝑟∗, any 𝜃 𝐿𝐵
(𝜃ℎ𝑜𝑚𝑜, 𝜌ℎ𝑜𝑚𝑜) satisfying the Euler equations (4.1), (4.2), with 𝜃ℎ𝑜𝑚𝑜 (𝑟𝑚𝑖𝑑) > 𝜃 𝐿𝐵

𝑚𝑖𝑑 ≥ 1, any homogeneous background solution

𝑚𝑖𝑑, and any width

𝑊0 > 0 of initial perturbation defined by

𝑊0 = |{𝜃 (𝑟0) ≠ 𝜃ℎ𝑜𝑚𝑜 (𝑟0)}|,

there exists 𝜖0 > 0 so that as long as

∥𝑎(𝑟0) ∥ 𝐿∞ + ∥𝑏(𝑟0)∥ 𝐿∞ < 𝜖0,

we have ∥𝑎(𝑟𝑚𝑖𝑑) ∥ 𝐿∞, ∥𝑏(𝑟𝑚𝑖𝑑)∥ 𝐿∞ are finite and

𝜃 (𝑟𝑚𝑖𝑑) > 𝜃 𝐿𝐵
𝑚𝑖𝑑.

Proof. We will use a bootstrap argument here. Our bootstrap assumption is

∥𝑎(𝑟) ∥ 𝐿∞ + ∥𝑏(𝑟) ∥ 𝐿∞ < 𝜖

∥𝜃 (𝑟) ∥ 𝐿∞ ≤ ∥𝜃ℎ𝑜𝑚𝑜 (𝑟) ∥ 𝐿∞ + 1

for 𝑟 ∈ [𝑟𝑚𝑖𝑑, 𝑟0] and for some 0 < 𝜖 < 1 that will be chosen later. Using the evolution equations

for 𝑎, 𝑏 derived in Section 3.3, we have

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

− 𝜕𝑟

(cid:12)
(cid:12)𝑋1

𝑎

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

≤

1 − 𝛾
√𝛾𝜃
√
1+𝜃2

(cid:0)1 +

)2

√

·

𝛼 ·

1
1 + 𝜃2

· (2𝜖) · |𝑎| +

1
4

·

1 − 𝛾
√𝛾𝜃
√
1+𝜃2

(cid:0)1 +

·

1
1 + 𝜃2

(cid:1) 2

(|𝑎| + |𝑏|)(𝜕𝑟 ln 𝛽)

+

𝑛𝛾

2

·

·

1
1 + 𝜃2

(cid:1) 2

1
√𝛾𝜃
√
1+𝜃2

(cid:0)1 +

(|𝑎| + |𝑏|)(𝜕𝑟 ln 𝑟) +

1
2

· |𝑎|(𝜕𝑟 ln 𝛽)

≤ 𝐶1 · (2𝜖) · ∥𝑎(𝑟)∥ 𝐿∞ + 𝐶1 · (∥𝑎(𝑟) ∥ 𝐿∞ + ∥𝑏(𝑟)∥ 𝐿∞)

where 𝐶1 = 𝐶1(sup𝑟∈[𝑟𝑚𝑖𝑑,𝑟0]
equality along 𝑋1, we have

√

𝛼, sup𝑟∈[𝑟𝑚𝑖𝑑,𝑟0] (𝜕𝑟 ln 𝛽), sup𝑟∈[𝑟𝑚𝑖𝑑,𝑟0] (𝜕𝑟 ln 𝑟)). Integrating this in-

∥𝑎(𝑟) ∥ 𝐿∞ + ∥𝑏(𝑟) ∥ 𝐿∞

(cid:17)

𝑑𝑟,

|𝑎| ≤ ∥𝑎(𝑟0)∥ 𝐿∞ + 3𝐶1

∫ 𝑟0

(cid:16)

𝑟

89

and therefore

∥𝑎(𝑟)∥ 𝐿∞ ≤ ∥𝑎(𝑟0)∥ 𝐿∞ + 3𝐶1

∫ 𝑟0

(cid:16)

𝑟

∥𝑎(𝑟) ∥ 𝐿∞ + ∥𝑏(𝑟) ∥ 𝐿∞

(cid:17)

𝑑𝑟.

We can derive the analogous inequality for 𝑏, which together with the above inequality leads to

(cid:16)

∥𝑎(𝑟) ∥ 𝐿∞ + ∥𝑏(𝑟) ∥ 𝐿∞

(cid:17)

(cid:16)

≤

∥𝑎(𝑟0) ∥ 𝐿∞ + ∥𝑏(𝑟0) ∥ 𝐿∞

(cid:17)

+ 6𝐶1

∫ 𝑟0

(cid:16)

𝑟

∥𝑎(𝑟)∥ 𝐿∞ + ∥𝑏(𝑟)∥ 𝐿∞

(cid:17)

𝑑𝑟.

According to the Gronwall’s inequality, we arrive at an 𝐿∞ control of the derivatives

(cid:16)

∥𝑎(𝑟) ∥ 𝐿∞ + ∥𝑏(𝑟)∥ 𝐿∞

(cid:17)

(cid:16)

≤

∥𝑎(𝑟0) ∥ 𝐿∞ + ∥𝑏(𝑟0) ∥ 𝐿∞

(cid:17)

𝑒6𝐶1 (𝑟0−𝑟𝑚𝑖𝑑)

≤ 𝜖0 · 𝑒6𝐶1 (𝑟0−𝑟𝑚𝑖𝑑).

This means that if we choose the upper bound 𝜖0 of the initial perturbation to be sufficiently small

depending on 𝜖 and 𝐶1, we can ensure (∥𝑎(𝑟)∥ 𝐿∞ + ∥𝑏(𝑟)∥ 𝐿∞) < 1
2

𝜖, an improved estimate for 𝑎

and 𝑏. For 𝜃, we have

|𝜃 (𝑟) − 𝜃ℎ𝑜𝑚𝑜 (𝑟)| ≤

∫

{𝑟=𝑟}

|𝜕𝑠𝜃|𝑑𝑠

∫

√︁

{𝑟=𝑟}
∫

{𝑟=𝑟}

1 + 𝜃2(cid:12)

(cid:12)(𝜕𝑠𝑣1) + (𝜕𝑠𝑣2)(cid:12)

(cid:12)𝑑𝑠

√︁

1 + 𝜃2 (cid:0)|𝜕𝑠𝑣1| + |𝜕𝑠𝑣2|(cid:1) 𝑑𝑠

√︃

1 + (cid:0)𝜃ℎ𝑜𝑚𝑜 (𝑟) + 1(cid:1) 2 · √︁𝛽(𝑟)

∫

{𝑟=𝑟}

(|𝑎| + |𝑏|)𝑑𝑠

√︃

1 + (cid:0)𝜃ℎ𝑜𝑚𝑜 (𝑟) + 1(cid:1) 2 · √︁𝛽(𝑟)𝜖 · 𝑊 (𝑟)

=

≤

≤

≤

where 𝑊 (𝑟) is the width of {𝑎 ≠ 0} ∪ {𝑏 ≠ 0} and is uniformly bounded during 𝑟 ∈ [𝑟𝑚𝑖𝑑, 𝑟0] by

finite speed of propagation (due to the uniform bounds of eigenvalues |𝜆1|, |𝜆2| ≤

√
√

𝛼
𝛽 ). We proceed

to ensure the largeness of 𝜃 (𝑟𝑚𝑖𝑑). If we choose 𝜖 so that

𝜖 = min

(cid:110)

1, 1
2

·

1

sup𝑟∈[𝑟𝑚𝑖𝑑,𝑟0]

√︃

1 + (cid:0)𝜃ℎ𝑜𝑚𝑜 (𝑟) + 1(cid:1) 2 · √︁𝛽(𝑟) · 𝑊 (𝑟)

(cid:111)

,

90

we can improve the estimate for 𝜃 and thus close the bootstrap argument. In order to ensure the

largeness of 𝜃, we do the same computation

|𝜃 (𝑟𝑚𝑖𝑑) − 𝜃ℎ𝑜𝑚𝑜 (𝑟𝑚𝑖𝑑)| ≤

√︃

1 + (cid:0)𝜃ℎ𝑜𝑚𝑜 (𝑟𝑚𝑖𝑑) + 1(cid:1) 2 · √︁𝛽(𝑟𝑚𝑖𝑑)𝜖 · 𝑊 (𝑟𝑚𝑖𝑑)

but this time we choose 𝜖 to be even smaller so that

|𝜃 (𝑟𝑚𝑖𝑑) − 𝜃ℎ𝑜𝑚𝑜 (𝑟𝑚𝑖𝑑)| ≤

(cid:0)𝜃ℎ𝑜𝑚𝑜 (𝑟𝑚𝑖𝑑) − 𝜃 𝐿𝐵
𝑚𝑖𝑑

(cid:1),

1
2

whic implies that 𝜃 (𝑟𝑚𝑖𝑑) > 𝜃 𝐿𝐵
𝑚𝑖𝑑.

□

Lemma 4.2.3. The solution (𝜃, 𝜌) satisfying the conditions both in Proposition 4.2.1 and Propo-

sition 4.2.2 exists in 𝑊 1,∞.

Proof. By the finite speed of propagation, we have

| ln 𝜌(𝑟) − ln 𝜌ℎ𝑜𝑚𝑜 (𝑟)| ≤

≤

∫

{𝑟=𝑟}
1 + 𝛾
√𝛾

1 + 𝛾
√𝛾

(cid:16)

· √︁𝛽

|𝜕𝑠 (𝑣1 − 𝑣2)|𝑑𝑠

∥𝑎∥ 𝐿∞ + ∥𝑏∥ 𝐿∞

(cid:17)

· 𝑊 (𝑟),

which is finite (but not bounded) in both 𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑] and 𝑟 ∈ [𝑟𝑚𝑖𝑑, 𝑟0] cases. 𝜃, |𝑎|, |𝑏| are also

bounded as shown in the previous propositions in both cases.

□

4.2.7 Main Theorem

Our main theorem in this paper is

Theorem 4.2.1. Fix a pair of metric components (𝛼, 𝛽) satisfying Assumption 4 with positive

blowup time 𝑟∗ > 0 and 𝑟0 > 𝑟∗, and fix any constant 𝑀 > 0. Then there exists 𝜃 𝐿𝐵
0
homogeneous solutions with 𝜃ℎ𝑜𝑚𝑜 (𝑟0) > 𝜃 𝐿𝐵

0 are stable. More specifically, there exists 𝜖0 > 0 so

so that the

that as long as

∥𝑎(𝑟0)∥ 𝐿∞ + ∥𝑏(𝑟0)∥ 𝐿∞ < 𝜖0,

shock will not form and the solution exists in 𝑊 1,∞ before the blowup time 𝑟 = 𝑟∗.

91

Proof. By Proposition 4.2.1, there exist 𝑟𝑚𝑖𝑑 and 𝜃 𝐿𝐵

𝑚𝑖𝑑 so that the homogeneous solutions with

𝜃ℎ𝑜𝑚𝑜 (𝑟𝑚𝑖𝑑) > 𝜃 𝐿𝐵

𝑚𝑖𝑑 are stable. Since the homogeneous solutions satisfy a system of ordinary

differential equations, there exists a corresponding 𝜃 𝐿𝐵
0
𝜃ℎ𝑜𝑚𝑜 (𝑟0) > 𝜃 𝐿𝐵
0
sition 4.2.1 (for (𝑟∗, 𝑟𝑚𝑖𝑑]), they are stable. The 𝑊 1,∞ claim is from Lemma 4.2.3.

𝑚𝑖𝑑, and by Proposition 4.2.2 (for [𝑟𝑚𝑖𝑑, 𝑟0]) and Propo-
□

so that the homogeneous solutions with

implies 𝜃ℎ𝑜𝑚𝑜 (𝑟𝑚𝑖𝑑) > 𝜃 𝐿𝐵

Remark. This theorem states that the homogeneous 𝜃 ≠ 0 solutions are dynamically stable as long

as the angle 𝜃 between the fluid velocity and time direction −𝜕𝑟 is sufficiently large. Intuitively, as

long as the homogeneous fluid drives away from time direction far enough, the blowup rate of 𝜃

beats the mechanism for shock formation, thus preventing the occurrence of shock. Notice that the

largeness of 𝜃 is essential in our proof when we improve the bootstrap estimate in 𝑟 ∈ (𝑟∗, 𝑟𝑚𝑖𝑑]

region. We introduced the parameter 𝑟𝑚𝑖𝑑 to ensure the monotonicity of the fluid variable 𝜃 and

the metric component 𝛽 in the (𝑟∗, 𝑟𝑚𝑖𝑑] region, relying on the asymptotic behavior described in

Assumption 4.

92

CHAPTER 5

STABILITY OF MEMBRANE EQUATIONS

This chapter aims to prove the global existence of membrane equations for sufficiently small initial

data. This is a separate work from previous chapters.

Membrane equation is a historically interesting problem. In Euclidean space, it describes a

membrane minimizing the area with a given boundary, while in Lorentzian spacetime, it represents

the world sheet of an extended object without external force (see [12]). This paper aims to prove

the global existence of the Lorentzian-type membrane equation

(cid:32)

𝜕𝑖

(cid:33)

𝑚𝑖 𝑗 𝜕𝑗 𝑢
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

√︁

= 0

(5.1)

for sufficiently small initial data, where (𝑥0, 𝑥1, 𝑥2, 𝑥3) = (𝑡, 𝑥1, 𝑥2, 𝜃) represents the coordinate for

R1,2 × T1 and 𝑚𝑖 𝑗 is the component of the Minkowski metric (including T1 as a periodic space

variable)

𝑚 = −𝑑𝑡2 + 𝑑 (𝑥1)2 + 𝑑 (𝑥2)2 + 𝑑𝜃2.

Previously in Lindblad’s work [20], he proved the global existence for small initial data on clas-

sical Minkowski spacetime R1,𝑛, where he used the vector field method, proposed by Klainerman

[18], to achieve the global existence of membrane equation with space dimension greater than 1 for

compactly supported initial data. We record the main ideas in his proof and address the difference

between his strategy and ours.

The vector field method is to use the appropriately chosen weighted vector fields to gain the

decay of derivatives. The general strategy of this method is

1. applying prior inequalities involving

a) the energy of weighted vector fields (acting on the unknown 𝑢),

b) the pointwise bound of the derivatives of 𝑢, and

c) the nonhomogeneous term from the differential equation being considered, and

93

2. running the bootstrap argument to argue the boundedness of the energy, and thus the bound-

edness of the derivatives of 𝑢.

According to the well-known criterion for the global existence of quasilinear wave equation, it is

sufficient to have the boundedness of |𝜕𝑢| + |𝜕2𝑢| to ensure the global existence (where 𝜕 may be

𝜕𝑡, 𝜕1, 𝜕2, or 𝜕𝜃 in our case). To close the bootstrap argument, one has to bound the nonhomogeneous

term by energy with an appropriate decay so that the integrand becomes integrable. In Lindblad’s

argument, he uses three different inequalities: two energy estimates and one 𝐿∞-𝐿1 estimate. He

also makes use of the null structure of the Lorentzian membrane equation to close the bootstrap

argument.

In order to apply the estimates mentioned above, Lindblad commutes the membrane equation

with Λ𝐼 and derive an equation for □Λ𝐼𝑢, in which the right hand side consists of terms falling into

three catogories: terms that are multilinear in 𝑢, of divergence form, and of null form. Here Λ𝐼

may be the composition of

𝜕𝑡, 𝜕𝑥1, ..., 𝜕𝑥𝑛

𝑆 = 𝑡𝜕𝑡 +

𝑛
∑︁

𝑖=1

𝑥𝑖𝜕𝑖

𝐿𝑖 = 𝑡𝜕𝑖 + 𝑥𝑖𝜕𝑡,

1 ≤ 𝑖 ≤ 𝑛

Ω𝑖 𝑗 = 𝑥𝑖𝜕𝑗 − 𝑥 𝑗 𝜕𝑖,

1 ≤ 𝑖, 𝑗 ≤ 𝑛.

Among these vector fields, the dilation field 𝑆 is an obstacle for generalizing the strategy to our

case R1,2 × T1 since there is no naturally analogous dilation field on T1. In order to resolve this

issue, we observe where Lindblad uses this dilation field. The 𝐿∞-𝐿1 estimate

|𝑤(𝑡, 𝑥)| ≤𝐶 (1 + 𝑡 + |𝑥|)−(𝑛−1)/2

∫ 𝑡

∑︁

|𝐼 |≤𝑛−1

0

(cid:169)
(cid:173)
(cid:171)

(cid:13)
(cid:13)

(cid:13)Λ𝐼 𝐹 (𝑠, .)/(1 + 𝑠 + |.|) (𝑛−1)/2(cid:13)

(cid:13)
(cid:13)𝐿1

,

𝑑𝑠 + 𝐶 ( 𝑓 , 𝑔)𝜖(cid:170)
(cid:174)
(cid:172)

requires the dilation field to work. Another place involving 𝑆 is when he takes advantage of the

94

null structure of the membrane equation, the estimate

|𝑄(𝜙, 𝜓)| ≤ 𝐶 (1 + 𝑡 + |𝑥|)−1(|𝜕𝜙||Λ𝜓| + |Λ𝜙||𝜕𝜓|)

also requires the dilation field to be one of the vector fields Λ. Based on these two observations,

we can not directly apply the same argument to R1,2 × T1 case. Instead, we only use

𝜕𝑡, 𝜕𝜃, 𝐿𝑖 = 𝑡𝜕𝑖 + 𝑥𝑖𝜕𝑡,

1 ≤ 𝑖 ≤ 2

to form a Lie algebra and apply the vector field method. Notice that Ω𝑖 𝑗 can be expressed in

terms of 𝐿𝑖 when 𝑡 ≠ 0. In addition, we foliate the spacetime region lying inside the future light

cone (𝑡 > |𝑥|) with the hyperbolic curves (𝑡2 − |𝑥|2 = const). This was introduced by LeFloch

and Ma in [19] where they established the global well-posedness of nonlinear wave equations and

Klein-Gordon equations on R1+3. This foliation helps us capture the decay along the 𝑡 − 𝑥 = const

direction and enables us to close the bootstrap argument.

Another related work is Ifrim and Stingo’s paper (see [14]). In their work, they proved the

global existence for a coupled Klein-Gordon equation on R1,2 with small data. In their paper, they
dyadically decomposed the spacetime region (cid:8)(𝑡, 𝑥)(cid:12)
(cid:12)|𝑡 − |𝑥|| ≲ 𝑡 + |𝑥|, 𝑡 > 0(cid:9) and used the constant-

time-slice energy to bound the weighted spacetime energy. The relation between their equation

and our membrane equation is that, by expanding our solution in Fourier series with respect to 𝜃:

𝑢 = (cid:205)∞

𝑛=−∞ 𝑢𝑛𝑒𝑖𝑛𝜃 and plugging this into our main equation (5.1) (see Lemma 5.1.2)

(cid:0)1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢(cid:1)𝑚𝑖 𝑗 𝜕𝑖𝜕𝑗 𝑢 = 𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑖𝜕𝑗 𝑢,

we would arrive at

□𝑡,𝑥𝑢𝑛 − 𝑛2𝑢𝑛 +

(cid:32)

∑︁

𝑛1+𝑛2+𝑛3=𝑛

− 𝜕𝑡𝑢𝑛1

𝜕𝑡𝑢𝑛2 +

2
∑︁

𝑗=1

𝜕𝑗 𝑢𝑛1

𝜕𝑗 𝑢𝑛2 − (𝑛1𝑛2)𝑢𝑛1

𝑢𝑛2

(cid:33)

× (□𝑡,𝑥𝑢𝑛3 − (𝑛3)2𝑢𝑛3)
(cid:32)

∑︁

=

𝑛1+𝑛2+𝑛3=𝑛

𝜕𝑡𝑢𝑛1

𝜕𝑡𝑢𝑛2

𝑡 𝑢𝑛3 − 2
𝜕2

2
∑︁

𝑗=1

𝜕𝑡𝑢𝑛1

𝜕𝑗 𝑢𝑛2

𝜕𝑡 𝜕𝑗 𝑢𝑛3 +

2
∑︁

𝑖, 𝑗=1

𝜕𝑖𝑢𝑛1

𝜕𝑗 𝑢𝑛2

𝜕𝑖𝜕𝑗 𝑢𝑛3

95

+ 2𝜕𝑡𝑢𝑛1 (𝑛2)𝑢𝑛2 (𝑛3)𝜕𝑡𝑢𝑛3 − 2

+ (𝑛1𝑛2𝑛2

3)𝑢𝑛1

𝑢𝑛2

𝑢𝑛3

(cid:33)

2
∑︁

𝑗=1

𝜕𝑗 𝑢(𝑛2)𝑢𝑛2 (𝑛3)𝜕𝑖𝑢𝑛3

This can be regarded as a strongly coupled system of Klein-Gordan and wave equations on R1,2.

In this paper, we are going to show the global existence of (5.1), which is recorded in Theorem

5.5.2. In section 2, we begin with the setup for the geometry of the spacetime and computation of

geometric quantities that are involved with the energy. In section 3, we prove the global Sobolev

inequality, which plays an essential role in this paper to derive the desired decay. This proof is

parallel to the proof in [28]. Section 4 works for all the derivative estimates we need later, using the

global Sobolev inequality and the exploit of the null structure in the Lorentzian membrane equation.

We treat 𝜕2

𝑡 𝑢 specially because our energy only involves 𝜕𝑡𝑢 but no 𝜕2

𝑡 𝑢. Section 5 establishes the

comparability of the two versions of energy introduced in Section 2. Section 6 shows the energy

estimate using the divergence theorem and the bootstrap mechanism.

5.1 Geometry and Energy

We will use the coordinate (𝑡, 𝑥1, 𝑥2, 𝜃) to represent the points in our spacetime throughout this

paper. In order to adapt the hyperboloidal foliation method, we parametrize the spacetime region
lying inside the future null cone, (cid:8)(𝑡, 𝑥1, 𝑥2, 𝜃)(cid:12)

(cid:12)|𝑥| < 𝑡(cid:9), with (𝜏, 𝜌, 𝜙, 𝜃):

𝑡 = 𝜏 cosh(𝜌)

𝑥1 = 𝜏 sinh(𝜌) cos 𝜙

𝑥2 = 𝜏 sinh(𝜌) sin 𝜙,

where 𝜏 ∈ [2, ∞), 𝜌 ∈ [0, ∞), 𝜙 ∈ [0, 2𝜋), 𝜃 ∈ [0, 2𝜋). The following Lorentz boost vector fields

would be used in this paper.

𝐿1 (cid:66) 𝑡𝜕1 + 𝑥1𝜕𝑡 = (cos 𝜙)𝜕𝜌 − (coth 𝜌 sin 𝜙)𝜕𝜙

𝐿2 (cid:66) 𝑡𝜕2 + 𝑥2𝜕𝑡 = (sin 𝜙)𝜕𝜌 + (coth 𝜌 cos 𝜙)𝜕𝜙.

96

Notice that the latter expression only works for 𝜌 > 0, but 𝐿1, 𝐿2 are well-defined even when 𝜌 = 0.

We introduce a basic computation for the inverse matrix.

Lemma 5.1.1. Let 𝑔𝑖 𝑗 = 𝑚𝑖 𝑗 + 𝜕𝑖𝑢𝜕𝑗 𝑢, then

𝑔𝑖 𝑗 = 𝑚𝑖 𝑗 −

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

(cid:16)

𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙𝑢

(cid:17)

.

Proof. Let 𝑀, 𝐺 be the matrix representations of 𝑚𝑖 𝑗 , 𝑔𝑖 𝑗 respectively, and let 𝑣 be the column

vector representing 𝜕𝑖𝑢. Then we have 𝐺 = 𝑀 + 𝑣𝑣𝑇 and, by the Sherman-Morrison formula,

𝐺−1 = 𝑀 −

𝑀𝑣𝑣𝑇 𝑀
1 + 𝑣𝑇 𝑀𝑣

,

which gives 𝑔𝑖 𝑗 .

□

This metric 𝑔 plays a role since our main equation can be rewritten using 𝑔 with a much simpler

structure than (5.1).

Lemma 5.1.2. The equation (5.1) is equivalent to

□𝑔𝑢 = 0,

where 𝑔𝑖 𝑗 = 𝑚𝑖 𝑗 + 𝜕𝑖𝑢𝜕𝑗 𝑢.

Proof. From the definition of □𝑔, we have

□𝑔𝑢 = 𝑔𝑖 𝑗 𝜕𝑖𝜕𝑗 𝑢 − 𝑔𝑖 𝑗 Γ𝑘

𝑖 𝑗 𝜕𝑘𝑢.

To calculate the second term on the right hand side, we calculate Γ𝑘

𝑖 𝑗 first.

Γ𝑘
𝑖 𝑗 =

=

1
2
1
2

𝑔𝑘𝑙 (𝜕𝑖𝑔𝑙 𝑗 + 𝜕𝑗 𝑔𝑖𝑙 − 𝜕𝑙𝑔𝑖 𝑗 )

𝑔𝑘𝑙 (𝜕𝑖 (𝜕𝑙𝑢𝜕𝑗 𝑢) + 𝜕𝑗 (𝜕𝑖𝑢𝜕𝑙𝑢) − 𝜕𝑙 (𝜕𝑖𝑢𝜕𝑗 𝑢))

= 𝑔𝑘𝑙 𝜕𝑙𝑢(𝜕𝑖𝜕𝑗 𝑢),

which implies

□𝑔𝑢 = 𝑔𝑖 𝑗 (𝜕𝑖𝜕𝑗 𝑢) (1 − 𝑔𝑘𝑙 𝜕𝑘𝑢𝜕𝑙𝑢).

97

Observe that

𝑔𝑖 𝑗 𝜕𝑖𝜕𝑗 𝑢 =

(cid:18)

𝑚𝑖 𝑗 −

𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙𝑢

(cid:17) (cid:19)

𝜕𝑖𝜕𝑗 𝑢

(cid:16)

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢
1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

= 𝑚𝑖 𝑗 (𝜕𝑖𝜕𝑗 𝑢) −

𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙𝑢(𝜕𝑖𝜕𝑗 𝑢)

is equivalent to (5.1) and

𝑔𝑖 𝑗 𝜕𝑖𝑢𝜕𝑗 𝑢 = 𝑚𝑖 𝑗 𝜕𝑖𝑢𝜕𝑗 𝑢 −

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

(𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙 𝜕𝑖𝑢𝜕𝑗 𝑢)

1
1 + 𝜎

𝜎2

= 𝜎 −

=

𝜎
1 + 𝜎

is never 1, where 𝜎 = 𝑚𝑖 𝑗 𝜕𝑖𝑢𝜕𝑗 𝑢.

□

Remark. Lemma 5.1.2 implies that 𝑢 could be regarded as a wave map. See [27] for more details.

Using the Minkowski metric 𝑚𝑖 𝑗 and the dynamic metric 𝑔𝑖 𝑗 , we can define the corresponding

tensors

and currents

where

𝑄𝑖

𝑗 [𝑢; 𝑚] = 𝑚𝑖𝑘 𝜕𝑘𝑢𝜕𝑗 𝑢 −

𝑄𝑖

𝑗 [𝑢; 𝑔] = 𝑔𝑖𝑘 𝜕𝑘𝑢𝜕𝑗 𝑢 −

𝜎[𝑢; 𝑚]𝛿𝑖
𝑗

1
2
1
𝜎[𝑢; 𝑔]𝛿𝑖
𝑗 ,
2

(𝑋) 𝐽 [𝑢; 𝑚] = 𝑄𝑖

𝑗 [𝑢; 𝑚] 𝑋 𝑗 𝜕𝑖

(𝑋) 𝐽 [𝑢; 𝑔] = 𝑄𝑖

𝑗 [𝑢; 𝑔] 𝑋 𝑗 𝜕𝑖,

𝜎[𝑢; 𝑚] = 𝑚𝑖 𝑗 𝜕𝑖𝑢𝜕𝑗 𝑢

𝜎[𝑢; 𝑔] = 𝑔𝑖 𝑗 𝜕𝑖𝑢𝜕𝑗 𝑢.

98

To apply the divergence theorem, we note that the divergence of the current (𝑋) 𝐽 is given by

𝑑𝑖𝑣𝑔

(cid:16)(𝑋) 𝐽

(cid:17)

= □𝑔𝑢(∇ 𝑗 𝑢) 𝑋 𝑗 + ∇𝑖𝑢∇ 𝑗 𝑢∇𝑖 𝑋 𝑗 −

∇𝑘𝑢∇𝑘𝑢∇𝑖 𝑋𝑖.

1
2

(5.2)

If we plug 𝜕𝑡 into 𝑋, the above expression simplifies to

𝑑𝑖𝑣𝑔

(cid:16)(𝜕𝑡 ) 𝐽

(cid:17)

= □𝑔𝑢(∇𝑡𝑢) + (∇𝑖𝑢∇ 𝑗 𝑢)Γ

𝑗
𝑖𝑡 −

(∇𝑘𝑢∇𝑘𝑢)Γ𝑖
𝑖𝑡 .

1
2

Since are going to apply the divergence theorem on

{(𝑡, 𝑥1, 𝑥2, 𝜃)|𝜏0 ≤ 𝜏 ≤ 𝜏1},

which is bounded by Σ𝜏 = {(𝑡, 𝑥1, 𝑥2, 𝜃)|√︁𝑡2 − |𝑥|2 = 𝜏} with 𝜏 equals 𝜏0 and 𝜏1 respectively, we

define our energy to be

E𝜏 [𝑢; 𝑚]2 = 2

∫

⟨(𝜕𝑡 ) 𝐽 [𝑢; 𝑚], 𝜕𝜏⟩𝑚𝑑𝑆𝑚

E𝜏 [𝑢; 𝑔]2 = 2

∫

Σ𝜏

⟨(𝜕𝑡 ) 𝐽 [𝑢; 𝑔], (cid:174)𝑛⟩𝑔𝑑𝑆𝑔

E ≤𝑠

𝜏 [𝑢; 𝑚] =

E ≤𝑠

𝜏 [𝑢; 𝑔] =

Σ𝜏
∑︁

E𝜏 [𝐿𝛾1 𝜕𝛾2

𝜃 𝑢; 𝑚]

|𝛾1|+|𝛾2|≤𝑠
∑︁

E𝜏 [𝐿𝛾1 𝜕𝛾2

𝜃 𝑢; 𝑔]

|𝛾1|+|𝛾2|≤𝑠

where (cid:174)𝑛 is the future-directed normal vector of Γ𝜏 with respect to 𝑔. In section 5, we are going to

establish the comparability of the energies with respect to 𝑚 and with respect to 𝑔. In other words,

we will be able to ensure the smallness of derivative of 𝑢 and thus smallness of 𝑔𝑖 𝑗 − 𝑚𝑖 𝑗 .

Observe that we can complete the square for the integrand in E𝜏 [𝑢; 𝑚]:

⟨(𝜕𝑡 ) 𝐽 [𝑢; 𝑚], 𝜕𝜏⟩

𝑗 (𝜕𝑡) 𝑗 𝜕𝑖, cosh(𝜌)𝜕𝑡 + sinh(𝜌) cos 𝜙𝜕𝑥1 + sinh(𝜌) sin 𝜙𝜕𝑥2⟩

= ⟨𝑄𝑖
(cid:18)

=

−∇𝑡𝑢∇𝑡𝑢 +

(cid:19)

𝜎

1
2

cosh(𝜌) + (∇1𝑢∇𝑡𝑢) sinh(𝜌) cos 𝜙 + (∇2𝑢∇𝑡𝑢) sinh(𝜌) sin 𝜙

99

=

(cid:16)

1
2

(∇𝑡𝑢)2 + (∇1𝑢)2 + (∇2𝑢)2 + (∇𝜃𝑢)2)

(cid:17)

cosh(𝜌)

+ (∇1𝑢∇𝑡𝑢) sinh(𝜌) cos 𝜙 + (∇2𝑢∇𝑡𝑢) sinh(𝜌) sin 𝜙

(cid:32)

√︁

cosh(𝜌)∇1𝑢 +

sinh(𝜌)
√︁cosh(𝜌)

(cid:33) 2

cos 𝜙∇𝑡𝑢

+

(cid:32)

√︁

1
2

cosh(𝜌)∇2𝑢 +

sinh(𝜌)
√︁cosh(𝜌)

(cid:33) 2

sin 𝜙∇𝑡𝑢

1
2

1
cosh(𝜌)
1
𝜏2 cosh(𝜌)

(∇𝑡𝑢)2 +

|𝐿1𝑢|2 +

1
2
1
2

cosh(𝜌) (∇𝜃𝑢)2

1
𝜏2 cosh(𝜌)

|𝐿2𝑢|2 +

1
2

1
cosh(𝜌)

|𝜕𝑡𝑢|2 +

1
2

cosh(𝜌)|𝜕𝜃𝑢|2,

=

=

1
2

+

1
2

where the ∇ above denotes the connection with respect to the Minkowski metric 𝑚𝑖 𝑗 .

5.2 Sobolev inequality

The following theorem is mostly following the analogous one in [28].

Theorem 5.2.1. (𝐿∞-𝐿2 estimate.) Let 𝑙 ∈ R. Then

| 𝑓 (𝜏, 𝜌, 𝜙, 𝜃)|2 ≲ 𝜏−2(cosh 𝜌)−1−𝑙 ∑︁

∫

|𝛾1|+|𝛾2|≤2

Σ𝜏

(cosh 𝜌)𝑙 |𝐿𝛾1 𝜕𝛾2
𝜃

𝑓 |2𝑑𝑣𝑜𝑙 ˜Σ𝜏

Proof. We discuss two cases separately.

Case 1: 𝜌 < 5

3. In this case, we are going to apply the standard Sobolev inequality with the metric

ℎ1 (on Σ𝜏), where

ℎ1 = 𝑑𝜌2 + sinh(𝜌)2𝑑𝜙2 + 𝑑𝜃2
𝑑𝜌2 + sinh(𝜌)2𝑑𝜙2(cid:17)
ℎ𝜏 = 𝜏2 (cid:16)

+ 𝑑𝜃2.

By the Sobolev inequality, we have

∫

| 𝑓 (𝜏, 𝜌, 𝜙, 𝜃)|2 ≲ ∑︁
|𝛾|≤2
≲ ∑︁

Σ𝜏∩{𝜌<2}
∫

|∇𝛾 𝑓 |2
ℎ1

𝑑𝑣𝑜𝑙ℎ1

|𝐿𝛾1 𝜕𝛾2
𝜃

𝑓 |2𝑑𝑣𝑜𝑙ℎ1

|𝐿𝛾1 𝜕𝛾2
𝜃

𝑓 |2𝑑𝑣𝑜𝑙ℎ𝜏

|𝛾1|+|𝛾2|≤2
= 𝜏−2 ∑︁

Σ𝜏∩{𝜌<2}

∫

Σ𝜏∩{𝜌<2}
|𝛾1|+|𝛾2|≤2
≈ 𝜏−2 cosh(𝜌)−1−𝑙 ∑︁

∫

Σ𝜏∩{𝜌<2}

cosh(𝜌)𝑙 |𝐿𝛾1 𝜕𝛾2
𝜃

𝑓 |2𝑑𝑣𝑜𝑙ℎ𝜏 ,

|𝛾1|+|𝛾2|≤2

100

where the reasons are as follows. The second ≲ is because

|∇ 𝑓 |2
ℎ1

= (𝜕𝜌 𝑓 )2 +

1
sinh(𝜌)2

(𝜕𝜙 𝑓 )2 + (𝜕𝜃 𝑓 )2

≤ |𝐿1 𝑓 |2 + |𝐿2 𝑓 |2 + |𝜕𝜃 𝑓 |2

and

|∇∇ 𝑓 |2
ℎ1

= (ℎ1)𝑖𝑘 (ℎ1) 𝑗𝑙∇∇ 𝑓 (𝜕𝑖, 𝜕𝑗 )∇∇ 𝑓 (𝜕𝑘 , 𝜕𝑙)
≲ ∑︁

𝑓 |2.

|𝐿𝛾1 𝜕𝛾2
𝜃

The ≈ is because we are focusing on a compact region (of (𝜌, 𝜙, 𝜃)).

|𝛾1|+|𝛾2|≤2

Case 2: 𝜌 > 4

3. In this case, we are going to use the metric ℎ0 instead, where

ℎ0 = 𝑑𝜌2 + 𝑑𝜙2 + 𝑑𝜃2.

By the Sobolev inequality,

| 𝑓 (𝜏, 𝜌, 𝜙, 𝜃) cosh(𝜌)𝑙/2 sinh(𝜌)1/2|2
∇𝛾 (cid:16)
≲ ∑︁
|𝛾|≤2
∑︁

Σ𝜏∩{𝜌>1}

∫

∫

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

𝑓 cosh(𝜌)𝑙/2 sinh(𝜌)1/2(cid:17)(cid:12)
(cid:12)
(cid:12)

2

ℎ0

𝑑𝑣𝑜𝑙ℎ0

cosh(𝜌)𝑙 |∇𝛾 𝑓 |2
ℎ0

𝑑𝑣𝑜𝑙ℎ1

≈

|𝛾|≤2
≲ ∑︁

Σ𝜏∩{𝜌>1}
∫

|𝛾1|+|𝛾2|≤2
= 𝜏−2 ∑︁

Σ𝜏∩{𝜌>1}

∫

|𝛾1|+|𝛾2|≤2

Σ𝜏∩{𝜌>1}

cosh(𝜌)𝑙 (cid:12)

(cid:12)𝐿𝛾1 𝜕𝛾2

𝜃

𝑓 (cid:12)
(cid:12) 𝑑𝑣𝑜𝑙ℎ1

cosh(𝜌)𝑙 (cid:12)

(cid:12)𝐿𝛾1 𝜕𝛾2

𝜃

𝑓 (cid:12)
(cid:12)

2 𝑑𝑣𝑜𝑙ℎ𝜏 ,

where the reasons are as follows. The ≈ is because we exclude an open neighborhood of 𝜌 = 0,

and thus sinh(𝜌) and cosh(𝜌) are comparable. The second ≲ is because

|∇ 𝑓 |2
ℎ0

= (𝜕𝜌 𝑓 )2 + (𝜕𝜙 𝑓 )2 + (𝜕𝜃 𝑓 )2

≤ |𝐿1 𝑓 |2 + |𝐿2 𝑓 |2 + |𝜕𝜃 𝑓 |2

101

and

|∇∇ 𝑓 |2
ℎ0

= (ℎ0)𝑖𝑘 (ℎ0) 𝑗𝑙 (𝜕𝑖𝜕𝑗 𝑓 ) (𝜕𝑘 𝜕𝑙 𝑓 )
|𝐿𝛾1 𝜕𝛾2
𝜃

𝑓 |2.

∑︁

≤

|𝛾1|+|𝛾2|≤2

□

5.3 Estimate for derivatives

Lemma 5.3.1. We calculate the following three terms in this lemma.

𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢 = 𝐴 − 1
1
cosh(𝜌)2

𝑚𝑖 𝑗 𝜕𝑖𝜕𝑗 𝑢 = −

𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑖𝜕𝑗 𝑢.

𝜕𝑡 𝜕𝑡𝑢 + 𝐵

Proof. Using the identity 𝜕𝑖 = 1

𝑡 𝐿𝑖 − 𝑥𝑖

𝑡 𝜕𝑡 for 𝑖 = 1, 2, we have

𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢 = −(𝜕𝑡𝑢)2 +

(cid:18) 1
𝑡

2
∑︁

𝑖=1

𝐿𝑖𝑢 −

(cid:19) 2

𝜕𝑡𝑢

𝑥𝑖
𝑡

+ (𝜕𝜃𝑢)2

= −

1
cosh(𝜌)2

(𝜕𝑡𝑢)2 +

2
∑︁

𝑖=1

1
𝑡2

|𝐿𝑖𝑢|2 − 2

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

(𝐿𝑖𝑢𝜕𝑡𝑢) + (𝜕𝜃𝑢)2

(cid:66) 𝐴 − 1,

𝑚𝑖 𝑗 𝜕𝑖𝜕𝑗 𝑢 = −𝜕𝑡 𝜕𝑡𝑢 +

= −𝜕𝑡 𝜕𝑡𝑢 +

−

1
𝑡

2
∑︁

𝑖=1
2
∑︁

𝑖=1
𝑥𝑖
𝑡2

(cid:18) 1
𝑡

𝐿𝑖 −

𝑥𝑖
𝑡

(cid:19) (cid:18) 1
𝑡

𝜕𝑡

𝐿𝑖𝑢 −

(cid:19)

𝜕𝑡𝑢

𝑥𝑖
𝑡

+ 𝜕𝜃 𝜕𝜃𝑢

(cid:16) 1
𝑡2

𝐿𝑖 𝐿𝑖𝑢 −

𝑥𝑖
𝑡

1
𝑡

𝐿𝑖𝜕𝑡𝑢 −

𝑥𝑖
𝑡

1
𝑡

𝜕𝑡 𝐿𝑖𝑢 +

𝑥𝑖
𝑡

𝑥𝑖
𝑡

𝜕𝑡 𝜕𝑡𝑢

1
𝑡2

𝑥𝑖
𝑡
𝑥𝑖
𝑡

1
𝑡

𝐿𝑖𝑢 −

𝐿𝑖𝜕𝑡𝑢 −

𝑥𝑖
𝑡
𝑥𝑖
𝑡

𝑥𝑖
𝑡2

1
𝑡

(cid:17)

𝜕𝑡𝑢

+ 𝜕𝜃 𝜕𝜃𝑢

𝜕𝑡 𝐿𝑖𝑢 −

(cid:17)

𝜕𝑡𝑢

1
𝑡

𝐿𝑖𝑢 −

1
𝑡

(1 −

𝑥𝑖𝑥𝑖
𝑡2

)𝜕𝑡𝑢 +

= −

1
cosh(𝜌)2

𝜕𝑡 𝜕𝑡𝑢 +

2
∑︁

𝑖=1

(cid:16) 1
𝑡2

𝐿𝑖 𝐿𝑖𝑢 −

+ 𝜕𝜃 𝜕𝜃𝑢

(cid:66) −

1
cosh(𝜌)2

𝜕𝑡 𝜕𝑡𝑢 + 𝐵

102

and

𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑖𝜕𝑗 𝑢 =

𝜕𝑡𝑢𝜕𝑡𝑢(𝜕𝑡 𝜕𝑡𝑢) − 2

𝜕𝑡𝑢

(cid:18) 1
𝑡

2
∑︁

𝑖=1

𝐿𝑖𝑢 −

𝑥𝑖
𝑡

𝜕𝑡𝑢

(cid:19) (cid:18) 1
𝑡

𝐿𝑖𝜕𝑡𝑢 −

(cid:19)

𝜕𝑡 𝜕𝑡𝑢

𝑥𝑖
𝑡

+

(cid:18) 1
𝑡

2
∑︁

𝑖, 𝑗=1

𝐿𝑖𝑢 −

𝑥𝑖
𝑡

𝜕𝑡𝑢

(cid:19) (cid:18) 1
𝑡

𝐿 𝑗 𝑢 −

𝑥 𝑗
𝑡

𝜕𝑡𝑢

(cid:19) (cid:18)(cid:18) 1
𝑡

𝐿𝑖 −

𝑥𝑖
𝑡

(cid:19) (cid:18) 1
𝑡

𝜕𝑡

𝐿 𝑗 𝑢 −

(cid:19)(cid:19)

𝜕𝑡𝑢

𝑥 𝑗
𝑡

− 2𝜕𝑡𝑢𝜕𝜃𝑢𝜕𝑡 𝜕𝜃𝑢 + 2

(cid:18) 1
𝑡

2
∑︁

𝑖=1

𝐿𝑖𝑢 −

(cid:19)

𝜕𝑡𝑢

𝑥𝑖
𝑡

𝜕𝜃𝑢

(cid:18) 1
𝑡

𝐿𝑖𝜕𝜃𝑢 −

(cid:19)

𝜕𝑡 𝜕𝜃𝑢

𝑥𝑖
𝑡

+ 𝜕𝜃𝑢𝜕𝜃𝑢𝜕𝜃 𝜕𝜃𝑢
(cid:32)

=

1
cosh(𝜌)4

𝜕𝑡𝑢𝜕𝑡𝑢 + 2

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

𝜕𝑡𝑢𝐿𝑖𝑢

+

2
∑︁

𝑖, 𝑗=1

𝑥𝑖
𝑡

𝑥 𝑗
𝑡

1
𝑡2

𝐿𝑖𝑢𝐿 𝑗 𝑢 − 2

2
∑︁

𝑖, 𝑗=1

𝑥𝑖
𝑡

1
𝑡

(𝑥 𝑗 )2
𝑡2

(cid:33)

𝐿𝑖𝑢𝜕𝑡𝑢

(𝜕𝑡 𝜕𝑡𝑢)

2
∑︁

− 2

+

𝑖=1
2
∑︁

𝑖, 𝑗=1
(cid:32)

1
𝑡2

𝜕𝑡𝑢𝐿𝑖𝑢(𝐿𝑖𝜕𝑡𝑢) + 2

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

𝜕𝑡𝑢𝜕𝑡𝑢(𝐿𝑖𝜕𝑡𝑢)

(cid:18) 1
𝑡

𝐿𝑖𝑢 −

𝑥𝑖
𝑡

𝜕𝑡𝑢

(cid:19) (cid:18) 1
𝑡

𝐿 𝑗 𝑢 −

(cid:19)

𝜕𝑡𝑢

𝑥 𝑗
𝑡

(𝐿𝑖 𝐿 𝑗 𝑢) −

𝑥 𝑗
𝑡

1
𝑡

(𝐿𝑖𝜕𝑡𝑢) −

𝑥𝑖
𝑡

1
𝑡

(𝜕𝑡 𝐿 𝑗 𝑢)

1
𝑡2

𝑥𝑖
𝑡2

1
𝑡

−

𝐿 𝑗 𝑢 −

1
𝑡

(𝛿𝑖 𝑗 −

𝑥𝑖𝑥 𝑗
𝑡2

)𝜕𝑡𝑢 +

𝑥𝑖
𝑡

1
𝑡2

𝐿 𝑗 𝑢 −

(cid:33)

𝜕𝑡𝑢

𝑥𝑖
𝑡

𝑥 𝑗
𝑡2

− 2𝜕𝑡𝑢𝜕𝜃𝑢(𝜕𝑡 𝜕𝜃𝑢) + 2

2
∑︁

𝑖=1

1
𝑡2

𝐿𝑖𝑢𝜕𝜃𝑢(𝐿𝑖𝜕𝜃𝑢) − 2

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

𝐿𝑖𝑢𝜕𝜃𝑢(𝜕𝑡 𝜕𝜃𝑢)

− 2

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

1
cosh(𝜌)4

(cid:66) (cid:169)
(cid:173)
(cid:171)
+ 𝐶,

𝜕𝑡𝑢𝜕𝜃𝑢(𝐿𝑖𝜕𝜃𝑢) +

2|𝑥|2
𝑡2

𝜕𝑡𝑢𝜕𝜃𝑢(𝜕𝑡 𝜕𝜃𝑢) + 𝜕𝜃𝑢𝜕𝜃𝑢(𝜕𝜃 𝜕𝜃𝑢)

|𝜕𝑡𝑢|2 +

2
cosh(𝜌)2

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

𝜕𝑡𝑢𝐿𝑖𝑢 +

2
∑︁

𝑖, 𝑗=1

𝑥𝑖
𝑡

𝑥 𝑗
𝑡

1
𝑡2

(𝜕2

𝑡 𝑢)

𝐿𝑖𝑢𝐿 𝑗 𝑢(cid:170)
(cid:174)
(cid:172)

103

where

𝐴 = 𝐴(𝑡, 𝑥, 𝜕𝑡𝑢, 𝐿𝑢, 𝜕𝜃𝑢)

𝐵 = 𝐵(𝑡, 𝑥, 𝜕𝑡𝑢, 𝐿𝑢, 𝐿𝐿𝑢, 𝐿𝜕𝑡𝑢)

𝐶 = 𝐶 (𝑡, 𝑥, 𝜕𝑡𝑢, 𝐿𝑢, 𝐿𝐿𝑢, 𝐿𝜕𝑡𝑢, 𝐿𝜕𝜃𝑢, 𝜕𝑡 𝜕𝜃𝑢, 𝜕𝜃 𝜕𝜃𝑢).

□

Remark. By the notation above, we could simplify the equation 𝑔𝑖 𝑗 𝜕𝑖𝜕𝑗 𝑢 = 0, or

𝑚𝑖 𝑗 𝜕𝑖𝜕𝑗 𝑢 =

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

(𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑖𝜕𝑗 𝑢),

to the following identity

(cid:18)

−

𝐴

1
cosh(𝜌)2

(cid:19)

𝑡 𝑢 + 𝐵
𝜕2

1
cosh(𝜌)4

|𝜕𝑡𝑢|2 +

2
cosh(𝜌)2

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

𝜕𝑡𝑢𝐿𝑖𝑢 +

2
∑︁

𝑖, 𝑗=1

𝑥𝑖
𝑡

𝑥 𝑗
𝑡

1
𝑡2

= (𝜕2

𝑡 𝑢) (cid:169)
(cid:173)
(cid:171)

+ 𝐶.

𝐿𝑖𝑢𝐿 𝑗 𝑢(cid:170)
(cid:174)
(cid:172)

The point is that, since our energy does not involve second derivative with respect to time, we need

the main equation to help control 𝜕2

𝑡 𝑢. Therefore, we solve for 𝜕2

𝑡 𝑢:

𝑡 𝑢 = cosh(𝜌)2
𝜕2

1 + (cid:205)2
𝑖=1

𝐴𝐵 − 𝐶

1

𝑡2 |𝐿𝑖𝑢|2 + |𝜕𝜃𝑢|2 + cosh(𝜌)2 (cid:205)2

𝑖, 𝑗=1

1
𝑡2

𝑥𝑖
𝑡

𝑥 𝑗
𝑡 𝐿𝑖𝑢𝐿 𝑗 𝑢

.

(5.3)

We begin with estimating pointwise upper bound for derivative with at most one 𝜕𝑡.

Lemma 5.3.2. We have

|𝐿𝛾1 𝜕𝛾2

𝜃 𝜕𝑡𝑢| ≲ 1
𝜏

E ≤𝑠

𝜏 [𝑢; 𝑚]

|𝐿𝛾1 𝜕𝛾2

𝜃 𝐿𝑖𝑢| ≲ E ≤𝑠

𝜏 [𝑢; 𝑚]
1
𝜏 cosh(𝜌)

E ≤𝑠

𝜏 [𝑢; 𝑚]

|𝐿𝛾1 𝜕𝛾2

𝜃 𝜕𝜃𝑢| ≲

for |𝛾1| + |𝛾2| + 3 ≤ 𝑠 + 1 and 𝑖 = 1, 2.

104

Proof. Using the global Sobolev inequality Theorem 5.2.1, we have

|𝐿𝛾1 𝜕𝛾2

𝜃 𝜕𝑡𝑢|2 ≲ 1
𝜏2

∑︁

∫

|𝛾′
1

|+|𝛾′
2

|≤2

Σ𝜏

1
cosh(𝜌)

|𝐿𝛾′

1 𝜕𝛾′

𝜃 𝐿𝛾1 𝜕𝛾2

2

𝜃 𝜕𝑡𝑢|2𝑑𝑣𝑜𝑙

(cid:32)

∫

∑︁

|𝛾′
1

|+|𝛾′
2

|≤𝑠

Σ𝜏

E ≤𝑠

𝜏 [𝑢; 𝑚]2,

≲ 1
𝜏2

≤

1
𝜏2

1
cosh(𝜌)

|𝜕𝑡 𝐿𝛾′

1 𝜕𝛾′

2

𝜃 𝑢|2 +

2
∑︁

𝑖=1

1
cosh(𝜌)

(cid:33)

|𝜕𝑖 𝐿𝛾′

1 𝜕𝛾′

2

𝜃 𝑢|2

where we use [𝐿𝑖, 𝜕𝑡] = −𝜕𝑖, [𝐿𝑖, 𝜕𝑗 ] = −𝛿𝑖 𝑗 𝜕𝑡 and 𝜕𝑖 = 1

𝑡 𝐿𝑖 − 𝑥𝑖

𝑡 𝜕𝑡,

|𝐿𝛾1 𝜕𝛾2

𝜃 𝐿𝑖𝑢|2 ≲ 1
𝜏2

2
∑︁

≤

∑︁

∫

|𝛾′
1

|+|𝛾′
2

|≤2

Σ𝜏

∑︁

∫

𝑗=1

|𝛾′
1

|+|𝛾′
2

|≤𝑠

Σ𝜏

≲ E ≤𝑠

𝜏 [𝑢; 𝑚]2,

1
cosh(𝜌)

|𝐿𝛾′

1 𝜕𝛾′

𝜃 𝐿𝛾1 𝜕𝛾2

𝜃 𝐿𝑖𝑢|2𝑑𝑣𝑜𝑙

2

1
𝜏2 cosh(𝜌)

|𝐿 𝑗 𝐿𝛾′

1 𝜕𝛾′

2

𝜃 𝑢|2𝑑𝑣𝑜𝑙

and

|𝐿𝛾1 𝜕𝛾2

𝜃 𝜕𝜃𝑢|2 ≲

1
𝜏2 cosh(𝜌)2

=

≤

1
𝜏2 cosh(𝜌)2

1
𝜏2 cosh(𝜌)2

∑︁

∫

|𝛾′
1

|+|𝛾′
2
∑︁

Σ𝜏

|≤2

∫

|𝛾′
1

|+|𝛾′
2

|≤𝑠

Σ𝜏

E ≤𝑠

𝜏 [𝑢; 𝑚]2.

cosh(𝜌)|𝐿𝛾′

1 𝜕𝛾′

𝜃 𝐿𝛾1 𝜕𝛾2

2

𝜃 𝜕𝜃𝑢|2𝑑𝑣𝑜𝑙

cosh(𝜌)|𝜕𝜃 𝐿𝛾′

1 𝜕𝛾′

2

𝜃 𝑢|2𝑑𝑣𝑜𝑙

□

Now we proceed to estimating the pointwise upper bound for second derivative with respect to

time. This requires the following bootstrap assumption, which we will assume from now on.

Bootstrap Assumption. Our bootstrap assumption is

E ≤𝑠

𝜏 [𝑢; 𝑚] ≤ 𝜖

(5.4)

for 𝜏 ∈ [𝜏1, 𝜏2], where 2 ≤ 𝜏1 < 𝜏2 are arbitrary, and 0 < 𝜖 ≤ 1 and 𝑠 will be chosen later.

105

Lemma 5.3.3. If □𝑔𝑢 = 0 and 𝑢 satisfies the bootstrap assumption 5.4, then

|𝐿𝛾1 𝜕𝑡 𝐿𝛾2 𝜕𝑡 𝐿𝛾3 𝜕𝛾4

𝜃 𝑢| ≲ cosh(𝜌)

𝜏

E ≤𝑠

𝜏 [𝑢; 𝑚]

for |𝛾| + 4 ≤ 𝑠 + 1 and 𝜖 sufficiently small, where |𝛾| = |𝛾1| + |𝛾2| + |𝛾3| + |𝛾4|.

Proof. From Remark 2, we have a pointwise estimate for 𝜕2

𝑡 𝑢:

|𝜕2

𝑡 𝑢| = cosh(𝜌)2

1 + (cid:205)2
𝑖=1

| 𝐴𝐵 − 𝐶 |

1

𝑡2 |𝐿𝑖𝑢|2 + |𝜕𝜃𝑢|2 + cosh(𝜌)2 (cid:205)2

𝑖, 𝑗=1

1
𝑡2

𝑥𝑖
𝑡

𝑥 𝑗
𝑡 𝐿𝑖𝑢𝐿 𝑗 𝑢

≲ cosh(𝜌)2 | 𝐴𝐵 − 𝐶 |
𝜖 2

1 − 4
𝜏2

≲ cosh(𝜌)2| 𝐴𝐵 − 𝐶 |

when 𝜖 is sufficiently small, where the first ≲ is because of the bootstrap assumption 5.4. Using

Lemma 5.3.2, we find that

𝐴𝐵 − 𝐶 ≲

1
𝜏 cosh(𝜌)

E ≤𝑠

𝜏 [𝑢; 𝑚],

which gives the result when |𝛾| = 0. To deal with the case |𝛾| > 0, we observe that

|𝐿𝛾1 𝜕𝛾2

𝜃 𝜕2

𝑡 𝑢| ≲ cosh(𝜌)2 ∑︁

|𝐿𝛾′

1 𝜕𝛾′

2

𝜃 ( 𝐴𝐵 − 𝐶)|

(cid:32)

|𝛾′ |+|𝛾′′ |≤|𝛾|

1

𝐿𝛾′′

1 𝜕𝛾′′
𝜃

2

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

𝑡2 |𝐿𝑖𝑢|2 + |𝜕𝜃𝑢|2 + cosh(𝜌)2 (cid:205)2
Since |𝛾| + 4 ≤ 𝑠 + 1 and 𝐴𝐵 − 𝐶 involves at most the second order derivatives, we get the pointwise

1 + (cid:205)2
𝑖=1

𝑥 𝑗
𝑡 𝐿𝑖𝑢𝐿 𝑗 𝑢

𝑖, 𝑗=1

𝑥𝑖
𝑡

1
𝑡2

1

(cid:33)(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

.

estimates

|𝐿𝛾′

1 𝜕𝛾′

2

𝜃 ( 𝐴𝐵 − 𝐶)| ≲

𝐿𝛾′′

1 𝜕𝛾′′
𝜃

2

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

(cid:32)

1 + (cid:205)2
𝑖=1

1
𝜏 cosh(𝜌)

E ≤𝑠

𝜏 [𝑢; 𝑚],

1

1

𝑡2 |𝐿𝑖𝑢|2 + |𝜕𝜃𝑢|2 + cosh(𝜌)2 (cid:205)2

𝑖, 𝑗=1

(cid:33)(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

1
𝑡2

𝑥𝑖
𝑡

𝑥 𝑗
𝑡 𝐿𝑖𝑢𝐿 𝑗 𝑢

≲ 1
𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚]

when |𝛾′′| > 0. This implies that

|𝐿𝛾1 𝜕𝛾2

𝜃 𝜕2

𝑡 𝑢| ≲ cosh(𝜌)

𝜏

E ≤𝑠

𝜏 [𝑢; 𝑚].

106

To do induction on |𝛾|, we observe that

𝐿𝛾1 𝜕𝑡 𝐿𝛾2 𝜕𝑡 𝐿𝛾3 𝜕𝛾4

𝜃 𝑢 = 𝐿𝛾1 𝐿𝛾2 𝐿𝛾3 𝜕𝛾4

𝜃 𝜕2

𝑡 𝑢 + 𝐿𝛾1 𝜕𝑡 𝐿𝛾2 [𝜕𝑡, 𝐿𝛾3]𝜕𝛾4
𝜃 𝑢

+ 𝐿𝛾1 [𝜕𝑡, 𝐿𝛾2 𝐿𝛾3]𝜕𝛾4

𝜃 𝜕𝑡𝑢.

Since [𝜕𝑡, 𝐿𝛾3] and [𝜕𝑡, 𝐿𝛾2 𝐿𝛾3] are linear combinations of 𝜕𝑡 and 𝜕𝑖 = 1

𝑡 𝐿𝑖 − 𝑥𝑖

𝑡 𝜕𝑡, and they decrease

the order of the derivatives, the result follows by induction hypothesis with the aid of Lemma

5.3.2.

□

Remark. The operator 𝐿𝑖 preserves the decays 1

𝑡 and 𝑥𝑖

𝑡 , which means that

and

𝐿𝑖

(cid:19)

𝑓

(cid:18) 1
𝑡

≤

1
𝑡

(|𝐿𝑖 𝑓 | + | 𝑓 |)

𝐿𝑖

(cid:19)

𝑓

(cid:18) 𝑥 𝑗
𝑡

≲ |𝐿𝑖 𝑓 | + | 𝑓 |.

It also preserves cosh(𝜌)2 by the following way:

𝐿𝑖 (cid:16)

cosh(𝜌)2(cid:17) ≲ cosh(𝜌)2.

Proposition 5.3.1. Let 𝑠 ≥ 5. If □𝑔𝑢 = 0 and 𝑢 satisfies the bootstrap assumption 5.4, then

∫

Σ𝜏

| (cid:0)[□𝑔, 𝐿𝛼1 𝜕𝛼2

𝜃 ]𝑢(cid:1) (cid:0)𝜕𝑡 𝐿𝛼1 𝜕𝛼2

𝜃 𝑢(cid:1) |𝑑𝑣𝑜𝑙 ≲ 1
𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚]4

for any |𝛼| = |𝛼1| + |𝛼2| ≤ 𝑠, where the implicit constant depends only on 𝑠.

Proof. We decompose the bracket first.

[𝐿𝛼1 𝜕𝛼2

𝜃 , □𝑔]𝑢 = [𝐿𝛼1 𝜕𝛼2

𝜃 , 𝑚𝑖 𝑗 𝜕𝑖𝜕𝑗 ]𝑢 + [𝐿𝛼1 𝜕𝛼2

𝜃 , (𝑔𝑖 𝑗 − 𝑚𝑖 𝑗 )𝜕𝑖𝜕𝑗 ]𝑢.

The first term on the right hand side vanishes since

[𝐿 𝑘 , 𝑚𝑖 𝑗 𝜕𝑖𝜕𝑗 ] = [𝑡𝜕𝑘 + 𝑥 𝑘 𝜕𝑡, −𝜕2

𝑡 + 𝜕2

1 + 𝜕2

2 + 𝜕2

𝜃 ] = 0.

The second term on the right hand side is

𝐿𝛼1 𝜕𝛼2
𝜃

(cid:18)

−1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑖𝜕𝑗 𝑢

(cid:19)

107

(cid:18)

−

−1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

(cid:19)

𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑖𝜕𝑗 (𝐿𝛼1 𝜕𝛼2

𝜃 𝑢)

The first term is of the form

(cid:18)

𝐿𝛾1 𝜕𝛾2
𝜃

−1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

(cid:19)

𝐿𝛾3 𝜕𝛾4

𝜃 (𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑖𝜕𝑗 𝑢).

By Lemma 5.3.1, the last part becomes

(cid:32)

(cid:16)

𝐿𝛾3 𝜕𝛾4
𝜃

1
cosh(𝜌)4

|𝜕𝑡𝑢|2 +

2
cosh(𝜌)2

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

𝜕𝑡𝑢𝐿𝑖𝑢 +

2
∑︁

𝑖, 𝑗=1

𝑥𝑖
𝑡

𝑥 𝑗
𝑡

1
𝑡2

𝐿𝑖𝑢𝐿 𝑗 𝑢

(cid:17)

(cid:33)

(𝜕2

𝑡 𝑢) + 𝐶

.

If the 𝜕2

𝑡 𝑢 absorbs the highest order of derivatives, the integral (neglecting the 𝐶 term for the

moment) would be bounded by

∫

Σ𝜏

1
𝜏2 cosh(𝜌)2

𝜏 [𝑢; 𝑚]2 (cid:12)
E ≤𝑠
(cid:12)
(cid:12)
√︄∫

≤

1
𝜏2

≲ 1
𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚]2

Σ𝜏

E ≤𝑠

𝜏 [𝑢; 𝑚]4,

𝐿𝛾′

3 𝜕𝛾′

4

𝑡 𝑢
𝜃 𝜕2

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

|𝜕𝑡 𝐿𝛼1 𝜕𝛼2

𝜃 𝑢|𝑑𝑆

1
cosh(𝜌)3

|𝐿𝛾′

3 𝜕𝛾′

4

𝜃 𝜕2

𝑡 𝑢|2𝑑𝑆

√︄∫

Σ𝜏

1
cosh(𝜌)

|𝜕𝑡 𝐿𝛼1 𝜕𝛼2

𝜃 𝑢|2𝑑𝑆

where we use (5.3) to replace 𝜕2

𝑡 𝑢 and Lemma 5.3.2 at the last step.

If the 𝜕2

𝑡 𝑢 does not involve the highest order of derivatives, the integral (again, neglecting the 𝐶

term), according to Lemma 5.3.3, is bounded by

(cid:32)

∫

Σ𝜏

1
𝜏 cosh(𝜌)2

|𝐿𝛾′

3 𝜕𝛾′

4

𝜃 𝜕𝑡𝑢| +

|𝐿𝛾′

3 𝜕𝛾′

4

𝜃 𝐿𝑖𝑢|

(cid:33)

cosh(𝜌)
𝜏

𝜏 [𝑢; 𝑚]2|𝜕𝑡 𝐿𝛼1 𝜕𝛼2
E ≤𝑠

𝜃 𝑢|𝑑𝑆

2
∑︁

𝑖=1

1
𝜏

=

1
𝜏2

≤

1
𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚]2

E ≤𝑠

𝜏 [𝑢; 𝑚]2

∫

Σ𝜏

(cid:118)(cid:117)(cid:116)∫

(cid:32)

1
cosh(𝜌)

|𝐿𝛾′

3 𝜕𝛾′

4

𝜃 𝜕𝑡𝑢| +

|𝐿𝛾′

3 𝜕𝛾′

4

𝜃 𝐿𝑖𝑢|

(cid:33)

|𝜕𝑡 𝐿𝛼1 𝜕𝛼2

𝜃 𝑢|𝑑𝑆

2
∑︁

𝑖=1

1
𝜏

(cid:32)

3
cosh(𝜌)

|𝐿𝛾′

3 𝜕𝛾′

4

𝜃 𝜕𝑡𝑢|2 +

|𝐿𝛾′

3 𝜕𝛾′

4

𝜃 𝐿𝑖𝑢|2

(cid:33)

𝑑𝑆

2
∑︁

𝑖=1

1
𝜏2

Σ𝜏

√︄∫

×

Σ𝜏

1
cosh(𝜌)

≲ 1
𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚]4.

|𝜕𝑡 𝐿𝛼1 𝜕𝛼2

𝜃 𝑢|2𝑑𝑆

108

Considering the 𝐶 term, the integral is bounded by

∫

Σ𝜏

1
𝜏2 cosh(𝜌)

(cid:32)

E ≤𝑠

𝜏 [𝑢; 𝑚]2

|𝐿𝛾′

3 𝜕𝛾′

4

𝜃 𝜕𝑡𝑢| +

2
∑︁

𝑖=1

1
𝜏

|𝐿𝛾′

3 𝜕𝛾′

4

𝜃 𝐿𝑖𝑢|

(cid:33)

|𝜕𝑡 𝐿𝛼1 𝜕𝛼2

𝜃 𝑢|𝑑𝑆

∫

+

Σ𝜏

1
𝜏2

𝜏 [𝑢; 𝑚]2 (cid:16)√︁
E ≤𝑠

cosh(𝜌)|𝐿𝛾′

3 𝜕𝛾′

4

𝜃 𝜕𝜃𝑢|

(cid:32)

(cid:17)

1
√︁cosh(𝜌)

|𝜕𝑡 𝐿𝛼1 𝜕𝛼2

𝜃 𝑢|

(cid:33)

𝑑𝑆

≲ 1
𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚]4,

where we use the Hölder inequality again as above.

□

Remark. In the above proof, the term

(cid:32)

𝐿𝛾1 𝜕𝛾2
𝜃

(cid:33)

−1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

is bounded by a constant depending only on 𝑠 when 𝜖 is sufficiently small.

Proposition 5.3.2. Let 𝑠 ≥ 5. If □𝑔𝑢 = 0 and 𝑢 satisfies the bootstrap assumption 5.4, then

and

∫

Σ𝜏

∫

Σ𝜏

|∇𝑖𝑣∇ 𝑗 𝑣Γ

𝑗

𝑖𝑡 |𝑑𝑣𝑜𝑙 ≲ 1
𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚]4

|∇𝑘 𝑣∇𝑘 𝑣Γ𝑖

𝑖𝑡 |𝑑𝑣𝑜𝑙 ≲ 1
𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚]4,

where 𝑣 = 𝐿𝛼1 𝜕𝛼2

𝜃 𝑢, 0 ≤ |𝛼| ≤ 𝑠, and ∇ is the connection with respect to 𝑔.

Proof.

∇𝑖𝑣∇ 𝑗 𝑣Γ

𝑗

𝑖𝑡 = 𝑔𝑖𝑘 𝜕𝑘 𝑣𝜕𝑗 𝑣

𝑔 𝑗𝑙 (𝜕𝑖𝑔𝑙𝑡 + 𝜕𝑡𝑔𝑖𝑙 − 𝜕𝑙𝑔𝑖𝑡)

(cid:19)

(cid:18) 1
2

=

1
2

𝑔𝑖𝑘 𝑔 𝑗𝑙 𝜕𝑘 𝑣𝜕𝑗 𝑣 (cid:0)𝜕𝑖 (𝜕𝑙𝑢𝜕𝑡𝑢) + 𝜕𝑡 (𝜕𝑖𝑢𝜕𝑙𝑢) − 𝜕𝑙 (𝜕𝑖𝑢𝜕𝑡𝑢)(cid:1)

= 𝑔𝑖𝑘 𝑔 𝑗𝑙 𝜕𝑘 𝑣𝜕𝑗 𝑣(𝜕𝑖𝜕𝑡𝑢)𝜕𝑙𝑢

Using Lemma 5.1.1, we could decompose the above expression into four terms:

𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘 𝑣𝜕𝑗 𝑣(𝜕𝑖𝜕𝑡𝑢)𝜕𝑙𝑢

109

(cid:18)

−

𝑚𝑖𝑘

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢
(cid:18)

−

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

𝑚𝑖 𝑝𝑚𝑘𝑞𝜕𝑝𝑢𝜕𝑞𝑢

(cid:19)

𝑚 𝑗𝑙 𝜕𝑘 𝑣𝜕𝑗 𝑣(𝜕𝑖𝜕𝑡𝑢)𝜕𝑙𝑢

𝑚 𝑗𝑟 𝑚𝑙𝑠𝜕𝑟𝑢𝜕𝑠𝑢

(cid:19)

𝜕𝑘 𝑣𝜕𝑗 𝑣(𝜕𝑖𝜕𝑡𝑢)𝜕𝑙𝑢

(cid:18)

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

𝑚𝑖 𝑝𝑚𝑘𝑞𝜕𝑝𝑢𝜕𝑞𝑢

(cid:19) (cid:18)

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

𝑚 𝑗𝑟 𝑚𝑙𝑠𝜕𝑟𝑢𝜕𝑠𝑢

(cid:19)

𝜕𝑘 𝑣𝜕𝑗 𝑣(𝜕𝑖𝜕𝑡𝑢)𝜕𝑙𝑢.

The last three terms have the desired estimate from the following naive estimates:

|𝜕𝑢| ≲ 1
𝜏

E ≤𝑠

𝜏 [𝑢; 𝑚],

where 𝜕𝑢 denotes 𝜕𝑡𝑢, 𝜕1𝑢, 𝜕2𝑢, or 𝜕𝜃𝑢.

For the first term, we have the following expansion:

𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘 𝑣𝜕𝑙𝑣(𝜕𝑖𝜕𝑡𝑢)𝜕𝑗 𝑢 =

(𝜕𝑡𝑣)2(𝜕2

𝑡 𝑢)𝜕𝑡𝑢 −

𝜕𝑡𝑣

(cid:18) 1
𝑡

2
∑︁

𝑗=1

𝐿 𝑗 𝑣 −

(cid:19)

𝜕𝑡𝑣

𝑥 𝑗
𝑡

(𝜕2

𝑡 𝑢)

(cid:18) 1
𝑡

𝐿 𝑗 𝑢 −

(cid:19)

𝜕𝑡𝑢

𝑥 𝑗
𝑡

−

+

2
∑︁

𝑖=1
2
∑︁

𝑖, 𝑗=1

(cid:18) 1
𝑡

𝐿𝑖𝑣 −

(cid:19)

𝜕𝑡𝑣

𝑥𝑖
𝑡

𝜕𝑡𝑣

(cid:18) 1
𝑡

𝐿𝑖 (𝜕𝑡𝑢) −

(cid:19)

𝑡 𝑢
𝜕2

𝑥𝑖
𝑡

𝜕𝑡𝑢

(cid:18) 1
𝑡

𝐿𝑖𝑣 −

𝑥𝑖
𝑡

𝜕𝑡𝑣

(cid:19) (cid:18) 1
𝑡

𝐿 𝑗 𝑣 −

𝑥 𝑗
𝑡

𝜕𝑡𝑣

(cid:19) (cid:18) 1
𝑡

𝐿𝑖 (𝜕𝑡𝑢) −

𝑥𝑖
𝑡

𝑡 𝑢
𝜕2

(cid:19) (cid:18) 1
𝑡

𝐿 𝑗 𝑢 −

(cid:19)

𝜕𝑡𝑢

𝑥 𝑗
𝑡

− 𝜕𝑡𝑣𝜕𝜃𝑣(𝜕2

𝑡 𝑢)𝜕𝜃𝑢 − 𝜕𝜃𝑣𝜕𝑡𝑣(𝜕𝜃 𝜕𝑡𝑢)𝜕𝑡𝑢

+

+

2
∑︁

𝑖=1
2
∑︁

𝑗=1

(cid:18) 1
𝑡

𝐿𝑖𝑣 −

(cid:19)

𝜕𝑡𝑣

𝑥𝑖
𝑡

𝜕𝜃𝑣

(cid:18) 1
𝑡

𝐿𝑖 (𝜕𝑡𝑢) −

(cid:19)

𝜕2
𝑡 𝑢

𝑥𝑖
𝑡

𝜕𝜃𝑢

𝜕𝜃𝑣

(cid:18) 1
𝑡

𝐿 𝑗 𝑣 −

(cid:19)

𝜕𝑡𝑣

𝑥 𝑗
𝑡

(𝜕𝜃 𝜕𝑡𝑢)

(cid:18) 1
𝑡

𝐿 𝑗 𝑢 −

(cid:19)

𝜕𝑡𝑢

𝑥 𝑗
𝑡

+ 𝜕𝜃𝑣𝜕𝜃𝑣(𝜕𝜃 𝜕𝑡𝑢)𝜕𝜃𝑢

= (𝜕2

𝑡 𝑢)

(cid:32)

1
cosh(𝜌)4

(𝜕𝑡𝑣)2𝜕𝑡𝑢

+

+

+

𝑗=1

2
∑︁

𝑖=1

2
∑︁

𝑖, 𝑗=1

2
∑︁

(cid:18)

−

1
𝑡2

𝜕𝑡𝑣𝐿 𝑗 𝑣𝐿 𝑗 𝑢 +

𝑥 𝑗
𝑡

1
𝑡

1
cosh(𝜌)2

𝜕𝑡𝑣𝐿 𝑗 𝑣𝜕𝑡𝑢 +

𝑥 𝑗
𝑡

1
𝑡

1
cosh(𝜌)2

(cid:19)

(𝜕𝑡𝑣)2𝐿 𝑗 𝑢

𝑥𝑖
𝑡

1
𝑡

1
cosh(𝜌)2

𝐿𝑖𝑣𝜕𝑡𝑣𝜕𝑡𝑢 −

𝑥𝑖
𝑡

𝑥 𝑗
𝑡

1
𝑡2

𝐿𝑖𝑣𝜕𝑡𝑣𝐿 𝑗 𝑢 +

2
∑︁

𝑖, 𝑗=1

2
∑︁

𝑖, 𝑗=1
𝑥𝑖
𝑡

1
𝑡2

𝑥𝑖
𝑡

1
𝑡3

𝐿𝑖𝑣𝐿 𝑗 𝑣𝐿 𝑗 𝑢 +

2
∑︁

𝑖, 𝑗=1

𝑥𝑖
𝑡

𝑥 𝑗
𝑡

1
𝑡2

𝐿𝑖𝑣𝐿 𝑗 𝑣𝜕𝑡𝑢

𝑥𝑖
𝑡

𝜕𝑡𝑣𝐿 𝑗 𝑣𝐿 𝑗 𝑢

110

− 𝜕𝑡𝑣𝜕𝜃𝑣𝜕𝜃𝑢 −

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

𝐿𝑖𝑣𝜕𝜃𝑣𝜕𝜃𝑢 +

1
𝑡2

𝐿𝑖𝑣𝜕𝑡𝑣(𝐿𝑖𝜕𝑡𝑢)𝜕𝑡𝑢 +

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

(cid:33)

𝜕𝑡𝑣𝜕𝜃𝑣𝜕𝜃𝑢

𝑥𝑖
𝑡

𝑥𝑖
𝑡

2
∑︁

𝑖=1

(𝜕𝑡𝑣)2(𝐿𝑖𝜕𝑡𝑢)𝜕𝑡𝑢

(cid:18) 1
𝑡

𝐿𝑖𝑣 −

𝑥𝑖
𝑡

𝜕𝑡𝑣

(cid:19) (cid:18) 1
𝑡

𝐿 𝑗 𝑣 −

𝑥 𝑗
𝑡

𝜕𝑡𝑣

(cid:19) (cid:18) 1
𝑡

(𝐿𝑖𝜕𝑡𝑢)

(cid:19) (cid:18) 1
𝑡

𝐿 𝑗 𝑢 −

(cid:19)

𝜕𝑡𝑢

𝑥 𝑗
𝑡

−

+

2
∑︁

𝑖=1
2
∑︁

𝑖, 𝑗=1

− 𝜕𝜃𝑣𝜕𝑡𝑣(𝜕𝜃 𝜕𝑡𝑢)𝜕𝑡𝑢 +

(cid:18) 1
𝑡

2
∑︁

𝑖=1

𝐿𝑖𝑣 −

(cid:19)

𝜕𝑡𝑣

𝑥𝑖
𝑡

𝜕𝜃𝑣

(cid:18) 1
𝑡

𝐿𝑖 (𝜕𝑡𝑢)

(cid:19)

𝜕𝜃𝑢

+

2
∑︁

𝑗=1

𝜕𝜃𝑣

(cid:18) 1
𝑡

𝐿 𝑗 𝑣 −

(cid:19)

𝜕𝑡𝑣

𝑥 𝑗
𝑡

(𝜕𝜃 𝜕𝑡𝑢)

(cid:18) 1
𝑡

𝐿 𝑗 𝑢 −

(cid:19)

𝜕𝑡𝑢

𝑥 𝑗
𝑡

+ 𝜕𝜃𝑣𝜕𝜃𝑣(𝜕𝜃 𝜕𝑡𝑢)𝜕𝜃𝑢,

where we combine the terms in the parenthesis after 𝜕2

𝑡 𝑢 due to its extraordinary decay. Using

Lemma 5.3.2, Lemma 5.3.3, and the Hölder inequality, we have proved the first inequality.

Similarly,

∇𝑘 𝑣∇𝑘 𝑣Γ𝑖

𝑖𝑡 = 𝑔𝑘𝑙 𝜕𝑙𝑣𝜕𝑘 𝑣

𝑔𝑖 𝑗 (𝜕𝑖𝑔 𝑗𝑡 + 𝜕𝑡𝑔𝑖 𝑗 − 𝜕𝑗 𝑔𝑖𝑡)

(cid:19)

(cid:18) 1
2

=

1
2

𝑔𝑖 𝑗 𝑔𝑘𝑙 𝜕𝑙𝑣𝜕𝑘 𝑣 (cid:0)𝜕𝑖 (𝜕𝑗 𝑢𝜕𝑡𝑢) + 𝜕𝑡 (𝜕𝑖𝑢𝜕𝑗 𝑢) − 𝜕𝑗 (𝜕𝑖𝑢𝜕𝑡𝑢)(cid:1)

= 𝑔𝑖 𝑗 𝑔𝑘𝑙 𝜕𝑙𝑣𝜕𝑘 𝑣(𝜕𝑖𝜕𝑡𝑢)𝜕𝑗 𝑢.

It could be decomposed into the following terms:

𝑚𝑖 𝑗 𝑚𝑘𝑙 𝜕𝑙𝑣𝜕𝑘 𝑣(𝜕𝑖𝜕𝑡𝑢)𝜕𝑗 𝑢
(cid:18)

−

𝑚𝑖 𝑗

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢
(cid:18)

−

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

𝑚𝑖 𝑝𝑚 𝑗 𝑞𝜕𝑝𝑢𝜕𝑞𝑢

(cid:19)

𝑚𝑘𝑙 𝜕𝑙𝑣𝜕𝑘 𝑣(𝜕𝑖𝜕𝑡𝑢)𝜕𝑗 𝑢

𝑚𝑘𝑟 𝑚𝑙𝑠𝜕𝑟𝑢𝜕𝑠𝑢

(cid:19)

𝜕𝑙𝑣𝜕𝑘 𝑣(𝜕𝑖𝜕𝑡𝑢)𝜕𝑗 𝑢

(cid:18)

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

𝑚𝑖 𝑝𝑚 𝑗 𝑞𝜕𝑝𝑢𝜕𝑞𝑢

(cid:19) (cid:18)

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

𝑚𝑘𝑟 𝑚𝑙𝑠𝜕𝑟𝑢𝜕𝑠𝑢

(cid:19)

𝜕𝑙𝑣𝜕𝑘 𝑣(𝜕𝑖𝜕𝑡𝑢)𝜕𝑗 𝑢.

As above, the last three terms have the desired decay. To deal with the first term, observe that

𝑚𝑘𝑙 𝜕𝑘 𝑣𝜕𝑙𝑣 = −(𝜕𝑡𝑣)2 +

(cid:18) 1
𝑡

2
∑︁

𝑖=1

𝐿𝑖𝑣 −

𝑥𝑖
𝑡

𝜕𝑡𝑣

(cid:19) (cid:18) 1
𝑡

𝐿𝑖𝑣 −

(cid:19)

𝜕𝑡𝑣

𝑥𝑖
𝑡

+ (𝜕𝜃𝑣)2

111

= −

1
cosh(𝜌)2

(𝜕𝑡𝑣)2 +

1
𝑡2

(|𝐿1𝑣|2 + |𝐿2𝑣|2) − 2

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

𝐿𝑖𝑣𝜕𝑡𝑣 + (𝜕𝜃𝑣)2

and

𝑚𝑖 𝑗 (𝜕𝑖𝜕𝑡𝑢)𝜕𝑗 𝑢 = −(𝜕2

𝑡 𝑢)𝜕𝑡𝑢 +

(cid:18) 1
𝑡

2
∑︁

𝑖=1

𝐿𝑖 (𝜕𝑡𝑢) −

𝑥𝑖
𝑡

𝑡 𝑢
𝜕2

(cid:19) (cid:18) 1
𝑡

𝐿𝑖𝑢 −

(cid:19)

𝜕𝑡𝑢

𝑥𝑖
𝑡

+ (𝜕𝜃 𝜕𝑡𝑢)𝜕𝜃𝑢

= −

1
cosh(𝜌)2

(𝜕2

𝑡 𝑢)𝜕𝑡𝑢 +

2
∑︁

𝑖=1

1
𝑡2

(𝐿𝑖𝜕𝑡𝑢)𝐿𝑖𝑢 −

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

(𝐿𝑖𝜕𝑡𝑢)𝜕𝑡𝑢

−

𝑥𝑖
𝑡

1
𝑡

2
∑︁

𝑖=1

(𝜕2

𝑡 𝑢)𝐿𝑖𝑢 + (𝜕𝜃 𝜕𝑡𝑢)𝜕𝜃𝑢.

Using Lemma 5.3.2, Lemma 5.3.3, and the Hölder inequality, we obtain the desired decay.

□

5.4 Energy Comparability

In this section, we aim to show the comparability of the two types of energy. The main reference

is [1].

Proposition 5.4.1. If 𝑢 satisfies the bootstrap assumption 5.4, then E ≤𝑠

𝜏 [𝑢; 𝑚] and E ≤𝑠

𝜏 [𝑢; 𝑔] are

comparable for 2 ≤ 𝜏1 ≤ 𝜏 ≤ 𝜏2.

From now on, we would always assume the bootstrap assumption 5.4. In order to prove this

result, we need some geometric computations. In this section, we denote the matrix associated to the

metric 𝑔𝑖 𝑗 by 𝐺, the matrix associated to the metric 𝑚𝑖 𝑗 by 𝑀, and the vector [𝑣0𝜕𝑡+𝑣1𝜕1+𝑣2𝜕2+𝑣3𝜕𝜃]

by

𝑣0












𝑣3


We begin by deriving the explicit expression for normal vectors on Σ𝜏 with respect to 𝑚 and 𝑔.















𝑣2

𝑣1

.

Lemma 5.4.1. For each surface Σ𝜏, the normal vectors with respect to 𝑚 and 𝑔 are

[ (cid:174)𝑛𝑚] = 𝑀 −1𝑤

112

[ (cid:174)𝑛𝑔] =

√

1
−𝑤𝑇 𝐺−1𝑤

𝐺−1𝑤,

𝑤 =

− cosh(𝜌)

sinh(𝜌) cos 𝜙

sinh(𝜌) sin 𝜙

0















.















where

Remark. 𝑤𝑇 𝑀 −1𝑤 = −1.

Proof. From

and

[𝜕𝜙] =

𝜏 sinh(𝜌)

𝜏 cosh(𝜌) cos 𝜙

𝜏 cosh(𝜌) sin 𝜙

[𝜕𝜌] =















0

0















,

−𝜏 sinh(𝜌) sin 𝜙

𝜏 sinh(𝜌) cos 𝜙

0





























we could check that [𝜕𝜌]𝑇 𝐺 [ (cid:174)𝑛𝑔] = 0 and [𝜕𝜙]𝑇 𝐺 [ (cid:174)𝑛𝑔] = 0. Furthermore, [ (cid:174)𝑛𝑔]𝑇 𝐺 [ (cid:174)𝑛𝑔] = −1 implies

that [ (cid:174)𝑛𝑔] is the desired (past-pointing) normal vector.

□

Note that the bootstrap assumption 5.4 ensures the negativity of 𝑤𝑇 𝐺−1𝑤. Intuitively, when the

𝜖 in the bootstrap assumption 5.4 is sufficiently small, 𝑤𝑇 𝐺−1𝑤 will be close to 𝑤𝑇 𝑀 −1𝑤 = −1, as

proved in the following lemma. From the definition of (𝑋) 𝐽, we have

(cid:104)(𝜕𝑡 ) 𝐽 [𝑣; 𝑔]

(cid:105)

=















∇𝑡𝑣∇𝑡𝑣 − 1

2 (∇𝑘 𝑣∇𝑘 𝑣)

∇1𝑣∇𝑡𝑣

∇2𝑣∇𝑡𝑣

∇𝜃𝑣∇𝑡𝑣

113

.















With the aid of Lemma 5.4.1, we are able to compute the product:

⟨(𝜕𝑡 ) 𝐽 [𝑣; 𝑔], (cid:174)𝑛𝑔⟩𝑔 =

(cid:104)(𝜕𝑡 ) 𝐽 [𝑣; 𝑔]

(cid:105)𝑇

𝐺

(cid:18)

√

1
−𝑤𝑇 𝐺−1𝑤

(cid:19)

𝐺−1𝑤

=

√

1
−𝑤𝑇 𝐺−1𝑤

(cid:32)

1
2

(∇𝑡𝑣∇𝑡𝑣 − ∇1𝑣∇1𝑣 − ∇2𝑣∇2𝑣 − ∇𝜃𝑣∇𝜃𝑣) (− cosh(𝜌))

+ (∇1𝑣∇𝑡𝑣) (sinh(𝜌) cos 𝜙) + (∇2𝑣∇𝑡𝑣) (sinh(𝜌) sin 𝜙)

,

(cid:33)

where the connection ∇ is with respect to 𝑔.

Lemma 5.4.2. Let 𝑣 = 𝐿𝛼1 𝜕𝛼2

𝜃 𝑢, where |𝛼| ≤ 𝑠. Then

and

⟨(𝜕𝑡 ) 𝐽 [𝑣; 𝑔], (cid:174)𝑛𝑔⟩𝑔

⟨(𝜕𝑡 ) 𝐽 [𝑣; 𝑚], (cid:174)𝑛𝑚⟩𝑚

are comparable provided that the 𝜖 in the bootstrap assumption 5.4 is sufficiently small.

Proof. We are going to show that

1. 𝑤𝑇 𝐺−1𝑤 and 𝑤𝑇 𝑀 −1𝑤 are comparable.

2. The two versions, with respect to 𝑔 and 𝑚, of

(∇𝑡𝑣∇𝑡𝑣 − ∇1𝑣∇1𝑣 − ∇2𝑣∇2𝑣 − ∇𝜃𝑣∇𝜃𝑣) (− cosh(𝜌))

1
2

+ (∇1𝑣∇𝑡𝑣) (sinh(𝜌) cos 𝜙) + (∇2𝑣∇𝑡𝑣) (sinh(𝜌) sin 𝜙)

are comparable.

For the first claim, observe that

(cid:12)𝑤𝑇 𝐺−1𝑤 − 𝑤𝑇 𝑀 −1𝑤(cid:12)
(cid:12)

(cid:12) =

𝑚𝑖𝑘 𝑚 𝑗𝑙 (𝜕𝑘𝑢𝜕𝑙𝑢)𝑤𝑖𝑤 𝑗

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

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢
E ≤𝑠

𝜏 [𝑢; 𝑚]2(cosh(𝜌)2)

(cid:12)
(cid:12)
−
(cid:12)
(cid:12)
≲ 1
𝜏2
≤ E ≤𝑠

𝜏 [𝑢; 𝑚]2.

114

Using the fact that 𝑤𝑇 𝑀 −1𝑤 = −1, we establish the first claim. For the second claim, by using

Lemma 5.1.1, it is sufficient to show that the following terms (approximately the difference of the

two versions) are relatively small compared to the 𝑚 version:

(cid:16) 1
2

(𝑚𝑡𝑘 𝑚 𝑝𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑝𝑣𝜕𝑡𝑣 − 𝑚1𝑘 𝑚 𝑝𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑝𝑣𝜕1𝑣 − 𝑚2𝑘 𝑚 𝑝𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑝𝑣𝜕2𝑣)

× (− cosh(𝜌))

(cid:17)

+ (𝑚1𝑘 𝑚 𝑝𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑝𝑣𝜕𝑡𝑣)(sinh(𝜌) cos 𝜙)

+ (𝑚2𝑘 𝑚 𝑝𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑝𝑣𝜕𝑡𝑣)(sinh(𝜌) sin 𝜙),

which is bounded by

cosh(𝜌)
𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚]2(|𝜕𝑡𝑣|2 + |𝜕1𝑣|2 + |𝜕2𝑣|2 + |𝜕𝜃𝑣|2)

up to a constant. On the other hand, the 𝑚 version is

1
2

(−|𝜕𝑡𝑣|2 − |𝜕1𝑣|2 − |𝜕2𝑣|2 − |𝜕𝜃𝑣|2) (− cosh(𝜌))

+ (𝜕1𝑣𝜕𝑡𝑣)(sinh(𝜌) cos 𝜙) + (𝜕2𝑣𝜕𝑡𝑣) (sinh(𝜌) sin 𝜙)

≥

≥

1
2

1
4

−

1
2
1
cosh(𝜌)

cosh(𝜌)(|𝜕𝑡𝑣|2 + |𝜕1𝑣|2 + |𝜕2𝑣|2 + |𝜕𝜃𝑣|2)

sinh(𝜌)

(cid:16)

|𝜕𝑡𝑣|2(cos2 𝜙) + |𝜕1𝑣|2 + |𝜕𝑡𝑣|2(sin2 𝜙) + |𝜕2𝑣|2(cid:17)

(|𝜕𝑡𝑣|2 + |𝜕1𝑣|2 + |𝜕2𝑣|2 + |𝜕𝜃𝑣|2).

Therefore, as long as E ≤𝑠

𝜏 [𝑢; 𝑚] is small enough, the second claim holds.

□

Remark. In the above proof, we use a rough estimate

|𝜕𝑢| ≲ 1
𝜏

E ≤𝑠

𝜏 [𝑢; 𝑚],

where 𝜕 may be 𝜕𝑡, 𝜕1, 𝜕2, or 𝜕𝜃. This is a rough version of Lemma 5.3.2.

Lemma 5.4.3. The two versions of volume form,

𝑑𝑣𝑜𝑙(Σ𝜏;𝑔)

115

and

𝑑𝑣𝑜𝑙(Σ𝜏;𝑚),

are comparable provided that the 𝜖 in the bootstrap assumption 5.4 is sufficiently small.

Proof. We will denote the column vector

𝜕𝑡𝑢

𝜕1𝑢

𝜕2𝑢

𝜕𝜃𝑢





























by 𝑣, as in the proof of Lemma 5.1.1. Observe that

and

det 𝐺 = det ((𝐼 + 𝑣𝑣𝑇 𝑀) 𝑀)

= (1 + 𝑣𝑇 𝑀𝑣) det(𝑀)

|𝑣𝑇 𝑀𝑣| = |𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢| ≲ 1
𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚],

we have the desired result provided that E ≤𝑠

𝜏 [𝑢; 𝑚] is sufficiently small.

Proof of Proposition 5.1. It follows from Lemma 5.4.2 and Lemma 5.4.3.

□

□

5.5 Global Existence

We now prove the energy estimate that is essential to our paper. We argue that the integrand in

the divergence theorem can be estimated by an integrable function of 𝜏 over [2, ∞), and thus prove

the claim.

Theorem 5.5.1. We have the energy inequality

max
𝜏1≤𝜏≤𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑔] ≲

(cid:18)

E ≤𝑠
𝜏1

[𝑢; 𝑔] + max
𝜏1≤𝜏≤𝜏2

(cid:19)

E ≤𝑠

𝜏 [𝑢; 𝑔]3

for any 𝜏2 ≥ 𝜏1 ≥ 2.

116

Proof. Let 𝑣 = 𝐿𝛼1 𝜕𝛼2

𝜃 𝑢 with |𝛼| ≤ 𝑠. We have

1
2

[𝑣; 𝑔]2 −
E𝜏′
2
∫

1
2

E𝜏1 [𝑣; 𝑔]2

⟨(𝜕𝑡 ) 𝐽, (cid:174)𝑛𝑔⟩𝑔𝑑𝑆𝑔 −

∫

Σ𝜏
1

⟨(𝜕𝑡 ) 𝐽, (cid:174)𝑛𝑔⟩𝑔𝑑𝑆𝑔

=

=

≲

≲

Σ𝜏′
2
∫ 𝜏′
2

∫

𝜏1
∫ 𝜏′
2

Σ𝜏
∫

𝜏1
∫ 𝜏′
2

𝜏1

Σ𝜏
1
𝜏2

𝑑𝑖𝑣𝑔 ( (𝜕𝑡 ) 𝐽)

1
√︁−⟨∇𝜏, ∇𝜏⟩𝑔

𝑑𝑆𝑔𝑑𝜏

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

□𝑔𝑣(∇𝑡𝑣) + (∇𝑖𝑣∇ 𝑗 𝑣)Γ

𝑗
𝑖𝑡 −

(∇𝑘 𝑣∇𝑘 𝑣)Γ𝑖
𝑖𝑡

1
2

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

𝑑𝑆𝑔𝑑𝜏

E ≤𝑠

𝜏 [𝑢; 𝑚]4𝑑𝜏

≤

1
2
≲ 1
2

max
𝜏1≤𝜏≤𝜏2

max
𝜏1≤𝜏≤𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚]4

E ≤𝑠

𝜏 [𝑢; 𝑔]4

for every 𝜏1 ≤ 𝜏′

2 ≤ 𝜏2. The first ≲ follows from the fact that

⟨∇𝜏, ∇𝜏⟩𝑔 = 𝑔𝑖 𝑗 𝜕𝑖𝜏𝜕𝑗 𝜏

= 𝑚𝑖 𝑗 𝜕𝑖𝜏𝜕𝑗 𝜏 −

(cid:18)

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

(cid:19)

𝑚𝑖𝑘 𝑚 𝑗𝑙 𝜕𝑘𝑢𝜕𝑙𝑢𝜕𝑖𝜏𝜕𝑗 𝜏

= −1 −

(cid:18)

1
1 + 𝑚𝑎𝑏𝜕𝑎𝑢𝜕𝑏𝑢

(cid:19) (cid:32)

(cid:18)

(𝜕𝑡𝑢)2

cosh(𝜌) −

(cid:19) 2

sinh(𝜌)2
cosh(𝜌)

+

2
∑︁

𝑗=1

(𝜕𝑡𝑢) (𝐿 𝑗 𝑢)

(cid:18) 2𝑥 𝑗
𝜏2

(cid:19)

|𝑥|2
𝑡2

2𝑥 𝑗
𝜏2

+

−

2
∑︁

𝑖, 𝑗=1

𝑥𝑖
𝜏

𝑥 𝑗
𝜏

1
𝑡2

(cid:33)

𝐿𝑖𝑢𝐿 𝑗 𝑢

≈ −1

provided that 𝜖 is small enough, the second ≲ follows from Proposition 4.1 and Proposition 4.2,

and the last ≲ follows from Proposition 5.1. This implies that

max
𝜏1≤𝜏≤𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑔]2 ≲

(cid:18)

E ≤𝑠
𝜏1

[𝑢; 𝑔]2 + max
𝜏1≤𝜏≤𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑔]4

(cid:19)

,

which gives the desired result.

□

It is clear that in the above proof, the implicit constants for ≲ do not depend on 𝜏2 thanks to the
𝜏2 . Therefore, we have the following corollary. There exists a constant 𝐶2 > 0 so

integrability of 1

117

that the energy inequality

E ≤𝑠

𝜏 [𝑢; 𝑚] ≤ 𝐶𝑠

(cid:18)

max
2≤𝜏≤𝜏2

E ≤𝑠
2 [𝑢; 𝑚] + max
2≤𝜏≤𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚]3

(cid:19)

holds for every 𝜏2 ≥ 2.

Theorem 5.5.2. There exists an 𝜖0 > 0 so that the equation (5.1)

(cid:32)

𝜕𝑖

(cid:33)

𝑚𝑖 𝑗 𝜕𝑗 𝑢
1 + 𝑚𝑘𝑙 𝜕𝑘𝑢𝜕𝑙𝑢

√︁

= 0

has a global solution in R1,2 × T1 provided that

E ≤𝑠
2 [𝑢; 𝑚] ≤ 𝜖0.

Proof. It is sufficient to show that there exists an 𝜖0 so that

E ≤𝑠

𝜏 [𝑢; 𝑚] ≤ 2𝐶𝑠𝜖0

for 𝜏 ≥ 2 since if this is true, Lemma 5.3.2 and Lemma 5.3.3 imply that

|𝜕𝜕𝑢| + |𝜕𝑢| ≲ E ≤𝑠

𝜏 [𝑢; 𝑚] ≤ 2𝐶𝑠𝜖0,

where 𝜕 may be 𝜕𝑡, 𝜕1, 𝜕2 or 𝜕𝜃, and therefore the solution can be continued according to the standard
local well-posedness results. To show the boundedness of E ≤𝑠

𝜏 [𝑢; 𝑚], it is sufficient to show that

implies

E ≤𝑠

𝜏 [𝑢; 𝑚] ≤ 4𝐶𝑠𝜖0

max
2≤𝜏≤𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚] ≤ 2𝐶𝑠𝜖0.

max
2≤𝜏≤𝜏2

If the 𝜖0 is chosen to be small enough so that 𝜖 = 4𝐶𝑠𝜖0 satisfies the small requirement in all the

previous lemmas and propositions, we have (by Corollary 5.5)

max
𝜏1≤𝜏≤𝜏2

E ≤𝑠

𝜏 [𝑢; 𝑚] ≤ 𝐶𝑠 (𝜖0 + (4𝐶𝑠𝜖0)3)

≤ 𝐶𝑠 (𝜖0 + 𝜖0)

provided that

Therefore we close the boostrap argument.

𝜖 2
0 ≤

1
43𝐶3
𝑠

.

118

□

CHAPTER 6

CONCLUSIONS

I will summarize my progress and compare our results with others in this chapter. My work is

restricted to warped product spacetimes.

state 𝑝 = 𝛾𝜌 where

In Section 1.2, we consider the homogeneous Einstein-Fluid equations with the equation of
√𝛾 is the sound speed within 3 ranges : 𝛾 = 0 (dust case), 0 < 𝛾 < 1 (fluid case),
and 𝛾 = 1 (stiff-fluid case). In each case, we classify all the physical solutions that have a Big Bang

singularity and derive the asymptotes of the unknowns close to the singularity 𝑟 = 𝑟∗. In the fluid

case, 𝑟∗ may be positive (𝑟∗ > 0) or zero (𝑟∗ = 0, such as the Friedmann-Lemaître-Robertson-Walker

spacetime).

In Chapter 3, we investigate the stability of those 𝑟∗ > 0 homogeneous solutions under nonho-

mogeneous, compactly supported perturbations on the initial slice {𝑟 = 𝑟0}. We proved that there
exists a sequence of initial perturbations that goes to 0 in 𝑊 1,∞ so that each perturbation generates

a shock before the Big Bang 𝑟 = 𝑟∗.

In other words, these 𝑟∗ > 0 homogeneous solutions are

unstable.

In Chapter 4, we consider Euler’s equations in a special relativity setting. That is, we do not

consider the full Einstein-Fluid equations; instead, we only focus on half of the system, dropping

the feedback from fluid variables to the metric. We assume the metric is a fixed function of time,

and consider the dynamic evolution of fluid variables. Surprisingly, these 𝑟∗ > 0 models are stable

under this special relativity setting. This means that the mechanism for generating shocks not

only relies on the structure of the fluid equations (specially relativistic fluids), but also involves the

evolution of metric components (Einstein-Fluid equations).

In Chapter 5, we investigate the stability of the membrane equation, an equation for the vanishing

mean curvature. We consider the space R1,2 × T1 involving a compact factor T1 and apply the

standard vector field method with the modification for T1. It turns out the extra compact factor

does not hurt the integrability of the coefficients and thus the energy remains small throughout the

time, estabilishing the global existence.

119

Regarding the stability of the Big Bang, in [24], Rodnianski and Speck proved the stability of

Friedmann-Lemaître-Robertson-Walker spacetimes with certain topology, governed by Einstein-

scalar field equations. In [11], Fournodavlos, Rodnianski, and Speck proved the stability of Kasner

solutions, governed by Einstein-vaccum or Einstein-scalar field equations. In both papers, they do

not consider Einstein-Fluie equations.

For the shock formation trend, Riemann introduced the concept of Riemann invariants in [23].

He considered an isentropic fluid with the plane symmetry and used the Riemann invariants to prove

that shocks can form from smooth initial data. In [15], John proposed a more general condition,

genuinely nonlinear condition, as a sufficient condition for shocks to form for a one-dimensional

hyperbolic system without source terms. He used the total variations of the unknowns to control

the solution and used the Riccati structure to prove the existence of shocks, which is introduced

in our Chapter 2. The first clear picture about specailly relativistic fluids was established in [6]

by Christodoulou. He provided sharp sufficient conditions to generate shocks for 3-dimensional

relativistic fluids and geometric information of the boundary of maximally extended classical

solutions. In [22], Rendall and Stahl proved the existence of shocks for a large class of solutions

governed by Einstein-Fluid equations under plane symmetry.

120

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