APPROXIMATION AND INCOMPRESSIBLE LIMIT OF INHOMOGENEOUS POROUS
MEDIUM EQUATIONS

By

Anthony Sulak

A DISSERTATION

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

Applied Mathematics—Doctor of Philosophy

2024

ABSTRACT

In this work we study the inhomogeneous porous medium equation (PME). In particular, we

introduce a deterministic particle method that approximates the PME in chapter 2 and how the PME

is used to model tumor growth and study the incompressible limit in chapter 3.

Particle methods aim to discretize a PDE into a system of ODEs for particles. In the absence of

diffusion, particle methods can approximate solutions while maintaining the Wasserstein gradient

flow structure. Diffusion smooths things out so that particles do not remain particles. This issue can

be dealt with via regularization of the diffusion term. Much has been developed for convex and

semi-convex energies. We would like to extend this particle method (blob method) to an energy with

more general convexity (𝜔-convexity) containing nonlinear porous medium diffusion (𝑚 = 2). We

connect this pde to chemotaxis via a Keller-Segel model for the Newtonian kernel and the Bessel

kernel. Then, we perform numerical simulations.

The porous medium equation can be used to model tumor growth. We study the incompressible

limit of an inhomogeneous porous medium equation (PME) with a cell division term that directly

depends on space, time, and the pressure. The incompressible limit connects the PME to a

Hele-Shaw free boundary problem (FBP). This relation is known as the complementarity condition.

We first achieve convergence to the limiting problem along with uniqueness. Then, we gain enough

compactness using 𝐿3 bound of the pressure gradient and 𝐿3 AB-estimate to get the complementarity

condition. To finish the connection to the FBP, the velocity law to the boundary of the tumor is

found. In particular, a novel inhomogeneous velocity of the free boundary is obtained.

ACKNOWLEDGEMENTS

I would like to thank my advisor Olga Turanova for her guidance and encouragement. I would also

like to thank Olga for her collaboration in chapter 3. I would like to thank my family for their love

and support. This research was partially supported by the TA Award for Excellence in Teaching

and Dr. Paul and Wilma Dressel Endowed Scholarship through the Department of Mathematics at

Michigan State University and NSF grant DMS-2204722.

iii

TABLE OF CONTENTS

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

1.1 Approximating Inhomogeneous Porous Medium with Aggregation via

Gradient Flow Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
Incompressible Limit of Inhomogeneous Porous Medium Equations

. . . . .

1.2

1

1
2

CHAPTER 2

.
.

.
.

APPROXIMATING INHOMOGENEOUS POROUS MEDIUM WITH
AGGREGATION VIA GRADIENT FLOW METHODS . . . . . . . . .
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1 Notation and Assumptions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
.
2.2 Main Results
.
. 14
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
2.3 Background .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Energy Properties .
.
2.5 An 𝐻1-type Bound .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
.
2.6 Convergence of Gradient Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.7 Convergence of the Confining Limit
. . . . . . . . . . . . . . . . . . . . . . . 55
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.8 Proofs of Main Results
2.9 Applications .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
.
2.10 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

. .

.

.

CHAPTER 3

.

.

.

Introduction .

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

INCOMPRESSIBLE LIMIT OF INHOMOGENEOUS POROUS MEDIUM
EQUATIONS .
.

. 70
.
3.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2 Notation and Assumptions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Main Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
.
3.4 Estimates
3.5 Proof of Convergence to Limiting Problem . . . . . . . . . . . . . . . . . . . . 79
3.6 Uniqueness of the Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.7 Complementarity Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.8 Velocity Law . .

. .
.
.

. .
.
.

. .

.

BIBLIOGRAPHY .

.

.

.

.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

APPENDIX A

MOLLIFIER AND AGGREGATION KERNELS . . . . . . . . . . .

. 107

APPENDIX B

CONSTRUCTION OF SUPERSOLUTION AND HEURISTICS . . .

. 110

iv

CHAPTER 1

INTRODUCTION

1.1 Approximating Inhomogeneous Porous Medium with Aggregation via

Gradient Flow Methods

We study the weighted porous medium equation,

(WPME)

𝜕𝑡 𝜇 = ∇ ·

(cid:18) 𝑎

∇

(cid:19)(cid:19)

(cid:18) 𝜇2
𝑎2
(cid:32)(cid: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
(cid:32)(cid: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)
Diffusion

(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)

+ ∇ · (𝜇∇(𝑉 + 𝑉𝑘 ))
(cid:32)(cid:32)(cid:32)(cid:32)(cid: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)
Drift

(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)

,
+ ∇ · (𝜇(∇𝑊 ∗ 𝜇))
(cid:32)(cid:32)(cid:32)(cid: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)
Aggregation

(cid:124)

by using Wasserstein gradient flow theory. The main focus is to show that solutions to the regularized

PDE,

(1.1)

𝜕𝑡 𝜇 = ∇ ·

(cid:18)

(cid:18)

𝜇

∇𝜁𝜖 ∗

(cid:19)(cid:19)

(cid:18) 𝜁𝜖 ∗ 𝜇
𝑎

+ 𝜇(∇𝜁𝜖 ∗ 𝑉) + 𝜇(∇𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊 ∗ 𝜇) + 𝜇∇𝑉𝑘

(cid:19)

,

converge to solutions of (WPME), where 𝜁𝜖 is a mollifier (see precise description in Assumption

2.1.1). Equivalently, we show that the gradient flow of the regularized energy functional converges

to the gradient flow of the unregularized energy functional. This gives a convergence result for a

deterministic particle method. Sometimes this particle method, developed by [9], is called blob

method as the regularization involves convolving a mollifier (blob function) with the gradient flow.

The particle method involves discretizing the initial data 𝜇0 = 𝜇(0) as a finite sum of Dirac masses.

Moreover, 𝜇 in (1.1) is a finite sum of Dirac masses so that we obtain a system of ODEs for the

particles. That is, the particle locations evolve based on this system of ODEs (written explicitly in

Theorem 2.2.2). The gradient flow structure is preserved in the limit. As 𝜖 → 0, the gradient flows

of (1.1) converge to gradient flows of (WPME). If there was no drift nor aggregation (𝑉 ≡ 0 ≡ 𝑊),

then the target or weight 𝑎(𝑥) is the steady-state of (WPME).

In 2000, [22] study convergence results of the solutions to the regularized version of 𝜇𝑡 = Δ𝜇2/2.

Due to assumptions made of the initial data, these result do not guarantee convergence of the particle

method. In [9], they establish convergence of the deterministic particle method for

𝜕𝑡 𝜇 = Δ𝜇𝑚 + ∇ · (𝜇∇𝑉) + ∇ · (𝜇(∇𝑊 ∗ 𝜇)), 𝑚 ≥ 1,

1

where the coinciding energy functional is semi-convex. In [14], they improve on the results by

considering the inhomogeneous PDE instead of the homogeneous. In particular, they studied

𝜕𝑡 𝜇 = ∇ ·

(cid:17) 2(cid:19)

(cid:18) 𝑎

2

∇

(cid:16) 𝜇
𝑎

+ ∇ · (𝜇∇𝑉),

where the weight (or inhomogeneity) 𝑎 = 𝑎(𝑥) is nice.

We will generalize this particle method from [14] by adding an aggregation term and more

importantly generalizing the convexity property of the energy so that 𝜔-convexity is sufficient rather

than semi-convexity (or 𝜆-convexity).

To preserve the gradient flow structure, we will show that the gradient flow of the regularized

energy converges to the gradient flow of the unregularized energy. To get the Γ-convergence of

gradient flows, we use Serfaty’s sufficient conditions. The two main conditions that we require are

1. Γ-convergence of the energies

2. Γ-convergence (lower semi-continuity) of the local slopes.

Furthermore, an another important necessity is getting an 𝐻1 bound on the mollified gradient flow

of the regularized energy. The key idea for the 𝐻1 bound is to use the flow interchange method.

Suppose that we have two energy functionals and their respective gradient flows. Differentiating

for a fixed time of the first energy at the gradient flow of the second is the same as differentiating

for a fixed time the second energy at the gradient flow of the first. This allows us to deal with the

“easier” energy functional. The PDE of interest has applications to chemotaxis as the Keller-Segel

model and crowd-motion models. Two common kernels for the aggregation in this application are

the Newtonian and Bessel kernel, in which, both satisfy the assumptions to get 𝜔-convexity of the

aggregation energy. Numerical simulations follow.

1.2

Incompressible Limit of Inhomogeneous Porous Medium Equations

The focus of our work is the inhomogeneous porous medium equation with reaction,

(1.2)

𝜕𝑡𝑢𝑚 = ∇ ·

(cid:18) 𝑢𝑚
𝑎(𝑥, 𝑡)

(cid:19)

+

∇𝑝𝑚

𝑢𝑚
𝑎(𝑥, 𝑡)

Φ(𝑥, 𝑡, 𝑝𝑚), where 𝑝𝑚 =

𝑚
𝑚 − 1

(cid:18) 𝑢𝑚
𝑏(𝑥, 𝑡)

(cid:19) 𝑚−1

.

In particular, we view (1.2) as a model for tumor growth and study the incompressible limit (𝑚 → ∞).

The cells tend to avoid over-crowding and move away from the congested regions. The density 𝑢𝑚

2

represents the cell population, in which, the pressure 𝑝𝑚 is generated from. The cell division rate is

controlled by Φ, where Φ depends on the pressure, space, and time. Given that the cells are less

willing to divide in packed areas, the division rate will decrease as the pressure increases and will be

zero once the pressure is high enough. We call this pressure the homeostatic pressure.

In terms of what laws to use, we can rewrite the PME so that we explicitly see the for the velocity

we use Darcy’s law and for the pressure we use the power law,

(PME)

𝜕𝑡𝑢𝑚 = ∇ · (𝑢𝑚𝑣) + 𝑢𝑚

𝑣 = ∇𝑝𝑚
𝑎(𝑥,𝑡)

, 𝑝𝑚 = 𝑚
𝑚−1

𝑎(𝑥,𝑡) Φ(𝑥, 𝑡, 𝑝𝑚),
(cid:17) 𝑚−1
(cid:16) 𝑢𝑚
𝑏(𝑥,𝑡)

.





From the power law, we see that taking the incompressible limit (𝑚 → ∞), the stiffness of the

pressure increases. The incompressible limit relates the PME and a Hele-Shaw free boundary

problem (FBP). This link is called the complementarity condition and is one of the goals achieved

here along with the inhomogeneous velocity law of the free boundary. The flow or velocity in both

models (PME and FBP) are induced by Darcy’s law. Now the free boundary velocity does not

only depend on the pressure gradient but on 𝑎(𝑥, 𝑡) as well. The novelty being an inhomogeneous

velocity law. The function 𝑏(𝑥, 𝑡) represents the max packing density of cells. That is, 𝑏 is the

largest the density can become. The ratio of the viscosity of the fluid and the permeability of the

medium is represented by 𝑎(𝑥, 𝑡). In other words, 𝑎(𝑥, 𝑡) describes the ease in which a fluid can

move through the medium.

We go back to 1981 where [4] established continuous dependence on 𝜑 of solutions of

𝜕𝑡𝑢 = Δ𝜑(𝑢) (filtration equation). They achieved the first incompressible limit result by letting

𝑚 → ∞ for 𝜑(𝑢) = 𝑢𝑚. Caffarelli and Friedman in 1987 ([7]) studied the incompressible limit

of 𝜕𝑡𝑢 = Δ𝑢𝑚 IVP on R𝑑. They showed that the IVP coincides with motionless or stationary

FBP. Gil and Quirós in 2001 ([19]) worked on the IVP on an open Ω ⊂ R𝑑 with boundary data

depending on space. With boundary data idendically equal to zero and large enough Ω, they got

the same results as Caffarelli and Friedman. In particular, a motionless or stationary boundary.

With nontrivial boundary data, they get a nonstationary FBP. In more recent work, [23] includes

the growth term 𝑢Φ( 𝑝). This results in a nonstationary free boundary in R𝑑. Also they showed

3

that the complementarity condition is equivalent to 𝐿2 strong convergence of the pressure gradient.

They used a 𝐿∞ AB-estimate (Aronson, Bénilan) to get the result. In 2021, [17] included nutrient

concentration in the growth term. They established a new way to get 𝐿2 strong compactness for

pressure gradient. They achieved a 𝐿3 AB-estimate and a 𝐿4 bound on the pressure gradient to gain

enough compactness. In [21], they included a term that can be either a source or sink term. They

prove the complementarity condition by connecting it to a obstacle problem, in which they show is

true. In all of the above prior work, they consider 𝑎 = 𝑏 ≡ 1. We generalize this so that 𝑎, 𝑏 are nice

functions that depend not only on space but on time as well.

4

CHAPTER 2

APPROXIMATING INHOMOGENEOUS POROUS MEDIUM WITH AGGREGATION
VIA GRADIENT FLOW METHODS

2.1 Notation and Assumptions

2.1.1 Definition of Energy Functionals

We study

(WPME)

𝜕𝑡 𝜇 = ∇ ·

(cid:18) 𝑎

∇

(cid:19)(cid:19)

(cid:18) 𝜇2
𝑎2
(cid:32)(cid: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
(cid:32)(cid: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)
Diffusion

(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)

+ ∇ · (𝜇∇(𝑉 + 𝑉𝑘 ))
(cid:32)(cid:32)(cid:32)(cid:32)(cid: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)
Drift

(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)

+ ∇ · (𝜇(∇𝑊 ∗ 𝜇))
(cid:32)(cid:32)(cid:32)(cid: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)
Aggregation

(cid:124)

by using the Wasserstein gradient flow, 𝜇 : [0, 𝑇] → P2(R𝑑), of the energy

F : P2(R𝑑) → R ∪ {+∞}, F (𝜇) = E (𝜇) + V (𝜇) + VΩ(𝜇) + W (𝜇)

where we have diffusion (E), potential (or drift V), confining potential (V𝑘 ), and interaction (or

aggregation W) energies

E (𝜇) =

∫

R𝑑

𝑢2(𝑥)
2𝑎(𝑥)

𝑑𝑥, V (𝜇) =

∫

R𝑑

𝑉 (𝑥) 𝑑𝜇(𝑥),

VΩ(𝜇) =

0, supp(𝑢) ⊆ Ω

+∞, otherwise

, W (𝜇) =

∫

R𝑑

1
2

𝑊 ∗ 𝜇(𝑥) 𝑑𝜇(𝑥),





respectively. Note 𝑑𝜇 = 𝜇(𝑥) 𝑑𝑥 and 𝑉, 𝑊: R𝑑 → R. Here, Ω ⊆ R𝑑 is nonempty, open, and convex.

A probability measure 𝜇 ∈ P (R𝑑) lies in the domain of an energy F , denoted 𝐷 (F ), if F (𝜇) < ∞.

The second moment of a probability measure 𝜇 is

∫

𝑀2(𝜇) =

|𝑥|2 𝑑𝜇(𝑥).

R𝑑
We are interested in the space P2(R𝑑) = P (R𝑑) ∩ 𝐷 (𝑀2), probability measures with finite second
moment, with the 2-Wasserstein distance 𝑊2 (see Remark 2.3.4). The space (P2(R𝑑), 𝑊2) is a

metric space and in particular a geodesic space [24]. We will use the flow interchange method (see

Remark 2.5.3) to prove a 𝐻1 bound, in which, we define the heat entropy

𝜇 log 𝜇 𝑑𝑥.

S(𝜇) =

∫

R𝑑

5

We assume that the weight 𝑎 ∈ 𝐶1(R𝑑) and there exists a constant 𝐶 > 0 such that 1/𝐶 ≤ 𝑎(𝑥) ≤ 𝐶

for all 𝑥 ∈ R𝑑. For the well-posedness of the gradient flows of E (and therefore F ), 𝑎(𝑥) is

log-concave on Ω. In particular, the functional E is convex and its local slope is a strong upper

gradient. Note that if Ω = R𝑑, then 𝑎(𝑥) would be a constant. For E to be well-defined, we insist that

∥𝜇∥ 𝐿2 (R𝑑) ≤ 𝐶. This correlates well with Assumption 2.1.3. As in [9, Corollary 5.5], for particles

to remain particles, regularization is used.

To illustrate why regularization is the solution [9] proposes to solve their issue, let us look at the

continuity equation

𝜕𝑡 𝜇 = ∇ · (𝜇𝑣) ,




(cid:1)(cid:1) + ∇𝑉 + ∇𝑊 ∗ 𝜇, then the continuity equation is the same

𝜇0(𝑥) = 𝜇(𝑥, 0).

If we define the velocity as 𝑣 = (cid:0)𝜇∇ (cid:0) 𝜇
𝑎

as (WPME) barring the confinement variable. If 𝑣 is nice (that is, there is no diffusion term), then

the particle method works without any regularization required. However, including diffusion makes

𝑣 not nice. To make 𝑣 nice or to give 𝑣 stronger regularity, we can regularize.

In particular, we regularize the energies by convolving a mollifier with 𝜇. That is,

F𝜖,𝑘 (𝜇) = E𝜖 (𝜇) + V𝜖 (𝜇) + V𝑘 (𝜇) + W𝜖 (𝜇)

where

E𝜖 (𝜇) =

∫

R𝑑

V𝑘 (𝜇) =

(𝜁𝜖 ∗ 𝜇(𝑥))2
2𝑎(𝑥)

∫

R𝑑

𝑑𝑥, V𝜖 (𝜇) =

𝑉 (𝑥) 𝑑𝜁𝜖 ∗ 𝜇(𝑥)

𝑉𝑘 (𝑥) 𝑑𝜇(𝑥), W𝜖 (𝜇) =

𝑊 ∗ 𝜁𝜖 ∗ 𝜇(𝑥) 𝑑𝜁𝜖 ∗ 𝜇(𝑥).

∫

R𝑑
∫
1
2

R𝑑

The corresponding regularized PDE is

𝜕𝑡 𝜇 = ∇ ·

(cid:18)

(cid:18)

𝜇

∇𝜁𝜖 ∗

(cid:19)(cid:19)

(cid:18) 𝜁𝜖 ∗ 𝜇
𝑎

+ 𝜇(∇𝜁𝜖 ∗ 𝑉) + 𝜇(∇𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊 ∗ 𝜇) + 𝜇∇𝑉𝑘

(cid:19)

.

The particle method starts by approximating the initial data 𝜇0 as a finite sum of Dirac masses. We

have a system of ODEs for the particles, where the particle locations evolve in time based on the

6

regularized version of the velocity, say 𝑣𝜖 . So, we get the gradient flow, 𝜇𝑁

𝜖 (𝑡), of the regularized

energy. Taking 𝜖 → 0 gives the gradient flow of unregularized energy, which corresponds to the

original PDE.

2.1.2 Assumptions

There are various assumptions necessary of the functions 𝑎, 𝑉, 𝑊 and the mollifier 𝜁𝜖 .

Assumption 2.1.1 (Mollifier). We assume that the mollifier satisfies the following:

(2.1)

(2.2)

𝜁 ∈ 𝐶2(R𝑑) is even, nonnegative, ∥𝜁 ∥ 𝐿1 (R𝑑) = 1, 𝐷2𝜁 ∈ 𝐿∞(R𝑑),

𝜁 (𝑥) ≤ 𝐶𝜁 |𝑥|−𝑞, |∇𝜁 (𝑥)| ≤ 𝐶𝜁 |𝑥|−𝑞′, for 𝐶𝜁 > 0, 𝑞 > 𝑑 + 1, 𝑞′ > 𝑑.

Assumption 2.1.2 (Target function). Let Ω ⊆ R𝑑 be nonempty, open, and convex. The weight

𝑎 ∈ 𝐶1(R𝑑) is log-concave on Ω and there exists a constant 𝐶 > 0 such that 1/𝐶 ≤ 𝑎(𝑥) ≤ 𝐶 for all

𝑥 ∈ R𝑑.

We will consider 𝑉 and 𝑊 satisfying assumptions 4.1 and 4.2 from [13] so that the aggregation

and drift functionals are 𝜔-convex.

Assumption 2.1.3 (Aggregation/Interaction). There exists a constant 𝐶 > 0 (not necessarily the

same constant) such that

1. For all 𝜇 ∈ P2(R𝑑) with ∥𝜇∥ 𝐿 𝑝 ≤ 𝐶𝑝, 𝑊 − ∗ 𝜇(𝑥) ≤ 𝐶.
2. For all 𝜇, 𝜈 ∈ P2(R𝑑) with ∥𝜇∥ 𝐿 𝑝 ≤ 𝐶𝑝 , we have ∥∇𝑊 ∗ 𝜇∥ 𝐿2 (𝜈) ≤ 𝐶.
3. For all ∥𝜇∥ 𝐿 𝑝 ≤ 𝐶𝑝, the kernel 𝑊 ∗𝜇 is continuously differentiable and there exists a continuous,

nondecreasing, concave function 𝜓 : [0, ∞) → [0, ∞) satisfying 𝜓(0) = 0, 𝜓(𝑥) ≥ 𝑥, and
∫ 1
0

𝑑𝑥
𝜓(𝑥) = ∞ so that

|∇𝑊 ∗ 𝜇(𝑥) − ∇𝑊 ∗ 𝜇(𝑦)|2 ≤ 𝐶2𝜓(|𝑥 − 𝑦|2).

4. For all ∥𝜇∥ 𝐿 𝑝 , ∥𝜈∥ 𝐿 𝑝 , ∥ 𝜌∥ 𝐿 𝑝 ≤ 𝐶𝑝,

∥∇𝑊 ∗ 𝜇 − ∇𝑊 ∗ 𝜈∥ 𝐿2 (𝜌) ≤ 𝐶𝑊2(𝜇, 𝜈).

5. 𝑊 is lower semi-continuous.

7

6. 1 < 𝑝 ≤ +∞.

Assumption 2.1.4 (Drift Potential). There exists a constant 𝐶 > 0 (not necessarily the same

constant) such that

1. 𝑉 ≥ −𝐶.

2. For all 𝜇 ∈ 𝐷 (V), we have ∥∇𝑉 ∥ 𝐿2 (𝜇) ≤ 𝐶.

3. The kernel 𝑉 is continuously differentiable and there exists a continuous, nondecreasing,

concave function 𝜓 : [0, ∞) → [0, ∞) satisfying 𝜓(0) = 0, 𝜓(𝑥) ≥ 𝑥, and ∫ 1

0

𝑑𝑥
𝜓(𝑥) = ∞ so

that

|∇𝑉 (𝑥) − ∇𝑉 (𝑦)|2 ≤ 𝐶2𝜓(|𝑥 − 𝑦|2).

Assumption 2.1.5 (Confining Potential). Let Ω ⊆ R𝑑 is nonempty, open, and convex. The confining

potential 𝑉𝑘 (𝑥) ≥ 0, for 𝑘 ∈ N is convex and twice differentiable with 𝐷2𝑉𝑘 ∈ 𝐿∞(R𝑑). Furthermore,

𝑉𝑘 = 0 on Ω and lim𝑘→∞(inf𝑥∈𝐵 𝑉𝑘 (𝑥)) = +∞ for all 𝐵 ⊂⊂ Ω𝑐.

Remark 2.1.6. Assumption 2.1.5 implies that 𝑉𝑘 ∈ 𝐿1(𝜇) and ∇𝑉𝑘 ∈ 𝐿2(𝜇) for 𝜇 ∈ P2(R𝑑) by
taking the 𝐿∞ norm. In particular, we get well-posedness of the gradient flow and the correct

limiting dynamics as 𝑘 → ∞.

Assumption 2.1.7 (Additional Assumptions). For any 𝜇 ∈ P2(R𝑑) with ∥𝜇∥ 𝐿 𝑝 (R𝑑) ≤ 𝐶, we require

that either

1. ∃𝑅 > 0 such that, ∥∇𝑊 ∗ 𝜇∥ 𝐿2 (R𝑑\𝐵𝑅) ≤ 𝐶𝑅 and ∥∇𝑉 ∥ 𝐿2 (R𝑑\𝐵𝑅) ≤ 𝐶𝑅.
2. ∥𝐷2𝑊 ∗ 𝜇∥ 𝐿2 (R𝑑) ≤ 𝐶 ∥𝜇∥ 𝐿2 (R𝑑) and ∥𝐷2𝑉 ∥ 𝐿2 (R𝑑) ≤ 𝐶.

Remark 2.1.8. Given Assumption 2.1.3, for any 𝑅 > 0, choosing 𝜈 =

L 𝑑 |𝐵𝑅 gives us
∥∇𝑊 ∗ 𝜇∥ 𝐿2 (L 𝑑 | 𝐵𝑅 ) ≤ 𝐶𝑅. We have that item 1 in Assumption 2.1.7 gives ∥∇𝑊 ∗ 𝜇∥ 𝐿2 (R𝑑) ≤ 𝐶
after we fix 𝑅. Regarding item 2, recall that ∥𝜇∥ 𝐿2 (𝑅𝑑) ≤ 𝐶. Thus we have a bound of 𝐶 after

1
L 𝑑 (𝐵𝑅)

combining constants. Analogous arguments are made for the assumptions on 𝑉.

We can get the same bound if we convolve ∇𝑊 ∗ 𝜇 (or ∇𝑉) with 𝜁𝜖 by the same method as

Remark 2.4.10 or by using Young’s convolution inequality. These additional assumptions are used

in Proposition 2.5.10.

8

We give examples that satisfy the assumptions required for the aggregation kernel.

2.1.3 Newtonian Potential

There are multiple examples where 𝜔-convex energies have the Newtonian potential,

N (𝑥) =

1

2𝜋 log(|𝑥|), 𝑑 = 2

|𝑥|2−𝑑
𝑑 (2−𝑑)𝛼(𝑑)

, 𝑑 ≠ 2





as their kernel. It is only interesting when 𝑑 ≥ 2 as when 𝑑 = 1 the Newtonian potential is convex

and many energies (see example 2.19 of [13]) are 𝜔-convex with 𝜆𝜔 = 0 (i.e. convex). For any extra

assumptions added for 𝑊, we would like the Newtonian potential to satisfy them as well.

2.1.4 Kernels of Aggregation

There are two generalization of the Newtonian potential. The first is the Riesz potential

R 𝛽,𝑑 (𝑥) = 𝐶𝑑,𝛽|𝑥| 𝛽−𝑑

with 2 ≤ 𝛽 < 𝑑 and 𝑑 ≥ 3. See that the Newtonian and the Riesz kernels are equivalent when 𝛽 = 2

([8]). The Bessel kernel is given by

B𝛼,𝑑 (𝑥) =

∫ ∞

0

1
(4𝜋𝑡)𝑑/2

𝑒 − | 𝑥 |2

4𝑡 −𝛼𝑡 𝑑𝑡

with 𝛼 ≥ 0. The Newtonian and the Bessel kernels are equivalent when 𝛼 = 0 ([8]). Both of Riesz

and Bessel kernels satisfy the assumptions for 𝜔-convexity.

2.2 Main Results

Theorem 2.2.1 (Convergence of gradient flows as 𝑘 → ∞, 𝜖 = 𝜖 (𝑘) → 0). Suppose Assumptions
2.1.3, 2.1.4, 2.1.5, 2.1.7 hold. Fix 𝑇 > 0 and 𝜇(0) ∈ 𝐷 (F ) ∩ 𝐷 (S) ∩ P2(R𝑑). For 𝜖 > 0 and
𝑘 ∈ N, let 𝜇𝜖,𝑘 ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)) be a gradient flow of F𝜖,𝑘 with the initial data 𝜇(0). Then as
𝑘 → ∞, there exists a sequence 𝜖 = 𝜖 (𝑘) → 0 so that

𝑊1(𝜇𝜖,𝑘 (𝑡), 𝜇(𝑡)) = 0, uniformly for 𝑡 ∈ [0, 𝑇],

lim
𝑘→∞

where 𝜇 ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)) is the unique gradient flow of F with initial condition 𝜇(0).

9

Before we give the result for convergence with particle initial data, we need to briefly discuss

Osgood’s criterion and the discretization of the PDE. The well-posedness of 𝜔-convexity functionals

were inspired by the Osgood’s criterion of well-posedness of ODEs in Euclidean space. Furthermore,

𝜔 being an Osgood modulus of convexity (see [13]) ensures the ODE




𝑑
𝑑𝑡 𝐹𝑡 (𝑥) = 𝜆𝜔𝜔(𝐹𝑡 (𝑥)),

𝐹0(𝑥) = 𝑥,

is well-posed locally in time. We will review some properties of the ODE. The solutions to the ODE

is

𝐹𝑡 (𝑥) = 𝜙−1(𝜙(𝑥) + 𝑡𝜆𝜔),

𝜙(𝑥) =

∫ 𝑥

1

𝑑𝑦
𝜔(𝑦)

.

where 𝜙 : (0, ∞) → R. If 𝜆𝜔 ≤ 0, then 𝐹𝑡 (𝑥) is a solution for all 𝑡 ≥ 0. If 𝜆𝜔 > 0, then 𝐹𝑡 (𝑥) is a

solution for 0 ≤ 𝑡 < (𝜙(+∞) − 𝜙(𝑥))/𝜆𝜔. The function 𝐹𝑡 (𝑥) and its spatial inverse 𝐹−1

𝑡

(𝑥) = 𝐹−𝑡 (𝑥)

are continuous and strictly increasing in 𝑥. If 𝜆𝜔 ≤ 0, then 𝐹𝑡 (𝑥) is nonincreasing in 𝑡. If 𝜆𝜔 > 0,

then 𝐹𝑡 (𝑥) is nondecreasing in 𝑡. We also know that 𝜙′(𝑥) = 1/𝜔(𝑥) and (𝜙−1(𝑥))′ = 𝜔(𝜙−1(𝑥)) ≥ 0.

In the particle method 𝜆𝜔 is a function of 𝜖, denoted 𝜆𝜔,𝜖 . To this end 𝐹𝑡 (𝑥) is also a function of 𝜖,

denoted 𝐹𝑡,𝜖 (𝑥) = 𝜙−1(𝜙(𝑥) + 𝑡𝜆𝜔,𝜖 ).

After mollifying the gradient flow, the regularized PDE is

𝜕𝑡 𝜇 = ∇ ·

(cid:18)

(cid:18)

𝜇

∇𝜁𝜖 ∗

(cid:19)(cid:19)

(cid:18) 𝜁𝜖 ∗ 𝜇
𝑎

+ 𝜇(∇𝜁𝜖 ∗ 𝑉) + 𝜇(∇𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊 ∗ 𝜇) + 𝜇∇𝑉𝑘

(cid:19)

.

Given that the gradient flow of F𝜖 solves the above PDE in the weak sense, we can discretize the

problem by letting

𝜇(𝑡) =

𝑁
∑︁

𝑖=1

𝛿𝑋 𝑖 (𝑡)𝑚𝑖,

𝜇0 =

𝑁
∑︁

𝑖=1

𝑚𝑖,

𝛿𝑋 𝑖

0

𝑁
∑︁

𝑖=1

𝑚𝑖 = 1,

where {𝑋𝑖 (𝑡)}𝑁

𝑖=1 are the location of the particles. We then obtain a system of ODEs for the particle,

(cid:164)𝑋𝑖 (𝑡) = −

𝑁
∑︁

𝑗=1

𝑓 (𝑋𝑖, 𝑋 𝑗 )𝑚 𝑗 − ∇𝜁𝜖 ∗ 𝑉 (𝑋𝑖) − ∇𝑉𝑘 (𝑋𝑖) −

𝑁
∑︁

𝑗=1

𝑚 𝑗 ∇𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊 (𝑋𝑖 − 𝑋 𝑗 )

with 𝑋𝑖 (0) = 𝑋𝑖

0 where

𝑓 (𝑋𝑖, 𝑋 𝑗 ) =

∫

𝑅𝑑

∇𝜁𝜖 (𝑋𝑖 − 𝑧)𝜁𝜖 (𝑋 𝑗 − 𝑧)
𝑎(𝑧)

𝑑𝑧.

10

Theorem 2.2.2 (Convergence with particle initial data). Suppose Assumptions 2.1.3, 2.1.4, 2.1.5,
𝜔 . Fix 𝑇 > 0 and 𝜇(0) ∈ 𝐷 (F ) ∩ 𝐷 (S) ∩ P2(R𝑑). For 𝑘, 𝑁 ∈ N, 𝜖 > 0,

2.1.7 hold. Let 𝜆𝜔,𝜖 := 𝜆F𝜖

and 𝑡 ∈ [0, 𝑇], consider the evolving empirical measure,

𝜇𝑁
𝜖,𝑘 (𝑡) =

𝑁
∑︁

𝑖=1

𝛿𝑋 𝑖

𝜖 ,𝑘 (𝑡)𝑚𝑖, 𝑚𝑖 ≥ 0,

𝑁
∑︁

𝑖=1

𝑚𝑖 = 1,

where 𝑋𝑖

𝜖,𝑘 ∈ 𝐶1([0, 𝑇]; R𝑑) solves,

𝑚 𝑗 ∫

𝑅𝑑 ∇𝜁𝜖 (𝑋𝑖

𝑚 𝑗 ∇𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊 (𝑋𝑖

𝜖,𝑘 − 𝑧)𝜁𝜖 (𝑋 𝑗
𝜖,𝑘 − 𝑋 𝑗

𝜖,𝑘 )

𝜖,𝑘 − 𝑧) 1
𝑎(𝑧)

𝑑𝑧 − ∇𝜁𝜖 ∗ 𝑉 (𝑋𝑖

𝜖,𝑘 ) − ∇𝑉𝑘 (𝑋𝑖

𝜖,𝑘 )

(cid:164)𝑋𝑖
𝜖,𝑘 (𝑡)

= − (cid:205)𝑁
𝑗=1

− (cid:205)𝑁

𝑗=1

𝜖,𝑘 (0) = 𝑋𝑖
𝑋𝑖

0,𝜖 .






Suppose that as 𝜖 → 0 there exists 𝑁 = 𝑁 (𝜖) → ∞, such that, for all 𝑘 ∈ N, 𝜇𝑁

𝜖,𝑘 (0) = (cid:205)𝑁
𝑖=1

𝑚𝑖

𝛿𝑋 𝑖
0, 𝜖

converge to 𝜇(0) with the rate,

𝐹−2𝑡,𝜖 (𝑊 2

2 (𝜇𝑁

𝜖,𝑘 (0), 𝜇(0))) = 0.

lim
𝑘→∞

Then as 𝑘 → ∞, there exists 𝜖 = 𝜖 (𝑘) → 0 and 𝑁 = 𝑁 (𝜖) → ∞, for which 𝜇𝑁

𝜖,𝑘 (𝑡) = (cid:205)𝑁
𝑖=1

𝛿𝑋 𝑖

𝜖 ,𝑘 (𝑡)𝑚𝑖

satisfies

𝑊1(𝜇𝑁

𝜖,𝑘 (𝑡), 𝜇(𝑡)) = 0, uniformly for 𝑡 ∈ [0, 𝑇],

lim
𝑘→∞

where 𝜇 ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)) is the unique weak solution of (WPME) with initial condition
𝜇(0).

Corollary 2.2.3 (Long time limit). Suppose Assumptions 2.1.3, 2.1.4, 2.1.5, 2.1.7 hold. Define the

empirical measure 𝜇𝑁

𝜖,𝑘 (𝑡) = (cid:205)𝑁
𝑖=1

𝛿𝑋 𝑖

𝜖 ,𝑘 (𝑡)𝑚𝑖. Assume 𝑉 = 𝑊 = 0, Ω is bounded, and ∫

Ω

𝑎 𝑑L𝑑 = 1.

Then there exists 𝑘 = 𝑘 (𝑡) → ∞, 𝜖 = 𝜖 (𝑘) → 0, and 𝑁 = 𝑁 (𝜖) → ∞ so that

𝑊1(𝜇𝑁

𝜖,𝑘 (·, 𝑡), 𝑎1

Ω) = 0.

lim
𝑡→∞

We can get the convergence results without diffusion on R𝑑 instead of on Ω. That is, no

confinement is necessary.

11

Theorem 2.2.4 (Convergence of gradient flow of Drift and Aggregation on R𝑑). Suppose Assumptions

2.1.1, 2.1.3, 2.1.4, 2.1.7 hold. Define G𝜖 = V𝜖 + W𝜖 and G = V + W. Fix 𝑇 > 0, 𝜇(0) ∈
𝐷 (G) ∩ P2(R𝑑). For 𝜖 > 0, let 𝜇𝜖 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) be the gradient flow of G𝜖 with initial
data 𝜇(0). Then as 𝜖 → 0, 𝜇𝜖 (𝑡) → 𝜇(𝑡) narrowly for 𝑡 ∈ [0, 𝑇] where 𝜇 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑))
is the unique gradient flow of G with initial condition 𝜇(0).

2.2.1 Remarks on Convergence Rate

There are two interesting examples from [13, Example 2.17] worth discussing when it pertains

to Theorem 2.2.2.

Remark 2.2.5 (Convergence when 𝜔(𝑥) = 𝑥). For 𝜔(𝑥) = 𝑥, 𝐹𝑡 (𝑥) = 𝑥 exp(𝜆𝜔,𝜖 𝑡). Given the

approximation via an empirical measure 𝑊 2

2 (𝜇𝑁

𝜖,𝑘 (0), 𝜇(0)) ≤ 𝛿𝜖 ∼ 1/𝑁𝜖 ([14, Lemma A.4]), we

want to analyze

Using the semi-convexity of E𝜖 , 𝜆𝜔,𝜖 ∼ −1/𝜖 𝑑+2, as 𝜖 → 0,

𝐹−2𝑡,𝜖 (𝛿𝜖 ) = 0.

lim
𝑘→∞

0 ← 𝐹−2𝑡,𝜖 (𝛿𝜖 ) = 𝛿𝜖 exp(−2𝑡𝜆𝜔,𝜖 ) =

𝛿𝜖
exp(−2𝑡/𝜖 𝑑+2)

.

Thus, we get the same convergence as [14] as 𝜖 → 0,

1
𝑁 (𝜖, 𝑘)

= 𝑜

(cid:18)

exp

(cid:18) −1
𝜖 𝑑+2

(cid:19)(cid:19)

.

Remark 2.2.6 (Convergence for log-Lipschitz modulus of convexity). Some explicit examples are

mention later with the Newtonian potential and Bessel kernel for the aggregation energy. Both are

𝜔-convex where 𝜔 is log-Lipschitz. In particular, 𝐹𝑡 (𝑥) = 𝑥exp(−𝑡𝜆 𝜔, 𝜖 ). Given the approximation via

an empirical measure 𝑊 2

𝜖,𝑘 (0), 𝜇(0)) ≤ 𝛿𝜖 ∼ 1/𝑁𝜖 ([14, Lemma A.4]), we want to analyze

2 (𝜇𝑁

Using the semi-convexity of E𝜖 , 𝜆𝜔,𝜖 ∼ −1/𝜖 𝑑+2, as 𝜖 → 0,

𝐹−2𝑡,𝜖 (𝛿𝜖 ) = 0.

lim
𝑘→∞

0 ← 𝐹−2𝑡,𝜖 (𝛿𝜖 ) = exp

(cid:18)

log(𝛿𝑒)
exp(2𝑡/𝜖 𝑑+2)

(cid:19)

.

12

This requires that as 𝜖 → 0,

Equivalently, as 𝜖 → 0,

log(𝛿𝑒)
exp(2𝑡/𝜖 𝑑+2)

→ −∞.

1/log(𝑁𝑒)
exp(−2𝑡/𝜖 𝑑+2)

→ 0.

Thus, we get what we expect based on the previous convergence result that as 𝜖 → 0,

1
log(𝑁 (𝜖, 𝑘))

= 𝑜

(cid:18)

exp

(cid:18) −1
𝜖 𝑑+2

(cid:19)(cid:19)

.

Neither of the previous convergence rates are desirable as in some numerical simulations in [14],

𝑁 ∼ 𝜖 −1.01. We cannot show this in particular, but do improve on the qualitative rates above.

Remark 2.2.7 (Improved Qualitative Convergence Rate). Given the approximation via an empirical

measure 𝑊 2

2 (𝜇𝑁

𝜖,𝑘 (0), 𝜇(0)) ≤ 𝛿𝜖 ∼ 1/𝑁𝜖 ([14, Lemma A.4]), we want to analyze

𝐹−2𝑡,𝜖 (𝛿𝜖 ) = 0.

lim
𝑘→∞

In particular, we want to have some convergence rate for 𝑁 (𝜖) to achieve the above limit. Given that

𝜔 is nondecreasing and positive, we use a discrete approximation so that,

∫ 𝛿 𝜖

1

𝑑𝑦
𝜔(𝑦)

≤

=

≤

1
𝜔(𝑥𝑖)

+ 𝐸𝑛

−(1 − 𝛿𝜖 )
𝑛

−(1 − 𝛿𝜖 )
𝑛

𝑛
∑︁

𝑖=1
1
𝜔(𝛿𝜖 )

+

−(1 − 𝛿𝜖 )
𝑛

𝑛
∑︁

𝑖=2

1
𝜔(𝑥𝑖)

+ 𝐸𝑛

−(1 − 𝛿𝜖 )
𝑛

1
𝜔(𝛿𝜖 )

+ 𝐸𝑛,

for a fixed 𝑛, where 𝐸𝑛 is the error. We use the fact that we know the explicit form of 𝐹𝑡 (𝑥), 𝜙−1 and

𝜔 are nondecreasing, and that the semi-convexity of E𝜖 with 𝜆𝜔,𝜖 ∼ −1/𝜖 𝑑+2,

𝐹−2𝑡,𝜖 (𝛿𝜖 ) = 𝜙−1(𝜙(𝛿𝜖 ) − 2𝑡𝜆𝜔,𝜖 )

= 𝜙−1(𝜙(𝛿𝜖 ) + 2𝑡/𝜖 𝑑+2)
2𝑡
𝜖 𝑑+2

= 𝜙−1

(cid:18)∫ 𝛿 𝜖

+

(cid:19)

1

𝑑𝑦
𝜔(𝑦)
(cid:18) −(1 − 𝛿𝜖 )
𝑛

≤ 𝜙−1

1
𝜔(𝛿𝜖 )

+ 𝐸𝑛 +

(cid:19)

.

2𝑡
𝜖 𝑑+2

13

If ∫ 1

0

𝑑𝑦
𝜔(𝑦) = ∞ (see [13, Definition 2.15]), then 𝜙(0) = −∞. Given that 𝜙−1(𝜙(0)) = 0,

−(1 − 𝛿𝜖 )
𝜔(𝛿𝜖 )

+

2𝑡
𝜖 𝑑+2

lim
𝜖→0

= −∞.

Note that since 𝑛 does not depend on 𝜖 and is fixed, we can ignore it as it is a constant. Taking

lim
𝜖→0

log (cid:169)
(cid:173)
(cid:173)
(cid:171)

lim
𝜖→0

exp

(cid:16) −(1−𝛿 𝜖 )
𝜔(𝛿 𝜖 )
(cid:17)
(cid:16) −2𝑡
𝜖 𝑑+2

exp

= −∞.

(cid:17)

(cid:170)
(cid:174)
(cid:174)
(cid:172)

exp

(cid:17)
(cid:17) = 0.

(cid:16) −(1−𝛿 𝜖 )
𝜔(𝛿 𝜖 )
(cid:16) −2𝑡
𝜖 𝑑+2

exp

exp

(cid:19)

(cid:18) −(1 − 𝛿𝜖 )
𝜔(𝛿𝜖 )

= 𝑜

(cid:18)

exp

(cid:18) −2𝑡
𝜖 𝑑+2

(cid:19)(cid:19)

,

exp

(cid:19)

(cid:18) −(1 − 𝛿𝜖 )
𝜔(𝛿𝜖 )

= 𝑜 (cid:0)exp (cid:0)2𝑡𝜆𝜔,𝜖 (cid:1)(cid:1) .

exp(1 − 𝑁 (𝜖, 𝑘)) = 𝑜

(cid:18)

exp

(cid:18) −1
𝜖 𝑑+2

(cid:19)(cid:19)

,

exp

(cid:18) 1 − 𝑁 (𝜖, 𝑘)
| log(𝑁 (𝜖, 𝑘))|

(cid:19)

= 𝑜

(cid:18)

exp

(cid:18) −1
𝜖 𝑑+2

(cid:19)(cid:19)

.

log exp,

It follows that,

Thus as 𝜖 → 0,

or with full generality,

As 𝜖 → 0, for 𝜔(𝑥) = 𝑥,

and for 𝜔(𝑥) = 𝑥| log(𝑥)|,

2.3 Background

2.3.1 Preliminaries

We discuss numerous definitions and lemmas used throughout the chapter. The first allows us to

move the mollifier from the measure to the integrand.

Lemma 2.3.1 (mollifier exchange, [9] Lemma 2.2). Let 𝑓 : R𝑑 → R be Lipschitz continuous with

constant 𝐿 𝑓 > 0, and let 𝜎 and 𝜈 be finite, signed Borel measures on R𝑑. There is 𝑝 = 𝑝(𝑞, 𝑑) > 0

so that
(cid:12)
∫
(cid:12)
(cid:12)
(cid:12)

𝜁𝜖 ∗ ( 𝑓 𝜈) 𝑑𝜎 −

∫

(𝜁𝜖 ∗ 𝜈) 𝑓 𝑑𝜎

(cid:18)∫

≤ 𝜖 𝑝 𝐿 𝑓

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

(𝜁𝜖 ∗ |𝜈|) 𝑑|𝜎| + 𝐶𝜁 |𝜎|(R𝑑)|𝜈|(R𝑑)

(cid:19)

14

for all 𝜖 > 0.

Narrow convergence is one of the main notions of convergence in this chapter.

Definition 2.3.2 (narrow convergence). A sequence 𝜇𝑛 in P (R𝑑) is said to narrowly converge to
𝜇 ∈ P (R𝑑) if ∫ 𝑓 𝑑𝜇𝑛 → ∫ 𝑓 𝑑𝜇 for all bounded and continuous functions 𝑓 .

Lemma 2.3.3 (narrow convergence and mollifiers). Suppose 𝜁𝜖 is a mollifier satisfying Assumption

2.1.1 and let 𝜇𝜖 be a sequence in P (R𝑑) converging narrowly to 𝜇 ∈ P (R𝑑). Then 𝜁𝜖 ∗ 𝜇𝜖 narrowly

converges to 𝜇.

Another main notion of convergence used here is via distance. In particular, the 2-Wasserstein

metric. This relates to optimal transport.

Remark 2.3.4 (optimal transport and Wasserstein metric). For 𝜇, 𝜈 ∈ P (R𝑑), the set of transport

plans from 𝜇 to 𝜈 is given by

Γ(𝜇, 𝜈) = {𝛾 ∈ P (R𝑑 × R𝑑)| 𝜋1
#

𝛾 = 𝜇, 𝜋2
#

𝛾 = 𝜈}

where 𝜋1, 𝜋2 : R𝑑 × R𝑑 → R𝑑 are projections of R𝑑 × R𝑑 onto the first and second copy of R𝑑,

respectively. For 𝑝 ≥ 1, the 𝑝-Wasserstein distance between 𝜇, 𝜈 ∈ P𝑝 (R𝑑) is given by

𝑊𝑝 (𝜇, 𝜈) =

(cid:18)∫

R𝑑×R𝑑

|𝑥 − 𝑦| 𝑝 𝑑𝛾(𝑥, 𝑦)

(cid:19) 1/𝑝

with 𝛾 ∈ Γ0(𝜇, 𝜈) where Γ0(𝜇, 𝜈) is the set of optimal transport plans. The 𝑝th moment is defined
as 𝑀𝑝 (𝜇) = ∫

R𝑑 |𝑥| 𝑝 𝑑𝜇 and thus define the space

P𝑝 (R𝑑) = {𝜇 ∈ P (R𝑑)| 𝑀𝑝 (𝜇) < ∞}.

Remark 2.3.5 (geodesics and generalized geodesics). Given 𝜇0, 𝜇1 ∈ P2(R𝑑), a geodesic connecting
𝜇0 to 𝜇1 are curves of the form

𝜇𝛼 = ((1 − 𝛼)𝜋1 + 𝛼𝜋2)#𝛾

for 𝛼 ∈ [0, 1], 𝛾 ∈ Γ0(𝜇0, 𝜇1). Given 𝜇0, 𝜇1, 𝜇2 ∈ P2(R𝑑), a generalized geodesic from 𝜇2 to 𝜇3
with base 𝜇1 is given by

𝜇2→3
𝛼

= ((1 − 𝛼)𝜋2 + 𝛼𝜋3)#𝛾

15

for 𝛼 ∈ [0, 1] and 𝛾 ∈ P (R𝑑 × R𝑑 × R𝑑) such that 𝜋1,2
#

𝛾 ∈ Γ0(𝜇1, 𝜇2) and 𝜋1,3

#

𝛾 ∈ Γ0(𝜇1, 𝜇3). For

short hand, sometimes 𝜇𝛼 will be used for the generalized geodesic.

Often times we require a curve to be absolutely continuous.

Definition 2.3.6 (absolutely continuous). We say 𝜇(𝑡) is absolutely continuous on [0, 𝑇], and write

𝜇 ∈ 𝐴𝐶2

loc((0, 𝑇); P2(𝑅𝑑)), if there exists 𝑓 ∈ 𝐿2

loc((0, 𝑇)) so that,

𝑊2(𝜇(𝑡), 𝜇(𝑠)) ≤

∫ 𝑡

𝑠

𝑓 (𝑟) 𝑑𝑟

for all 𝑡, 𝑠 ∈ (0, 𝑇) with 𝑠 ≤ 𝑡.

The minimal 𝑓 to satisfy this is the metric derivative of 𝜇.

Definition 2.3.7 (metric derivative). Given 𝜇 ∈ 𝐴𝐶2

loc((0, 𝑇); P2(𝑅𝑑)), the limit

|𝜇′|(𝑡) := lim
𝑠→𝑡

𝑊2(𝜇(𝑡), 𝜇(𝑠))
|𝑡 − 𝑠|

exists for a.e. 𝑡 ∈ (0, 𝑇) and is called the metric derivative of 𝜇.

A main point of this work is to generalize results of [14] to 𝜔-convex functionals. Here we

provide a definition as well as recall other notions of convexity and how they relate.

Definition 2.3.8 (𝜔-convexity). Given an energy 𝐸 : P2(R𝑑) → R ∪ {+∞}, a curve 𝜇𝛼 ∈ P2(R𝑑),
and a distance function 𝑑 : P2(R𝑑) × P2(R𝑑) → [0, ∞), we say that 𝐸 is 𝜔-convex along 𝜇𝛼 w.r.t.
𝑑 if for some 𝜔 : [0, +∞) → [0, +∞) and 𝜆𝜔 ∈ R,

𝐸 (𝜇𝛼) ≤(1 − 𝛼)𝐸 (𝜇0) + 𝛼𝐸 (𝜇1)
𝜆𝜔
2

(1 − 𝛼)𝜔

−

(cid:16)

(cid:16)

𝛼2𝑑 (𝜇0, 𝜇1)2(cid:17)

+ 𝛼𝜔

(cid:16)

(1 − 𝛼)2𝑑 (𝜇0, 𝜇1)2(cid:17)(cid:17)

.

where the modulus of convexity 𝜔(𝑥) is continuous, nondecreasing, and vanishes only at 𝑥 = 0. If

𝜇𝛼 is a geodesic, then 𝑑 = 𝑊2. If 𝜇𝛼 is a generalized geodesic from 𝜇2 to 𝜇3 with base 𝜇1, then

𝑑 = 𝑊2,𝛾 where

𝑊2,𝛾 (𝜇2, 𝜇3) =

(cid:18)∫

|𝜋2 − 𝜋3|2 𝑑𝛾

(cid:19) 1/2

.

16

Remark 2.3.9 (Discussion of notions of convexity). In [13, Definition 2.4], convexity, semi-

convexity, and 𝜔-convexity are discussed. It is sometimes easier to compare the different notions

of convexity using the negative part of 𝜆, 𝜆− = max{0, −𝜆}. It is clear that when 𝜆 = 0, convex

and semi-convexity are equivalent. If 𝜆 > 0, then semi-convexity implies convexity. Conversely, if

𝜆 < 0, then convexity implies semi-convexity. We have similar implications for 𝜔-convexity and

semi-convexity (and therefore convexity). Semi-convexity and 𝜔-convexity are equivalent when we

have the identity map, 𝜔(𝑥) = 𝑥. Semi-convexity implies 𝜔-convexity when 𝜔(𝑥) ≥ 𝑥 and 𝜆−

𝜔 ≥ 𝜆−.

The first requirement is used so that

𝛼(1 − 𝛼)𝑑2(𝜇0, 𝜇1) = (1 − 𝛼)𝛼2𝑑2(𝜇0, 𝜇1) + 𝛼(1 − 𝛼)2𝑑2(𝜇0, 𝜇1)

≤ (1 − 𝛼)𝜔(𝛼2𝑑2(𝜇0, 𝜇1)) + 𝛼𝜔((1 − 𝛼)2𝑑2(𝜇0, 𝜇1)).

Requiring 𝜆−

𝜔 ≥ 𝜆− gives

−𝜆
2

𝛼(1 − 𝛼)𝑑2(𝜇0, 𝜇1) ≤

(cid:16)

−𝜆𝜔
2

(1 − 𝛼)𝜔(𝛼2𝑑2(𝜇0, 𝜇1)) + 𝛼𝜔((1 − 𝛼)2𝑑2(𝜇0, 𝜇1))

(cid:17)

.

Using the same reasoning, we get the converse. That is, if 𝜔(𝑥) ≤ 𝑥 and 𝜆−

𝜔 ≤ 𝜆−, then 𝜔-convexity

implies semi-convexity.

It can be quite difficult to check if an energy functional is 𝜔-convex by the definition. Here we

have an criterion for 𝜔-convexity using the above the tangent line property.

Proposition 2.3.10 (above the tangent line property and 𝜔-convexity, [13] Proposition 2.7). Suppose

that for all generalized geodesic 𝜇𝛼 from 𝜇0 to 𝜇1 with base 𝜈 such that 𝜇0, 𝜇1 ∈ 𝐷 (𝐸), 𝐸 (𝜇𝛼) is
differentiable for 𝛼 ∈ [0, 1], 𝑑

𝑑𝛼 𝐸 (𝜇𝛼) ∈ 𝐿1( [0, 1]), and
𝑑
𝑑𝛼

𝐸 (𝜇𝛼)|𝛼=0 ≥

𝐸 (𝜇1) − 𝐸 (𝜇0) −

𝜆𝜔
2

𝜔(𝑊 2

2,𝛾 (𝜇0, 𝜇1)).

Then 𝐸 is 𝜔-convex along generalized geodesics. Furthermore, if 𝐸 merely satisfies these assumptions

in the specific case that 𝜈 = 𝜇0 or 𝜈 = 𝜇1, then 𝐸 is 𝜔-convex along geodesics.

Definition 2.3.11 (local slope). Given 𝐸 : P2(R𝑑) → (−∞, ∞], for any 𝜇 ∈ 𝐷 (𝐸), the local slope

is

|𝜕𝐸 |(𝜇) = lim sup

𝜈→𝜇

(𝐸 (𝜇) − 𝐸 (𝜈))+
𝑊2(𝜇, 𝜈)

17

where 𝜆+ = max{𝜆, 0} is the positive part of 𝜆.

Many methods in this chapter take advantage of gradient flow. Moreover, the preservation of the

gradient flow structure in the limit.

Definition 2.3.12 (gradient flow). Suppose 𝐸 : (−∞, ∞] is proper, lower semi-continuous, and
𝜔-convex along generalized geodesics. A curve 𝜇(𝑡) ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) is a gradient flow of
𝐸 in the Wasserstein metric if 𝜇(𝑡) is a weak solution of the continuity equation

𝜕𝑡 𝜇(𝑡) + ∇ · (𝑣(𝑡)𝜇(𝑡)) = 0

in the sense of distributions and 𝑣(𝑡) = −∇ 𝛿F

𝛿𝜇 for L1-a.e. 𝑡 > 0.

We state the characterization of the gradient flow, in which, are multiples results of [1].

Theorem 2.3.13 (well-posedness and characterization of gradient flow). Suppose 𝐸 : (−∞, ∞] is

proper, lower semi-continuous, and 𝜔-convex along generalized geodesics and 𝜇(0) ∈ 𝐷 (𝐸). Then

there exists 𝜇(𝑡) an unique gradient flow of 𝐸 such that as 𝑡 → 0+, 𝑊2(𝜇(𝑡), 𝜇(0)) → 0. Moreover,
𝜇(𝑡) ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)) is a gradient flow of 𝐸 if and only if 𝜇(𝑡) satisfies one the following

equivalent conditions:

1. Curve of Maximal Slope: For all 0 < 𝑠 ≤ 𝑡,

∫ 𝑡

𝑠

1
2

|𝜇′|2(𝑟) 𝑑𝑟 +

∫ 𝑡

𝑠

1
2

|𝜕𝐸 |2(𝜇(𝑟)) 𝑑𝑟 ≤ 𝐸 (𝜇(𝑠)) − 𝐸 (𝜇(𝑡)).

2. Evolution Variational Inequality: For all 𝜈 ∈ P2(R𝑑) and for L1-a.e. 𝑡 ≥ 0,

𝑑
𝑑𝑡

1
2

𝑊 2

2 (𝜇(𝑡), 𝜈) +

𝜆𝜔
2

𝜔(𝑊 2

2 (𝜇(𝑡), 𝜈)) ≤ 𝐸 (𝜈) − 𝐸 (𝜇(𝑡)).

We will show that curves of maximal curves coincide with gradient flows of 𝜔-convex energies

in Theorem 2.3.23. The EVI condition comes from [13]. Note that the 1. is also referred to as the

Energy Dissipation Inequality (EDI).

2.3.2 Sufficient Conditions via Serfaty’s Theorem

Serfaty in [25], establishes a framework for convergence of gradient flows. We provide the

definition of this type of convergence and adjust the framework of Serfaty for 𝜔-convex functionals.

18

Definition 2.3.14 (Γ-convergence of energies). We say that G𝜖 : P2(R𝑑) → R ∪ {+∞} Γ-converges
to G : P2(R𝑑) → R ∪ {+∞} if

1. For any 𝜇𝜖 ∈ P (R𝑑) converging narrowly to 𝜇 ∈ P (R𝑑),

(2.3)

lim inf
𝜖→0

G𝜖 (𝜇𝜖 ) ≥ G(𝜇).

2. For any 𝜇 ∈ P2(R𝑑), there exists 𝜇𝜖 ∈ P (R𝑑) converging narrowly to 𝜇 such that

(2.4)

G𝜖 (𝜇) ≤ G(𝜇).

lim sup
𝜖→0

The reason why Theorem 2.2.1 as uniform convergence in 𝑊1 and not 𝑊2 is because of the

compactness of absolutely continuous curves.

Lemma 2.3.15 (compactness of absolutely continuous curves, [14] Lemma 2.15). Fix 𝑇 > 0.
Suppose we have a sequence {𝜇𝜖 }𝜖 >0 ⊂ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) and

∫ 𝑇

0

sup
𝜖 >0

|𝜇′

𝜖 |2(𝑟) 𝑑𝑟 < ∞,

𝑀2(𝜇𝜖 (0)) < ∞.

sup
𝜖 >0

Then there exists 𝜇 ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)) such that, along a subsequence 𝜖 → 0,
𝑊1(𝜇𝜖 (𝑡), 𝜇(𝑡)) → 0 uniformly in 𝑡 ∈ [0, 𝑇], and

lim inf
𝜖→0

∫ 𝑡

0

|𝜇′

𝜖 |2(𝑟) 𝑑𝑟 ≥

∫ 𝑡

0

|𝜇′|2(𝑟) 𝑑𝑟

for every 𝑡 ∈ [0, 𝑇].

Proof. We use Proposition 2.5.9 with the hypothesis of this lemma to get existence of a 𝐶 = 𝐶 (𝑇) > 0,

so that for all 𝑡 ∈ [0, 𝑇] and 𝜖 > 0, 𝜇𝜖 (𝑡) belongs to the set {𝜇 : 𝑀2(𝜇) ≤ 𝐶}. This set is narrowly

sequentially compact [1, Remark 5.1.5, Lemma 5.1.7] and uniformly integrable 1st moments [1,

equation 5.1.20]. So it is relatively compact in the 1-Wasserstein metric [1, Proposition 7.1.5]. Thus

pointwise in time, {𝜇𝜖 (𝑡)}𝜖 >0 is relatively compact with respect to the 1-Wasserstein metric. By

19

Hölder’s inequality, for all 0 ≤ 𝑠 ≤ 𝑡 ≤ 𝑇,

sup
𝜖 >0

𝑊1(𝜇𝜖 (𝑠), 𝜇𝜖 (𝑡)) ≤ sup
𝜖 >0

𝑊2(𝜇𝜖 (𝑠), 𝜇𝜖 (𝑡))

∫ 𝑡

𝑠

|𝜇′

𝜖 |(𝑟) 𝑑𝑟

≤ sup
𝜖 >0
√

≤

𝑡 − 𝑠

(cid:18)

∫ 𝑡

𝑠

sup
𝜖 >0

|𝜇′

𝜖 |2(𝑟) 𝑑𝑟

(cid:19) 1/2

.

The equicontinuity with respect to the 1-Wasserstein metric means we can apply Arzelá-Ascoli
so that there exits 𝜇 : [0, 𝑇] → P2(R𝑑), such that, up to a subsequence, 𝑊1(𝜇𝜖 (𝑡), 𝜇(𝑡)) → 0
uniformly in 𝑡 ∈ [0, 𝑇].

The hypothesis ensures {|𝜇′

𝜖 |(𝑟)}𝜖 >0 is bounded in 𝐿2( [0, 𝑇]). Therefore, up to another

subsequence, it is weakly convergent to some 𝜈(𝑟) ∈ 𝐿2( [0, 𝑇]). For all 0 ≤ 𝑠 ≤ 𝑡 ≤ 𝑇, using

lower semi-continuity of 2-Wasserstein metric with respect to narrow convergence (and therefore

1-Wasserstein convergence),

𝑊2(𝜇(𝑠), 𝜇(𝑡)) ≤ lim inf
𝜖→0

𝑊2(𝜇𝜖 (𝑠), 𝜇𝜖 (𝑡)) ≤ lim inf
𝜖→0

∫ 𝑡

𝑠

|𝜇′

𝜖 |(𝑟) 𝑑𝑟 =

∫ 𝑡

𝑠

𝜈(𝑟) 𝑑𝑟.

This gives us 𝜇 ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)). By [1, Theorem 1.1.2], we have |𝜇′|(𝑟) ≤ 𝜈(𝑟) for a.e.
𝑟 ∈ [0, 𝑇]. We finish by acknowledging that the 𝐿2( [0, 𝑇]) norm is lower semi-continuous with

respect to weak convergence.

Due to the fact that 𝑎(𝑥) is log-concave on Ω instead of R𝑑, we require a weaker framework of

Serfaty’s result. Once we show that 𝜔-convex functionals are regular in Theorem 2.3.23, then we

obtain the following.

Theorem 2.3.16 (Weak Serfaty Framework for 𝜔-convex functionals, [14] Proposition 2.16). Let
F , F𝜖 : P2(R𝑑) → R be functionals that are proper, lower semi-continuous, 𝜔-convex along
generalized geodesics, and bounded from below uniformly in 𝜖 and suppose F𝜖 Γ-converges to F as

𝜖 → 0. Fix 𝑇 > 0. Suppose that for all 𝜖 > 0 there exists 𝜇𝜖 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) and for almost
all 𝑟 ∈ [0, 𝑇] there exists 𝜂𝜖 (𝑟) ∈ 𝐿2(𝜇𝜖 (𝑟)) such that

1
2

∫ 𝑡

0

|𝜇′

𝜖 |2(𝑟) 𝑑𝑟 +

1
2

∫ 𝑡

∫

0

R𝑑

|𝜂𝜖 (𝑟)|2 𝑑𝜇𝜖 (𝑟) 𝑑𝑟 ≤ F𝜖 (𝜇𝜖 (0)) − F𝜖 (𝜇𝜖 (𝑡))

20

for all 0 ≤ 𝑡 ≤ 𝑇. Suppose there exists 𝜇(0) ∈ 𝐷 (F ) ∩ P2(R𝑑) such that sup𝜖 >0
and as 𝜖 → 0

𝑀2(𝜇𝜖 (0)) < ∞

𝜇𝜖 (0) → 𝜇(0) narrowly, F𝜖 (𝜇𝜖 (0)) → F (𝜇(0)).

Then, there exists 𝜇 ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)) so that up to a subsequence

𝑊1(𝜇𝜖 (𝑡), 𝜇(𝑡)) = 0, uniformly for 𝑟 ∈ [0, 𝑇].

lim
𝜖→0

Furthermore, we have

1
2

∫ 𝑡

0

|𝜇′|2(𝑟) 𝑑𝑟 +

1
2

∫ 𝑡

0

lim inf
𝜖→0

∫

R𝑑

|𝜂𝜖 (𝑟)|2 𝑑𝜇𝜖 (𝑟) 𝑑𝑟 ≤ F (𝜇(0)) − F (𝜇(𝑡))

for all 0 ≤ 𝑡 ≤ 𝑇.

Proof. As the initial data is well prepared, we may assume

F𝜖 (𝜇𝜖 (0)) < ∞.

sup
𝜖 >0

With the assumption that F𝜖 is bounded from below uniformly in 𝜖, then

1
2

∫ 𝑡

0

|𝜇′

𝜖 |2(𝑟) 𝑑𝑟 +

1
2

∫ 𝑡

∫

0

R𝑑

|𝜂𝜖 (𝑟)|2 𝑑𝜇𝜖 (𝑟) 𝑑𝑟,

is bounded from above uniformly in 𝜖. It follows that, sup𝜖 >0

∫ 𝑡
0

|𝜇′

𝜖 |2(𝑟) 𝑑𝑟 < ∞. We may apply

Lemma 2.3.15 so that,

1
2

∫ 𝑡

0

|𝜇′|2(𝑟) 𝑑𝑟 +

1
2

lim inf
𝜖→0

∫ 𝑡

∫

0

R𝑑

|𝜂𝜖 (𝑟)|2 𝑑𝜇𝜖 (𝑟) 𝑑𝑟 ≤ lim inf
𝜖→0

(F𝜖 (𝜇𝜖 (0)) − F𝜖 (𝜇𝜖 (𝑡))).

We use Fatou’s Lemma to control the second term on the left-hand side. For the right-hand side, use

use Γ-convergence of the energy functional and narrow convergence of the density to obtain the

result.

If the log-concavity of 𝑎(𝑥) were not required for the well-posedness of the gradient flows of F

or if we did not have to restrict the log-concavity of 𝑎(𝑥) to Ω, then we could use the stronger result

that would resemble [25, Theorem 2] much more. In fact, we can use this version to prove Theorem

2.2.4.

21

Theorem 2.3.17 (Serfaty’s Sufficient Conditions for 𝜔-convex functionals). Let G𝜖 : P2(𝑅𝑑) →
R ∪ {+∞} and G : P2(R𝑑) → R ∪ {+∞} be proper, lower semi-continuous that are 𝜔-convex along
generalized geodesics. Let |𝜕G𝜖 | and |𝜕G| be strong upper gradients of G𝜖 and G, respectively.

For all 𝜖 > 0, let 𝜇𝜖 ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)) be a gradient flow of G𝜖 and that there exists
𝜇 : [0, 𝑇] → P2(𝑅𝑑) such that, 𝜇𝜖 (𝑡) → 𝜇(𝑡) narrowly for 𝑡 ∈ [0, 𝑇] and

(2.5)

𝜇(0) ∈ 𝐷 (G),

lim
𝜖→0

G𝜖 (𝜇𝜖 (0)) = G(𝜇(0)).

Assume that (2.3) holds and for almost every 𝑡 ∈ [0, 𝑇],

(2.6)

(2.7)

∫ 𝑡

0

|𝜇′

𝜖 |2(𝑠) 𝑑𝑠 ≥

∫ 𝑡

0

|𝜇′|(𝑠) 𝑑𝑠,

|𝜕G𝜖 |2(𝜇𝜖 (𝑡)) ≥ |𝜕G|2(𝜇(𝑡)).

lim inf
𝜖→0

lim inf
𝜖→0

Then 𝜇 ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)) and 𝜇 is a gradient flow of G with initial data 𝜇(0).

Proof. Based on [25, Theorem 2], it suffices to show that the local slopes of 𝜔-convex energies

are strong upper gradients (Theorem 2.3.18) and that 𝜔-convex energies are regular (Theorem

2.3.23).

We wish to prove the Γ -convergence of the gradient flows of 𝜔-convex energies using Serfaty’s

results [25, Theorem 2]. We must first show this holds for 𝜔-convex energies instead of the usually

semi-convex energies. The first thing to check is that local slopes are strong upper gradients.

Theorem 2.3.18 (Local slopes are strong upper gradients). Let 𝐸 be 𝜔-convex along geodesics.

Then |𝜕𝐸 | is a strong upper gradient of 𝐸.

Proof. Define the global slope 𝐼𝐸 (𝜇) = sup𝜈≠𝜇
𝐼𝐸 (𝜇). By the HWI inequality (see Proposition 2.5 of [14]),

(𝐸 (𝜇)−𝐸 (𝜈))+
𝑊2 (𝜇,𝜈)

. By definition of local slope, |𝜕𝐸 |(𝜇) ≤

𝜔(𝑊 2

2 (𝜇, 𝜈))

𝑊2(𝜇, 𝜈)

(cid:33) +

.

|𝜕𝐸 |(𝜇) = sup
𝜈≠𝜇

(cid:32) 𝐸 (𝜇) − 𝐸 (𝜈)
𝑊2(𝜇, 𝜈)

+

𝜆𝜔
2

22

Using that equality,

𝐼𝐸 (𝜇) = sup
𝜈≠𝜇

(cid:32) 𝐸 (𝜇) − 𝐸 (𝜈)
𝑊2(𝜇, 𝜈)

+

𝜆𝜔
2

−

𝜆𝜔
2

𝜔(𝑊 2

2 (𝜇, 𝜈))

𝑊2(𝜇, 𝜈)

(cid:33) +

𝜔(𝑊 2

2 (𝜇, 𝜈))

𝑊2(𝜇, 𝜈)
2 (𝜇, 𝜈))

𝜔(𝑊 2

𝑊2(𝜇, 𝜈)

≤ |𝜕𝐸 |(𝜇) +

(−𝜆𝜔)+
2

sup
𝜈≠𝜇
𝜔(𝑊 2

= |𝜕𝐸 |(𝜇) +

𝜆−
𝜔
2

sup
𝜈≠𝜇

2 (𝜇, 𝜈))

.

𝑊2(𝜇, 𝜈)

If 𝜆𝜔 ≥ 0, then we get that the local slope is identical to the global slope. As the global slope is a

strong upper gradient ([1, Theorem 1.2.5]), we get that the local slope is a strong upper gradient.
Thus, we only need to consider when 𝜆𝜔 < 0. For 𝜇, 𝜈 ∈ P2(R𝑑),

𝑊 2

2 (𝜇, 𝜈) ≤ 𝑀2(𝜇) + 𝑀2(𝜈) < ∞.

Since sup𝜈≠𝜇

(𝜇,𝜈))

𝜔(𝑊 2
2
𝑊2 (𝜇,𝜈)

is finite, then we can apply [1, Theorem 1.2.5]. We get that the local slope

is a strong upper gradient as in [1, Corollary 2.4.10].

The next thing we must check is that curves of maximal slope coincide with gradient flows of

𝜔-convex energies. It suffices to show that 𝜔-convex energies are regular. In particular, we must

show the subdifferential of an 𝜔-convex energy is closed. We first characterize the subdifferential

for 𝜔-convex energies similarly to semi-convex energies.

Theorem 2.3.19 (subdifferential characterization of 𝜔-convex functionals). Suppose 𝐸 : P2(R𝑑) →
(−∞, +∞] that is proper, lower semi-continuous, and 𝜔-convex along geodesics. Let 𝜇 ∈ 𝐷 (𝐸)

and 𝜉 : R𝑑 → R𝑑 with 𝜉 ∈ 𝐿2(𝜇). Then 𝜉 ∈ 𝜕𝐸 (𝜇) if and only if for all 𝜈 ∈ P2(R𝑑)

𝐸 (𝜈) − 𝐸 (𝜇) ≥

∫

R𝑑×R𝑑

⟨𝜉 (𝑥), 𝑦 − 𝑥⟩ 𝑑𝛾(𝑥, 𝑦) +

𝜆𝜔
2

𝜔(𝑊 2

2 (𝜇, 𝜈)), ∀𝛾 ∈ Γ0(𝜇, 𝜈).

Proof. The 𝜔-convexity characterization implies the definition ([1, Defintion 10.1.1]) as

𝜔(𝑊 2

2 (𝜇, 𝜈)) = 𝑜(𝑊2(𝜇, 𝜈)) as 𝜈 → 𝜇 in P2(R𝑑). The converse mostly follows from ([1, B
)#𝛾 for

in Section 10.1.1]). Conversely, suppose 𝜇0, 𝜇1 ∈ P2(R𝑑), 𝛾 ∈ Γ0(𝜇0, 𝜇1), 𝜇𝛼 = (𝜋1→2

𝛼

23

𝑡 ∈ [0, 1]. By the definition of subdifferential at 𝜇0 for 𝜉 ∈ 𝜕𝐸 (𝜇0) and 𝛾𝑡 ∈ Γ0(𝜇0, 𝜇𝑡)

𝐸 (𝜇𝑡) − 𝐸 (𝜇0) ≥

=

∫

∫

⟨𝜉 (𝑥), 𝑦 − 𝑥⟩ 𝑑𝛾𝑡 (𝑥, 𝑦) + 𝑜(𝑊 2

2 (𝜇0, 𝜇𝑡))

⟨𝜉 (𝑥), 𝑡𝑦 − 𝑡𝑥⟩ 𝑑𝛾(𝑥, 𝑦) + 𝑜(𝑡𝑊 2

2 (𝜇0, 𝜇1))

∫

= 𝑡

⟨𝜉 (𝑥), 𝑦 − 𝑥⟩ 𝑑𝛾(𝑥, 𝑦) + 𝑜(𝑡)

where the first equality is a result of change of variables. Now we divide by t and compute the lim inf

𝐸 (𝜇𝑡) − 𝐸 (𝜇0)
𝑡

∫

≥

lim inf
𝑡→0

⟨𝜉 (𝑥), 𝑦 − 𝑥⟩ 𝑑𝛾(𝑥, 𝑦).

By the definition of 𝜔-convexity,

𝐸 (𝜇𝑡) ≤ (1 − 𝑡)𝐸 (𝜇0) + 𝑡𝐸 (𝜇 + 1) −

(cid:16)

𝜆𝜔
2

((1 − 𝑡)𝜔(𝑡2𝑊 2

2 (𝜇0, 𝜇1)) + 𝑡𝜔((1 − 𝑡)2𝑊 2

2 (𝜇0, 𝜇1))

(cid:17)

.

Rearranging and dividing by 𝑡 we get a bound for the difference quotient

𝐸 (𝜇𝑡) − 𝐸 (𝜇0)
𝑡

≤ 𝐸 (𝜇1) − 𝐸 (𝜇0) −

𝜆𝜔
2

(cid:18)

(

1
𝑡

We take the lim inf

− 1)𝜔(𝑡2𝑊 2

2 (𝜇0, 𝜇1)) + 𝜔((1 − 𝑡)2𝑊 2

2 (𝜇0, 𝜇1))

(cid:19)

.

𝐸 (𝜇𝑡) − 𝐸 (𝜇0)
𝑡

lim inf
𝑡→0

≤ 𝐸 (𝜇1) − 𝐸 (𝜇0) −

𝜆𝜔
2

𝜔(𝑊 2

2 (𝜇0, 𝜇1))

and get the right-hand side by using that 𝜔 is continuous, 𝜔(0) = 0, and 𝜔(𝑥) = 𝑜(

√

𝑥) as 𝑥 → 0.

Rearranging and using the lower bound for the difference quotient gives the result.

Definition 2.3.20 (weak convergence of varying measure). Let 𝜇𝑛 be a sequence in P (R𝑑) be

narrowly converging to 𝜇 in P (R𝑑) and let 𝑣𝑛 ∈ 𝐿1(𝜇𝑛; R𝑚). We say that 𝑣𝑛 weakly converges to

𝑣 ∈ 𝐿1(𝜇; R𝑚) if for all 𝜁 ∈ 𝐶∞

𝑐 (R𝑑),

∫

𝑅𝑑

lim
𝑛→∞

𝜁 (𝑥)𝑣𝑛 (𝑥) 𝑑𝜇𝑛 (𝑥) =

∫

𝑅𝑑

𝜁 (𝑥)𝑣(𝑥) 𝑑𝜇(𝑥).

The main goal to show Theorem 2.3.21 is to take the lim inf as 𝑛 → ∞ of the inequality in the

characterization of the subdifferential of an 𝜔-convex energy. We have no issue with the energy term

as 𝐸 is lower semi-continuous and no issue with the 𝜔-convexity term as it is continuous. The main

24

issue is with the integral term, which has nothing to do with 𝐸 being 𝜔-convex. Thus, Theorem

2.3.21 follows from the standard result [1, Lemma 10.1.3] and noting that it is not restrictive to use

a transport map as it can be adjusted so that no map is necessary (see [1, Remark 10.3.3]).

Theorem 2.3.21 (closure of subdifferential, [1] Lemma 10.1.3). Let 𝐸 be proper, lower semi-
continuous, and 𝜔-convex functional. Let 𝜇𝑛 converge to 𝜇 ∈ 𝐷 (𝐸) in P2(R𝑑) and let 𝜉 ∈ 𝜕𝐸 (𝜇𝑛)

satisfying

∫

sup
𝑛

|𝜉 (𝑥)|2 𝑑𝜇𝑛 (𝑥) < ∞,

and converge to 𝜉 weakly (with varying measure). Then 𝜉 ∈ 𝜕𝐸 (𝜇).

Definition 2.3.22 (regular functional). A functional 𝐸 : P2(R𝑑) → (−∞, ∞] is proper, lower semi-
continuous, and 𝐷 (|𝜕𝐸 |) ⊆ P2(R𝑑). We say that 𝐸 is regular if whenever the strong differentials
𝜉𝑛 ∈ 𝜕𝐸 (𝜇𝑛), 𝜑𝑛 = 𝐸 (𝜇𝑛) satisfy

𝜇𝑛 → 𝜇 in P2(R𝑑),

𝜑𝑛 → 𝜑,

sup𝑛 ∥𝜉𝑛 ∥ 𝐿2 (𝜇𝑛;R𝑑) < ∞

𝜉𝑛 → 𝜉 weakly (of varying measure),





then 𝜉 ∈ 𝜕𝐸 (𝜇) and 𝜑 = 𝐸 (𝜇).

Theorem 2.3.23 (𝜔-convex functionals are regular). Suppose 𝐸 : P2(R𝑑) → (−∞, +∞] that is
proper, lower semi-continuous, and 𝐷 (|𝜕𝐸 |) ⊆ P2(R𝑑). Then 𝐸 is regular.

Proof. By Theorem 2.3.21, we have 𝜉 ∈ 𝜕𝐸 (𝜇). As 𝐸 is lower semi-continuous, it suffices to show

lim sup
𝑛→∞

𝐸 (𝜇𝑛) ≤ 𝐸 (𝜇),

lim inf
𝑛→∞

(𝐸 (𝜇) − 𝐸 (𝜇𝑛)) ≥ 0.

which is equivalent to

By Theorem 2.3.19,

𝐸 (𝜇) − 𝐸 (𝜇𝑛) ≥

∫

R𝑑×R𝑑

⟨𝜉𝑛 (𝑥), 𝑦 − 𝑥⟩ 𝑑𝛾𝑛 (𝑥, 𝑦) +

𝜆𝜔
2

𝜔(𝑊 2

2 (𝜇𝑛, 𝜇)), ∀𝛾𝑛 ∈ Γ0(𝜇𝑛, 𝜇).

25

As in Theorem 2.3.21, having 𝜇𝑛 converge to 𝜇 in P2(𝑅𝑑) (with respect to 𝑊2), the right-hand side

goes to zero. Thus,

lim inf
𝑛→∞

(𝐸 (𝜇) − 𝐸 (𝜇𝑛)) ≥ 0,

and moreover by lower semi-continuity lim𝑛→∞ 𝐸 (𝜇𝑛) = 𝐸 (𝜇).

2.4 Energy Properties

2.4.1 Lower semi-continuity of Energies

The first step in showing that the gradient flows of the energy functionals are well-posed is to

show they are lower semi-continuous.

Proposition 2.4.1 (lower semi-continuity). Let 𝜖 > 0 and 𝑉, 𝑊 satisfy Assumptions 2.1.3 and 2.1.4,

respectively. Then V𝜖 , W𝜖 are lower semi-continuous with respect to narrow convergence in P (R𝑑).

Proof. By Lemma 5.1.7 of [1], V and W are lower semi-continuous with respect to narrow

convergence. By definition V𝜖 (𝜇𝑛) = V (𝜁𝜖 ∗ 𝜇𝑛) and W𝜖 (𝜇𝑛) = W (𝜁𝜖 ∗ 𝜇𝑛). Thus we have,

lim inf
𝑛→∞

V𝜖 (𝜇𝑛) ≥ V𝜖 (𝜇),

lim inf
𝑛→∞

W𝜖 (𝜇𝑛) ≥ W𝜖 (𝜇).

This completes the proof.

The following proposition is standard and follows from [1] and [14].

Proposition 2.4.2 (Lower semi-continuity of E, E𝜖 , V, V𝑘 , VΩ, W). Suppose Assumptions 2.1.3,

2.1.4, 2.1.5 hold. The for all 𝜖 > 0, the functionals E, E𝜖 , V, V𝑘 , VΩ, W are lower semi-continuous

with respect to narrow convergence.

2.4.2 Directional derivatives of Energies

The derivatives for the regularized drift and aggregation functionals follow from the first parts

of the proofs of Proposition 4.6 and 4.7 in [13].

Proposition 2.4.3 (Directional Derivatives for V𝜖 , W𝜖 ). Suppose Assumptions 2.1.3 and 2.1.4
holds. Fix 𝜖 > 0, 𝜇0, 𝜇1 ∈ P2(R𝑑) satisfying ∥𝜇𝑖 ∥ 𝐿2 ≤ 𝐶 for 𝑖 = 1, 2 , and 𝛾 ∈ Γ(𝜇0, 𝜇1) such that

26

𝜇𝛼 = ((1 − 𝛼)𝜋1 + 𝛼𝜋2)#𝛾 satisfying ∥𝜇𝛼 ∥ 𝐿2 ≤ 𝐶 for all 𝛼 ∈ [0, 1]. Then V𝜖 (𝜇𝛼), W𝜖 (𝜇𝛼) are

continuously differentiable and

V𝜖 (𝜇𝛼)|𝛼=0 =

∫

W𝜖 (𝜇𝛼)|𝛼=0 =

∫

𝑑
𝑑𝛼
𝑑
𝑑𝛼

⟨∇(𝜁𝜖 ∗ 𝑉) (𝑦0), 𝑦1 − 𝑦0⟩ 𝑑𝛾,

⟨∇(𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊) ∗ 𝜇0(𝑦0), 𝑦1 − 𝑦0⟩ 𝑑𝛾.

Proof. As the calculations are similar, we elect to only show the one for the regularized drift energy.

By Proposition 2.4.9, 𝜁𝜖 ∗ 𝑉 is continuously differentiable. Define 𝜋𝛼 = (1 − 𝛼)𝜋1 + 𝛼𝜋2 where

𝜋1, 𝜋2 are projections on the first and second axis, respectively. Then,

∫

𝑑
𝑑𝛼

𝜁𝜖 ∗ 𝑉 𝑑𝜇𝛼 = lim
ℎ→0
∫

(cid:18)∫

1
ℎ

𝜁𝜖 ∗ 𝑉 ◦ 𝜋𝛼+ℎ 𝑑𝛾 −

∫

𝜁𝜖 ∗ 𝑉 ◦ 𝜋𝛼 𝑑𝛾

(cid:19)

=

⟨∇(𝜁𝜖 ∗ 𝑉) ◦ 𝜋𝛼, 𝑦1 − 𝑦0⟩ 𝑑𝛾.

Thus,

This finishes the proof.

V𝜖 (𝜇𝛼)|𝛼=0 =

∫

𝑑
𝑑𝛼

⟨∇(𝜁𝜖 ∗ 𝑉) (𝑦0), 𝑦1 − 𝑦0⟩ 𝑑𝛾.

Corollary 2.4.4 (Directional Derivatives for V, V𝑘 , W). There are analogous derivatives for

V, V𝑘 , W,

V𝑘 (𝜇𝛼)|𝛼=0 =

V (𝜇𝛼)|𝛼=0 =

∫

∫

W (𝜇𝛼)|𝛼=0 =

∫

𝑑
𝑑𝛼
𝑑
𝑑𝛼
𝑑
𝑑𝛼

⟨∇𝑉𝑘 (𝑦0), 𝑦1 − 𝑦0⟩ 𝑑𝛾

⟨∇𝑉 (𝑦0), 𝑦1 − 𝑦0⟩ 𝑑𝛾,

⟨∇𝑊 ∗ 𝜇0(𝑦0), 𝑦1 − 𝑦0⟩ 𝑑𝛾.

Proof. As the calculations are similar, we elect to only show the one for the drift energy. By

Assumption 2.1.4, 𝑉 is continuously differentiable. Define 𝜋𝛼 = (1 − 𝛼)𝜋1 + 𝛼𝜋2 where 𝜋1, 𝜋2 are

projections on the first and second axis, respectively. Then,

∫

𝑑
𝑑𝛼

𝑉 𝑑𝜇𝛼 = lim
ℎ→0
∫

(cid:18)∫

1
ℎ

𝑉 ◦ 𝜋𝛼+ℎ 𝑑𝛾 −

∫

(cid:19)

𝑉 ◦ 𝜋𝛼 𝑑𝛾

=

⟨∇𝑉 ◦ 𝜋𝛼, 𝑦1 − 𝑦0⟩ 𝑑𝛾.

27

Thus,

V (𝜇𝛼)|𝛼=0 =

∫

𝑑
𝑑𝛼

⟨∇𝑉 (𝑦0), 𝑦1 − 𝑦0⟩ 𝑑𝛾.

This completes the proof.

We restate a result from [14].

Proposition 2.4.5 (Directional Derivatives of E𝜖 , [14] Proposition 3.4). Suppose Assumptions 2.1.1,
2.1.2 hold. Fix 𝜖 > 0, 𝜈1, 𝜈2, 𝜈3 ∈ P2(R𝑑) and 𝛾 ∈ P2(R𝑑 × R𝑑 × R𝑑) with 𝜋𝑖
#
curve, 𝜇𝛼 = (cid:0)(1 − 𝛼)𝜋2 + 𝛼𝜋3(cid:1)

𝛾 for 𝛼 ∈ [0, 1]. Then,

𝛾 = 𝜈𝑖. Consider the

#

𝑑
𝑑𝛼

E𝜖 (𝜇)|𝛼=0 =

1
2

∫ 𝜁𝜖 ∗ 𝜈2(𝑥)

∫

𝑎(𝑥)

⟨∇𝜁𝜖 (𝑥 − 𝑦2), 𝑦3 − 𝑦2⟩ 𝑑𝛾(𝑦1, 𝑦2, 𝑦3) 𝑑𝑥.

Proof. Let 𝑥 ∈ R𝑑 and 𝛼 ∈ [0, 1]. By the dominated convergence theorem,

1
𝛼

lim
𝛼→0

(𝜁𝜖 ∗ 𝜇𝛼 (𝑥) − 𝜁𝜖 ∗ 𝜇0(𝑥)) =

∫

⟨∇𝜁𝜖 (𝑥 − 𝑦2), 𝑦3 − 𝑦2⟩ 𝑑𝛾(𝑦1, 𝑦2, 𝑦3),

as ∥∇𝜁𝜖 ∥ 𝐿∞ |𝑦3 − 𝑦2| ∈ 𝐿1(𝛾) and 𝑀1(𝛾) ≤ 𝑀2(𝛾)1/2 < ∞. Again by dominated convergence

theorem,

1
𝛼

(E𝜖 (𝜇𝛼) − E𝜖 (𝜇0))

lim
𝛼→0
∫

=

1
(𝜁𝜖 ∗ 𝜇𝛼 (𝑥) − 𝜁𝜖 ∗ 𝜇0(𝑥)) (𝜁𝜖 ∗ 𝜇𝛼 (𝑥) + 𝜁𝜖 ∗ 𝜇0(𝑥)) 𝑑𝑥
lim
2𝛼𝑎(𝑥)
𝛼→0
∫
∫ 𝜁𝜖 ∗ 𝜇0(𝑥)

⟨∇𝜁𝜖 (𝑥 − 𝑦2), 𝑦3 − 𝑦2⟩ 𝑑𝛾(𝑦1, 𝑦2, 𝑦3) 𝑑𝑥.

=

𝑎(𝑥)

This finishes the proof.

2.4.3 Convexity of Energies

The second step in showing that the gradient flows of the energy functionals are well-posed is to

show they are convex. We start with the main purpose of [13]. Namely, the conditions in which an

energy functional is 𝜔-convex. We restate Proposition 4.6 and 4.7 in [13] as one result.

28

Proposition 2.4.6. Let 𝜇0, 𝜇1 ∈ P2(R𝑑) satisfying ∥𝜇𝑖 ∥ 𝐿2 ≤ 𝐶 for 𝑖 = 1, 2 , and 𝛾 ∈ Γ(𝜇0, 𝜇1) such
that 𝜇𝛼 = ((1 − 𝛼)𝜋1 + 𝛼𝜋2)#𝛾 satisfying ∥𝜇𝛼 ∥ 𝐿2 ≤ 𝐶 for all 𝛼 ∈ [0, 1]. Then for 𝜔(𝑥) = √︁𝑥𝜓(𝑥),

𝑑
𝑑𝛼

V (𝜇0) − V (𝜇1) +
(cid:12)
(cid:12)
(cid:12)
(cid:12)

W (𝜇0) − W (𝜇1) +

V (𝜇𝛼)|𝛼=0 ≤ 2𝐶𝜔
𝑑
𝑑𝛼

W (𝜇𝛼)|𝛼=0

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

(cid:16)

∥𝑥 − 𝑦∥2

𝐿2 (𝛾)

(cid:17)

≤ 2𝐶𝜔

(cid:16)

∥𝑥 − 𝑦∥2

𝐿2 (𝛾)

(cid:17)

Proof. We only show the proof of the drift as they are similar. Define 𝜋𝛼 = (1 − 𝛼)𝜋1 + 𝛼𝜋2

where 𝜋1, 𝜋2 are projections on the first and second axis, respectively. As we know the directional

derivative,

V (𝜇1) = V (𝜇0) +

𝑑
𝑑𝛼

V (𝜇𝛼)|𝛼=0 +

∬ ∫ 1

0

⟨∇𝑉 ((1 − 𝛼𝑥 + 𝛼𝑦) − ∇𝑉 (𝑥), 𝑦 − 𝑥⟩ 𝑑𝛼 𝑑𝛾.

So we only need to control the final term by the correct bound. By Hölder’s inequality, item 3. of

Assumption 2.1.4, and Jensen’s inequality for concave 𝜓(𝑥),

∬

|∇𝑉 ◦ 𝜋𝛼 − ∇𝑉 (𝑥)| |𝑥 − 𝑦| 𝑑𝛾(𝑥, 𝑦) ≤ ∥∇𝑉 ◦ 𝜋𝛼 − ∇𝑉 ∥ 𝐿2 (𝛾) ∥𝑥 − 𝑦∥ 𝐿2 (𝛾)

≤ 2𝐶 ∥𝑥 − 𝑦∥ 𝐿2 (𝛾)

≤ 2𝐶 ∥𝑥 − 𝑦∥ 𝐿2 (𝛾)

√︃

√︃

∥𝜓(|𝜋𝛼 − 𝜋0|2) ∥ 𝐿1 (𝛾)

𝜓(∥|𝜋𝛼 − 𝜋0|2∥ 𝐿1 (𝛾))

√︃𝜓(𝛼2∥𝜋1 − 𝜋0∥2

𝐿2 (𝛾)

)

= 2𝐶 ∥𝑥 − 𝑦∥ 𝐿2 (𝛾)
(cid:16)

∥𝑥 − 𝑦∥2

≤ 2𝐶𝜔

(cid:17)

.

𝐿2 (𝛾)

Thus we have the result.

By Propositions 2.3.10, 2.4.6, we achieve the 𝜔-convexity of the drift and aggregation.

Proposition 2.4.7 (𝜔-convexity of V, W, [13] Theorem 4.3). Let 𝑊, 𝑉 satisfy Assumptions 2.1.3
and 2.1.4, respectively. Then V, W are 𝜔-convex along generalized geodesics, with 𝜔(𝑥) = √︁𝑥𝜓(𝑥),

𝜆𝜔 = 4𝐶.

Lemma 2.4.8. Let 𝜏𝑦 : R𝑑 → R𝑑 be the translation mapping 𝜏𝑦 (𝑥) = 𝑥 − 𝑦 and 𝜈 ∈ P2(R𝑑) with
∥𝜈∥ 𝐿 𝑝 ≤ 𝐶𝑝. Then, 𝜏𝑦#𝜈 ∈ P2(R𝑑) and ∥𝜏𝑦#𝜈∥ 𝐿 𝑝 ≤ 𝐶𝑝.

29

Proof. Given that 𝜏−1

𝑦 (R𝑑) = R𝑑, we have the following

𝜏𝑦#𝜈(R𝑑) = 𝜈(𝜏−1

𝑦 (R𝑑)) = 𝜈(R𝑑) = 1.

So,

Moreover,

∫

R𝑑

|𝑥|2 𝑑𝜏𝑦#𝜈(𝑥) =

∫

𝑦 (R𝑑)
𝜏−1

|𝑥|2 𝑑𝜈(𝑥) =

∫

R𝑑

|𝑥|2 𝑑𝜈(𝑥) < ∞.

∥𝜏𝑦#𝜈∥

𝑝
𝐿 𝑝 = ∥𝜈∥

𝑝
𝐿 𝑝 (𝜏−1

𝑦 (R𝑑))

=

∫

R𝑑

|𝑣(𝑥)| 𝑝 𝑑𝑥 ≤ 𝐶 𝑝
𝑝 .

Thus we obtain the result.

We use this lemma in a couple parts of the next proposition. Namely, that

∫

|∇𝑊 ∗ 𝜇(𝑥 − 𝑦)|2 𝑑𝜈(𝑥) =

∫

|∇𝑊 ∗ 𝜇(𝑥)|2 𝑑𝜏𝑦#𝜈(𝑥) ≤ 𝐶2.

We now show that the mollified versions of 𝑉 and 𝑊 satisfy Assumptions 2.1.3 and 2.1.4.

Proposition 2.4.9. If 𝑊, 𝑉 satisfies Assumptions 2.1.3 and 2.1.4, respectively. Then

1. for all 𝜇 ∈ P2(R𝑑) with∥𝜇∥ 𝐿 𝑝 ≤ 𝐶𝑝, (𝜁𝜖 ∗ 𝑊)− ∗ 𝜇(𝑥) ≤ 𝐶 and 𝜁𝜖 ∗ 𝑉 ≥ −𝐶
2. for all 𝜇, 𝜈 ∈ P2(R𝑑) with ∥𝜇∥ 𝐿 𝑝 ≤ 𝐶𝑝 , we have ∥𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇∥ 𝐿2 (𝜈) ≤ 𝐶 and with 𝜈 ∈ 𝐷 (V),

we have ∥𝜁𝜖 ∗ ∇𝑉 ∥ 𝐿2 (𝜈) ≤ 𝐶

3. for all ∥𝜇∥ 𝐿 𝑝 ≤ 𝐶𝑝, 𝜁𝜖 ∗ 𝑊 ∗ 𝜇 and 𝜁𝜖 ∗ 𝑉 are continuously differentiable and there

exists a continuous, nondecreasing, concave function 𝜓 : [0, ∞) → [0, ∞) satisfying
𝜓(0) = 0, 𝜓(𝑥) ≥ 𝑥, and ∫ 1

𝑑𝑥
𝜓(𝑥) = ∞ so that

0

|𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇(𝑥) − 𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇(𝑦)|2 ≤ 𝐶2𝜓(|𝑥 − 𝑦|2),

|𝜁𝜖 ∗ ∇𝑉 (𝑥) − 𝜁𝜖 ∗ ∇𝑉 (𝑦)|2 ≤ 𝐶2𝜓(|𝑥 − 𝑦|2)

4. for all ∥𝜇∥ 𝐿 𝑝 , ∥𝜈∥ 𝐿 𝑝 , ∥ 𝜌∥ 𝐿 𝑝 ≤ 𝐶𝑝,

∥𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇 − 𝜁𝜖 ∗ ∇𝑊 ∗ 𝜈∥ 𝐿2 (𝜌) ≤ 𝐶𝑊2(𝜇, 𝜈)

30

5. 𝜁𝜖 ∗ 𝑊 is lower semi-continuous

Proof. We will only show the case for 𝑊 as the case for 𝑉 is similar when we choose 𝑉 := 𝑊 ∗ 𝜇.

1. As 𝑊 − ∗ 𝜇 is bounded above by 𝐶,

𝜁𝜖 ∗ 𝑊 − ∗ 𝜇(𝑥) =

∫

R𝑑

𝜁𝜖 (𝑥 − 𝑦) (𝑊 − ∗ 𝜇(𝑦)) 𝑑𝑦 ≤ 𝐶.

As (𝜁𝜖 ∗ 𝑊)− = max{0, −𝜁𝜖 ∗ 𝑊 } and

−𝜁𝜖 ∗ 𝑊 ∗ 𝜇 ≤ 𝜁𝜖 ∗ max{0, −𝑊 } ∗ 𝜇 = 𝜁𝜖 ∗ 𝑊 − ∗ 𝜇

we get the result.

2. By Jensen’s inequality and Assumption 2.1.3,

∫

|𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇|2𝑑𝜈(𝑥) =

≤

∫

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

∇𝑊 ∗ 𝜇(𝑥 − 𝑦)𝜁𝜖 (𝑦) 𝑑𝑦

2

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

𝑑𝜈(𝑥)

|∇𝑊 ∗ 𝜇(𝑥 − 𝑦)|2𝜁𝜖 (𝑦) 𝑑𝑦 𝑑𝜈(𝑥)

∫ ∫

|∇𝑊 ∗ 𝜇(𝑥 − 𝑦)|2 𝑑𝜈(𝑥) 𝜁𝜖 (𝑦) 𝑑𝑦

≤ (𝐶)2.

3. By Jensen’s inequality and Assumption 2.1.3,

|𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇(𝑥) − 𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇(𝑦)|2 =

≤

∫

(cid:12)
(cid:12)
(cid:12)
(cid:12)
∫

(∇𝑊 ∗ 𝜇(𝑥 − 𝑧) − ∇𝑊 ∗ 𝜇(𝑦 − 𝑧))𝜁𝜖 (𝑧) 𝑑𝑧

2

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

|(∇𝑊 ∗ 𝜇(𝑥 − 𝑧) − ∇𝑊 ∗ 𝜇(𝑦 − 𝑧))|2𝜁𝜖 (𝑧) 𝑑𝑧

4. By Jensen’s inequality and Assumption 2.1.3,

≤ 𝐶2𝜓(|𝑥 − 𝑦|2)

∫

=

≤

=

|𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇 − 𝜁𝜖 ∗ ∇𝑊 ∗ 𝜈|2𝑑𝜌(𝑥)

∫

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

∫ ∫

(∇𝑊 ∗ 𝜇(𝑥 − 𝑦) − ∇𝑊 ∗ 𝜈(𝑥 − 𝑦))𝜁𝜖 (𝑦) 𝑑𝑦

2

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

𝑑𝜌(𝑥)

|∇𝑊 ∗ 𝜇(𝑥 − 𝑦) − ∇𝑊 ∗ 𝜈(𝑥 − 𝑦)|2𝜁𝜖 (𝑦) 𝑑𝑦 𝑑𝜌(𝑥)

|∇𝑊 ∗ 𝜇(𝑥 − 𝑦) − ∇𝑊 ∗ 𝜈(𝑥 − 𝑦)|2 𝑑𝜌(𝑥) 𝜁𝜖 (𝑦) 𝑑𝑦

≤ (𝐶𝑊2(𝜇, 𝜈))2 .

31

5. As 𝑊 is lower semi-continuous ∀𝜖 > 0 ∃𝛿 > 0 such that 𝑊 (𝑥0) < 𝑊 (𝑥) + 𝜖 for all 𝑥 ∈ 𝐵𝛿 (𝑥0).

So we can translate by 𝑦 so that,

𝜁𝜖 ∗ 𝑊 (𝑥0) =

<

∫

∫

𝑊 (𝑥0 − 𝑦)𝜁𝜖 (𝑦)𝑑𝑦

𝑊 (𝑥 − 𝑦)𝜁𝜖 (𝑦)𝑑𝑦 + 𝜖

∫

𝜁𝜖 (𝑦)𝑑𝑦

= 𝜁𝜖 ∗ 𝑊 (𝑥) + 𝜖 .

Therefore, 𝜁𝜖 ∗ 𝑊 is lower semi-continuous.

Thus we have the results.

Remark 2.4.10. We can generalize this further having the same statement hold with another

mollifier 𝜁𝜖 convolved against it by using the previous proposition and the proof along with it.

Now from Proposition 2.4.9, we obtained the conditions required to have V𝜖 , W𝜖 are 𝜔-convex

via the directional derivatives (Proposition 2.4.3) and the above the tangent line property (Proposition

2.3.10).

Proposition 2.4.11 (𝜔-convexity of V𝜖 , W𝜖 ). Let 𝑊, 𝑉 satisfy Assumptions 2.1.3 and 2.1.4,
respectively. Then V𝜖 , W𝜖 are 𝜔-convex along generalized geodesics, with 𝜔(𝑥) = √︁𝑥𝜓(𝑥),

𝜆𝜔 = 4𝐶.

The next propositions follow from [1] and [14].

Proposition 2.4.12 (Convexity properties of E). If Assumption 2.1.2 holds, then E +VΩ convex along

generalized geodesics. If Assumption 2.1.5 holds, then V𝑘 is convex along generalize geodesics.

Proposition 2.4.13 (Semi-convexity of E𝜖 , [14] Proposition 3.6). Suppose Assumptions 2.1.1, 2.1.2

hold. For all 𝜖 > 0, E𝜖 is 𝜆𝜖 -convex along generalized geodesics, where,

𝜆𝜖 = −𝜖 −𝑑−2∥1/𝑎∥ 𝐿∞ ∥𝐷2𝜁 ∥ 𝐿∞.

32

Proof. Let 𝜇𝛼 be a generalized geodesic with base 𝜇1 ∈ P2(R𝑑) connect 𝜇2, 𝜇3 ∈ P2(R𝑑). As the
mapping 𝑥 ↦→ 𝑥2 is convex, then the tangent inequality gives us,

E𝜖 (𝜇3) − E𝜖 (𝜇2) =

≥

=

∫ (𝜁𝜖 ∗ 𝜇3(𝑥))2

1
𝑎(𝑥)
2
∫ 𝜁𝜖 ∗ 𝜇2(𝑥)

𝑎(𝑥)

∫ 𝜁𝜖 ∗ 𝜇2(𝑥)

𝑎(𝑥)

𝑑𝑥 −

1
2

∫ (𝜁𝜖 ∗ 𝜇2(𝑥))2

𝑎(𝑥)

𝑑𝑥

(𝜁𝜖 ∗ 𝜇3(𝑥) − 𝜁𝜖 ∗ 𝜇2(𝑥)) 𝑑𝑥
∫

𝜁𝜖 (𝑥 − 𝑦3) − 𝜁𝜖 (𝑥 − 𝑦2) 𝑑𝛾(𝑦1, 𝑦2, 𝑦3) 𝑑𝑥.

By Proposition 2.4.5 and Taylor’s theorem,

𝑑
𝑑𝛼

E𝜖 (𝜇𝛼)|𝛼=0

𝜁𝜖 (𝑥 − 𝑦3) − 𝜁𝜖 (𝑥 − 𝑦2) − ⟨∇𝜁𝜖 (𝑥 − 𝑦2), 𝑦3 − 𝑦2⟩ 𝑑𝛾(𝑦1, 𝑦2, 𝑦3) 𝑑𝑥

E𝜖 (𝜇3) − E𝜖 (𝜇2) −
∫
∫ 𝜁𝜖 ∗ 𝜇2(𝑥)

≥

𝑎(𝑥)

∥𝐷2𝜁𝜖 ∥ 𝐿∞

≥ −

≥ −

1
2
1
2

∫ 𝜁𝜖 ∗ 𝜇2(𝑥)

∫

𝑎(𝑥)

|𝑦2 − 𝑦3|2 𝑑𝛾(𝑦1, 𝑦2, 𝑦3) 𝑑𝑥

∥1/𝑎∥ 𝐿∞ ∥𝐷2𝜁𝜖 ∥ 𝐿∞𝑊 2

2,𝛾 (𝜇2, 𝜇3).

The result follows from the above the tangent line property, Proposition 2.3.10.

With the previous results above for the lower semi-continuity and convexity, the gradient flows

of the energy functionals are well-posed. Now we move on to derivatives of the energies to obtain

their subdifferentials.

2.4.4 Subdifferential of Energies

We use the directional derivatives to characterize the minimal elements of the subdifferentials of

the energies.

Proposition 2.4.14 (Subdifferential of V𝜖 , W𝜖 ). Suppose Assumption 2.1.3 holds. If 𝜇 ∈ 𝐷 (W𝜖 ),

then ∇(𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊) ∗ 𝜇 ∈ 𝜕W𝜖 (𝜇). If 𝜇 ∈ 𝐷 (W), then ∇𝑊 ∗ 𝜇 ∈ 𝜕W (𝜇). Suppose Assumption

2.1.4 holds. If 𝜇 ∈ 𝐷 (V𝜖 ), then ∇(𝜁𝜖 ∗ 𝑉) ∈ 𝜕V𝜖 (𝜇). If 𝜇 ∈ 𝐷 (V), then ∇𝑉 ∈ 𝜕V (𝜇). Suppose

Assumption 2.1.5 holds. If 𝜇 ∈ 𝐷 (V𝑘 ), then ∇𝑉𝑘 ∈ 𝜕V𝑘 (𝜇).

Proof. As all cases are similar, we will elect to only write down the case for W𝜖 . Fix 𝜖 > 0, 𝜇0, 𝜇1 ∈
P2(R𝑑) satisfying ∥𝜇𝑖 ∥ 𝐿2 ≤ 𝐶 for 𝑖 = 1, 2 , and 𝛾 ∈ Γ(𝜇0, 𝜇1) such that 𝜇𝛼 = ((1 − 𝛼)𝜋1 + 𝛼𝜋2)#𝛾

33

satisfying ∥𝜇𝛼 ∥ ≤ 𝐶 for all 𝛼 ∈ [0, 1]. Given that W𝜖 is 𝜔-convex, it satisfies the above the tangent

line property,

W𝜖 (𝜇1) − W𝜖 (𝜇0) −

𝑑
𝑑𝛼

W𝜖 (𝜇𝛼)|𝛼=0 ≥

𝜆𝜔
2

𝜔(𝑊 2

2 (𝜇0, 𝜇1)).

As we calculated the directional derivatives (Proposition 2.4.3),

W𝜖 (𝜇1) − W𝜖 (𝜇0) ≥

∫

⟨∇(𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊) ∗ 𝜇0(𝑦0), 𝑦1 − 𝑦0⟩ 𝑑𝛾 +

𝜆𝜔
2

𝜔(𝑊 2

2 (𝜇0, 𝜇1)).

The results follow from the the characterization of the subdifferential of 𝜔-convex energies

(Proposition 2.3.19).

Proposition 2.4.15 (Subdifferential of E𝜖 , [14] Proposition 3.7 (i)). Suppose Assumptions 2.1.1 and

2.1.2 hold. For all 𝜖 > 0, and 𝜇 ∈ 𝐷 (E𝜖 ), we have

𝛿E𝜖
𝛿𝜇

∇

∈ 𝜕E𝜖 (𝜇), where

𝛿E𝜖
𝛿𝜇

= 𝜁𝜖 ∗

𝜁𝜖 ∗ 𝜇
𝑎

.

Proof. Fix 𝜇, 𝜈 ∈ P2(R𝑑) and 𝛾 ∈ Γ0(𝜇, 𝜈). Let 𝜇𝛼 = ((1 − 𝛼)𝜋1 + 𝛼𝜋2)#𝛾 be a geodesic from 𝜇
to 𝜈. By Proposition 2.4.13, E𝜖 is semi-convex along 𝜇𝛼 and by Proposition 2.3.10,

E𝜖 (𝜈) − E𝜖 (𝜇) −

𝑑
𝑑𝛼

E𝜖 (𝜇𝛼)|𝛼=0 ≥

𝜆𝜖
2

𝑊 2

2 (𝜇, 𝜈).

Applying Proposition 2.4.5 with ˜𝛾 = (𝜋1, 𝜋1, 𝜋2)#𝛾,

E𝜖 (𝜈) − E𝜖 (𝜇) ≥

=

=

=

∫ (cid:28)

∫ (cid:28)

𝑎(𝑥)

∇𝜁𝜖 ∗

𝛿E𝜖
𝛿𝜇

∇

∫ 𝜁𝜖 ∗ 𝜇(𝑥)

∫

𝑎(𝑥)

∫ 𝜁𝜖 ∗ 𝜇(𝑥)

∫

⟨∇𝜁𝜖 (𝑥 − 𝑦2), 𝑦3 − 𝑦2⟩ 𝑑 ˜𝛾(𝑦1, 𝑦2, 𝑦3) 𝑑𝑥 +
𝜆𝜖
2

⟨∇𝜁𝜖 (𝑥 − 𝑦1), 𝑦1 − 𝑦2⟩ 𝑑𝛾(𝑦1, 𝑦2) 𝑑𝑥 +

2 (𝜇, 𝜈)

𝑊 2

𝜆𝜖
2
2 (𝜇, 𝜈)

𝑊 2

(cid:18) 𝜁𝜖 ∗ 𝜇
𝑎

(cid:19)

(𝑦1)

, 𝑦2 − 𝑦1
(cid:29)

(𝑦1), 𝑦2 − 𝑦1

𝑑𝛾(𝑦1, 𝑦2) 𝑑𝑥 +

(cid:29)

𝑑𝛾(𝑦1, 𝑦2) 𝑑𝑥 +

𝜆𝜖
2

𝑊 2

2 (𝜇, 𝜈)

𝜆𝜖
2

𝑊 2

2 (𝜇, 𝜈)

This completes the proof.

Remark 2.4.16 (Minimal Selection and Chain rule). As long as the energy functional is regular,

then by [1, Lemma 10.1.5] the local slope is equivalent to the 𝐿2 norm of the minimal element of

the subdifferential. Therefore, 𝜔-convex energies satisfy [1, Lemma 10.1.5]. Similarly, 𝜔-convex

energies satisfy the chain rule (E of section 10.1.2 in [1]).

34

We mention to minimality of the regularized subdifferential as in [14, Proposition 3.8].

Proposition 2.4.17 (Minimal subdifferential of F𝜖,𝑘 ). Suppose Assumptions 2.1.3 and 2.1.4 hold.

For 𝜖 > 0 and 𝑘 ∈ N, 𝜇 ∈ 𝐷F𝜖,𝑘 ,

𝜕◦F𝜖,𝑘 (𝜇) = ∇

𝛿F𝜖,𝑘
𝛿𝜇

= ∇𝜁𝜖 ∗

𝜁𝜖 ∗ 𝜇
𝑎

+ ∇𝜁𝜖 ∗ 𝑉 + ∇𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊 ∗ 𝜇 + ∇𝑉𝑘 .

Proof. For ease, let

𝜉 = ∇𝜁𝜖 ∗

𝜁𝜖 ∗ 𝜇
𝑎

+ ∇𝜁𝜖 ∗ 𝑉 + ∇𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊 ∗ 𝜇 + ∇𝑉𝑘 .

By Propositions 2.4.14, 2.4.15 and additivity of the subdifferential, we have 𝜉 ∈ 𝜕F𝜖,𝑘 (𝜇). By [1,
Lemma 10.1.5], it suffices to show that ∥𝜉 | 𝐿2 (𝜇) ≤ |𝜕F𝜖,𝑘 |(𝜇). Fix 𝜓 ∈ 𝐶1(R𝑑) so that ∇𝜓 ∈ 𝐿2(𝜇).
Define 𝜇𝛼 = (𝑖𝑑 + 𝛼∇𝜓)#𝜇, where 𝑖𝑑 is the identity mapping. By definition of the 2-Wasserstein

distance,

𝑊2(𝜇𝛼, 𝜇) ≤ ∥(𝑖𝑑 + 𝛼∇𝜓) − 𝑖𝑑 ∥ 𝐿2 (𝜇) = 𝛼∥∇𝜓∥ 𝐿2 (𝜇).

By definition of local slope,

|𝜕F𝜖,𝑘 |(𝜇) ≥ lim sup

(F𝜖,𝑘 (𝜇) − F𝜖,𝑘 (𝜇𝛼))+
𝑊2(𝜇, 𝜇𝛼)

𝛼→0
1
∥∇𝜓∥ 𝐿2 (𝜇)

≥

(F𝜖,𝑘 (𝜇) − F𝜖,𝑘 (𝜇𝛼))+
𝛼

.

lim sup
𝛼→0

Choosing ∇𝜓 = −𝜉 and applying the directional derivatives Propositions 2.4.3, 2.4.5, we get that

|𝜕𝜖,𝑘 F𝜖,𝑘 |(𝜇)∥∇𝜓∥ 𝐿2 (𝜇) ≥ ∥∇𝜓∥2

𝐿2 (𝜇). Division by ∥∇𝜓∥2

𝐿2 (𝜇)

= ∥𝜉 ∥2

𝐿2 (𝜇) gives us the result.

The minimal subdifferential of F is standard as seen in [1, Theorems 10.4.9-10.4.13].

Proposition 2.4.18 (Minimal subdifferential of F ). Suppose Assumptions 2.1.2, 2.1.3, 2.1.4, 2.1.5
hold. Given 𝜇 ∈ 𝐷 (F ), we have |𝜕F |(𝜇) < ∞ if and only if (𝜇/𝑎)2 ∈ 𝑊 1,1

𝑙𝑜𝑐 (Ω) and there exists

𝜉 ∈ 𝐿2(𝜇) so that,

2
on Ω. In particular, 𝜉 is the minimal selection of 𝜕F . That is, 𝜉 = 𝜕𝑜F (𝜇).

𝜉 𝜇 =

(cid:17) 2

𝑎

∇

(cid:16) 𝜇
𝑎

+ 𝜇∇𝑉 + 𝜇(∇𝑊 ∗ 𝜇)

35

Proposition 2.4.19 (Long time behavior, [1] Corollary 4.0.6). Suppose Assumption 2.1.2 holds,

𝑉 = 𝑊 = 0, and Ω bounded. Let 𝜇0 ∈ 𝐷 (F ) and let 𝜇(𝑡) be the gradient flow of 𝜇 of F with initial

data 𝜇0. Then we have,

2.5 An 𝐻1-type Bound

(cid:32)

𝑊2

lim
𝑡→∞

𝜇(𝑡),

(cid:33)

𝑎1

Ω
𝑎 𝑑L𝑑

∫
Ω

= 0.

A key tool in the convergence of the gradient flows proof Proposition 2.6.6 is the 𝐻1 bound of

𝜁𝜖 ∗ 𝜇𝜖 by being an important hypothesis in the Γ-convergence (or lower semi-continuity) of the local

slopes, Proposition 2.6.4. In pursuit of the 𝐻1 bound of 𝜁𝜖 ∗ 𝜇𝜖 , we first start with the 𝐿2 bound.

Lemma 2.5.1 (𝐿2 bound of convolved gradient flow). Suppose Assumptions 2.1.3, 2.1.4, 2.1.7 hold.
For all 𝑇 > 0 and 𝜖 > 0, suppose that 𝜇𝜖 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) is a gradient flow of F𝜖,𝑘 . Then

∥𝜁𝜖 ∗ 𝜇𝜖 ∥2

𝐿2 (R𝑑) ≤ 2∥𝑎∥ 𝐿∞ (R𝑑)

(cid:0)F𝜖,𝑘 (𝜇𝜖 (0)) + 2𝐶(cid:1) .

Let 𝜇𝜏,𝜖 be the piecewise constant interpolation in the minimizing movement scheme of F𝜖 . Then we

get the same bound,

∥𝜁𝜖 ∗ 𝜇𝜏,𝜖 ∥2

𝐿2 (R𝑑) ≤ 2∥𝑎∥ 𝐿∞ (R𝑑)

(cid:0)F𝜖,𝑘 (𝜇𝜖 (0)) + 2𝐶(cid:1) .

Proof. For all 𝑇 > 0 and 𝜖 > 0, suppose that 𝜇𝜖 ∈ 𝐴𝐶2( [0, 𝑇]; P (R𝑑)) is a gradient flow of F𝜖,𝑘 .

Recall that

By definition,

E𝜖 (𝜇𝜖 ) + W𝜖 (𝜇𝜖 ) + V𝜖 (𝜇𝜖 ) ≤ F𝜖,𝑘 (𝜇𝜖 ) ≤ F𝜖,𝑘 (𝜇𝜖 (0)).

and by Assumption 2.1.3,

E𝜖 (𝜇𝜖 ) ≥

1
2∥𝑎∥ 𝐿∞ (R𝑑)

∥𝜁𝜖 ∗ 𝜇𝜖 ∥2

𝐿2 (R𝑑)

,

V𝜖 (𝜇𝜖 ) ≥ −𝐶.

As W𝜖 (𝜇𝜖 ) = W (𝜁𝜖 ∗ 𝜇𝜖 ) and W is lower semi-continuous, then we have

lim inf
𝜖→0

W𝜖 (𝜇𝜖 ) ≥ W (𝜇).

36

Moreover, sup𝜖 >0 W𝜖 (𝜇𝜖 ) ≥ W (𝜇). With Assumption 2.1.3,
1
2∥𝑎∥ 𝐿∞ (R𝑑)

∥𝜁𝜖 ∗ 𝜇𝜖 ∥2

𝐿2 (R𝑑) − 2𝐶 ≤ F𝜖,𝑘 (𝜇𝜖 (0)).

Let 𝜇𝜏,𝜖 be the piecewise constant interpolation in the minimizing movement scheme of F𝜖,𝑘 . We

get similar bounds for every term except for the interaction term. However, we can use the fact that

W𝜖 is lower semi-continuous and Assumption 2.1.3 so that

sup
𝜏>0

W𝜖 (𝜇𝜏,𝜖 ) ≥ lim inf
𝜏→0

W𝜖 (𝜇𝜏,𝜖 ) ≥ W𝜖 (𝜇𝜖 ) = W (𝜁𝜖 ∗ 𝜇𝜖 ) ≥ −𝐶.

Moreover, we get the result as stated.

2.5.1 𝐻1 bound on convolved gradient flow (formal/heuristic)

We first give a formal or heuristic argument of the 𝐻1 bound. After the formal argument, we

state some necessary definitions and lemmas to make the rigorous argument of the 𝐻1 bound.

Proposition 2.5.2 (𝐻1 bound on 𝜁𝜖 ∗ 𝜇𝜖 ). Let 𝑉, 𝑊 satisfy Assumptions 2.1.3, 2.1.4, and Assumption
2.1.7. For all 𝑇 > 0 and 𝜖 > 0, suppose that 𝜇𝜖 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) is a gradient flow of F𝜖,𝑘 .

Then,

∫ 𝑇

0

∥∇𝜁𝜖 ∗ 𝜇𝜖 (𝑠)∥2

𝐿2 (R𝑑)

𝑑𝑠 ≤ 𝐶 (cid:0)S(𝜇𝜖 (0)) + 𝑀2(𝜇𝜖 (0)) + F𝜖,𝑘 (𝜇𝜖 (0)) + 1(cid:1)

where 𝐶 = 𝐶 (𝑎, 𝑇, 𝑉, 𝑉𝑘 , 𝑊).

Proof. We start by differentiating formally along gradient flow of 𝜇𝜖

(cid:18)∫

R𝑑

𝜇𝜖 log 𝜇𝜖 𝑑𝑥 −

(cid:19)

𝜇𝜖 𝑑𝑥

∫

R𝑑

𝑑
𝑑𝑡

(cid:18)∫

R𝑑

(cid:19)

𝜇𝜖 log 𝜇𝜖 𝑑𝑥

=

=

=

𝑑
𝑑𝑡
∫

R𝑑

∫

R𝑑
∫

𝜕𝑡 𝜇𝜖 log 𝜇𝜖 𝑑𝑥

(cid:18)

𝜇𝜖 ∇

∇ ·

(cid:19)

𝛿F𝜖
𝛿𝜇𝜖
𝛿F𝜖
𝛿𝜇𝜖

= −

∇𝜇𝜖 · ∇

R𝑑

=

∫

R𝑑

−∇𝜇𝜖 · ∇𝜁𝜖 ∗

− ∇𝜇𝜖 ∇𝑉𝑘 𝑑𝑥

=: 𝐼 + 𝐼𝑉 + 𝐼𝑊 + 𝐼𝑉𝑘

log 𝜇𝜖 𝑑𝑥

− ∇𝜇𝜖 · ∇𝜁𝜖 ∗ 𝑉 − ∇𝜇𝜖 · ∇𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊 ∗ 𝜇𝜖

𝑑𝑥

(cid:19)

(cid:18) 𝜁𝜖 ∗ 𝜇𝜖
𝑎

37

where we use that 𝜇𝜖 formally satisfies the continuity equation and integration by parts. Moving the

gradient off 𝜁𝜖 and using the fact that 𝜁𝜖 is even,

(𝜁𝜖 ∗ ∇𝜇𝜖 )∇

(cid:19)

𝑑𝑥

∫

R𝑑

∫

𝐼 = −

= −

(∇𝜁𝜖 ∗ 𝜇𝜖 )

R𝑑
−(∇𝜁𝜖 ∗ 𝜇𝜖 )2
𝑎
−(∇𝜁𝜖 ∗ 𝜇𝜖 )2
𝑎

=

≤

∫

R𝑑
∫

R𝑑

(cid:18) 𝜁𝜖 ∗ 𝜇𝜖
𝑎
(cid:18) ∇𝜁𝜖 ∗ 𝜇𝜖
𝑎

∫

𝑑𝑥 +

𝑑𝑥 + 𝛿

−

(𝜁𝜖 ∗ 𝜇𝜖 )

∇𝑎
𝑎2
∇𝜁𝜖 ∗ 𝜇𝜖
𝑎1/2
(∇𝜁𝜖 ∗ 𝜇𝜖 )2
𝑎

R𝑑

R𝑑
∫

(cid:19)

𝑑𝑥

∇𝑎
𝑎3/2
∫

𝑑𝑥 +

(𝜁𝜖 ∗ 𝜇𝜖 )

≤ (𝛿 − 1)

∫

R𝑑

(∇𝜁𝜖 ∗ 𝜇𝜖 )2
𝑎

𝑑𝑥 +

1
4𝛿

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

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

∇𝑎
𝑎3/2

2(cid:13)
(cid:12)
(cid:13)
(cid:12)
(cid:13)
(cid:12)
(cid:13)
(cid:12)
(cid:13)𝐿∞ (R𝑑)

𝑑𝑥

(𝜁𝜖 ∗ 𝜇𝜖 )2
4𝛿

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

∇𝑎
𝑎3/2

(cid:12)
2
(cid:12)
(cid:12)
(cid:12)

𝑑𝑥

R𝑑

∫

R𝑑

sup
𝜖 >0

|𝜁𝜖 ∗ 𝜇𝜖 (0)|2 𝑑𝑥

where Cauchy’s inequality with 𝛿 > 0 is used at the first inequality. For 𝐼𝑉𝑘 ,

𝐼𝑉𝑘 =

∫

R𝑑

𝜇𝜖 Δ𝑉𝑘 𝑑𝑥

≤ ∥𝐷2𝑉𝑘 ∥ 𝐿∞.

Now we bound 𝐼𝑉 , 𝐼𝑊 using item 1 of Assumption 2.1.7. Moving the gradient on 𝜇𝜖 to 𝑉 by

integration by parts and again using Cauchy’s inequality with a 𝛿,

𝐼𝑉 =

(∇𝜁𝜖 ∗ ∇𝑉)𝜇𝜖 𝑑𝑥

∫

R𝑑
∫

(∇𝜁𝜖 ∗ 𝜇𝜖 )∇𝑉 𝑑𝑥

|∇𝑉 |𝑎1/2 𝑑𝑥

= −

∫

≤

≤ 𝛿

≤ 𝛿

≤ 𝛿

R𝑑
|∇𝜁𝜖 ∗ 𝜇𝜖 |
𝑎1/2
|∇𝜁𝜖 ∗ 𝜇𝜖 |2
𝑎
|∇𝜁𝜖 ∗ 𝜇𝜖 |2
𝑎
|∇𝜁𝜖 ∗ 𝜇𝜖 |2
𝑎

R𝑑

R𝑑

R𝑑

R𝑑
∫

∫

∫

|∇𝑉 |2𝑎 𝑑𝑥

∫

∫

1
4𝛿
R𝑑
∥𝑎∥ 𝐿∞ (R𝑑)
4𝛿
𝐶 ∥𝑎∥ 𝐿∞ (R𝑑)
4𝛿

R𝑑

.

|∇𝑉 |2 𝑑𝑥

𝑑𝑥 +

𝑑𝑥 +

𝑑𝑥 +

38

Using similar techniques with the intent of using item 1 of Assumption 2.1.7,

𝐼𝑊 =

(∇𝜁𝜖 ∗ 𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜖 )𝜇𝜖 𝑑𝑥

∫

R𝑑
∫

= −

= −

≤ 𝛿

≤ 𝛿

R𝑑

∫

R𝑑

∫

R𝑑

∫

R𝑑

(∇𝜁𝜖 ∗ 𝜇𝜖 ) (𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜖 ) 𝑑𝑥

(∇𝜁𝜖 ∗ 𝜇𝜖 )
𝑎1/2
(∇𝜁𝜖 ∗ 𝜇𝜖 )2
𝑎
(∇𝜁𝜖 ∗ 𝜇𝜖 )2
𝑎

𝑎1/2(𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜖 ) 𝑑𝑥

(𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜖 )2
4𝛿

𝑎 𝑑𝑥

∫

𝑑𝑥 +

𝑑𝑥 +

R𝑑
𝐶 ∥𝑎∥ 𝐿∞ (R𝑑)
4𝛿

.

Thus combining the results above with 𝛿 = 1/6 gives

𝑑
𝑑𝑡

(cid:18)∫

R𝑑

𝜇𝜖 log 𝜇𝜖 𝑑𝑥

(cid:19)

≤ −

∫

R𝑑

|∇𝜁𝜖 ∗ 𝜇𝜖 |2
2𝑎

𝑑𝑥 + 𝐶 (1 + F𝜖 (𝜇𝜖 (0)))

where 𝐶 is the combination of constants. For ease we define

S(𝜇𝜖 (𝑡)) =

∫

R𝑑

𝜇𝜖 log 𝜇𝜖 𝑑𝑥.

Integrating in time for 𝑡 ∈ [0, 𝑇] for 𝑇 > 0

S(𝜇𝜖 (𝑡)) − S(𝜇𝜖 (0)) ≤ −

∫ 𝑡

∫

0

R𝑑

(∇𝜁𝜖 ∗ 𝜇𝜖 )2
2𝑎

𝑑𝑥 𝑑𝑠 + 𝑡𝐶 (1 + F𝜖 (𝜇𝜖 (0)))

Given that we have the bound (from [9, Proposition 3.8])

S(𝜈) ≥ −(2𝜋)1/2 − 𝑀2(𝜈),

rearranging gives us

(2.8)

∫ 𝑡

∫

0

R𝑑

(∇𝜁𝜖 ∗ 𝜇𝜖 )2
2𝑎

𝑑𝑥 𝑑𝑠 ≤ 𝑀2(𝜇𝜖 (𝑡)) + (2𝜋)1/2 + S(𝜇𝜖 (0)) + 𝑡𝐶 (1 + F𝜖 (𝜇𝜖 (0)))

We will briefly pause here and see what happens when we apply Assumption 2.1.7 item 2 to 𝐼𝑉 , 𝐼𝑊 .

Alternatively, moving the gradient from the mollifier to ∇𝑉,

𝐼𝑉 =

∫

R𝑑

(𝜁𝜖 ∗ 𝜇𝜖 )Δ𝑉 𝑑𝑥

≤ ∥𝜁𝜖 ∗ 𝜇𝜖 ∥ 𝐿2 (𝑅𝑑) ∥𝐷2𝑉 ∥ 𝐿2 (𝑅𝑑)

≤ 𝐶.

39

In a similar manner,

𝐼𝑊 =

=

∫

R𝑑

∫

R𝑑

(𝜁𝜖 ∗ 𝜁𝜖 ∗ Δ𝑊 ∗ 𝜇𝜖 )𝜇𝜖 𝑑𝑥

(𝜁𝜖 ∗ 𝜇𝜖 ) (𝜁𝜖 ∗ Δ𝑊 ∗ 𝜇𝜖 ) 𝑑𝑥

≤ ∥𝜁𝜖 ∗ 𝜇𝜖 ∥ 𝐿2 (R𝑑) ∥𝜁𝜖 ∗ 𝐷2𝑊 ∗ 𝜇𝜖 ∥ 𝐿2 (R𝑑)

≤ 𝐶.

We instead choose 𝛿 = 1/2 , we get (2.8) with a different constant. What is left to show is that the

second moment at 𝜇𝜖 (𝑡) is uniformly bounded. We get this from Proposition 2.5.9. Therefore for

any 𝑡 ∈ [0, 𝑇], we have an bound

∫ 𝑡

∫

0

R𝑑

(∇𝜁𝜖 ∗ 𝜇𝜖 )2
2𝑎

𝑑𝑥 𝑑𝑠 ≤ (1 + 𝑇 𝑒𝑇 ) (𝑀2(𝜇𝜖 (0)) + F𝜖 (𝜇𝜖 (0)))

+ (2𝜋)1/2 + S(𝜇𝜖 (0)) + 𝑡𝐶 (1 + F𝜖 (𝜇𝜖 (0))) .

We can take 𝑡 as 𝑇 and take the sup in 𝜖 to get a uniform bound in 𝜖.

2.5.2 𝐻1 bound on convolved gradient flow (rigorous)

Remark 2.5.3 (flow interchange method). What makes the argument formal is the differentiation of

the entropy, S, along the gradient flow of the diffusion energy F𝜖 . Say that we have two energy

functionals 𝐸1, 𝐸2 with gradient flows 𝜇1, 𝜇2 respectively. The flow interchange method says that

the derivative with respect to time of 𝐸1 along 𝜇2 at 𝑡 = 0 is equivalent to the derivative with respect

to time of 𝐸2 along 𝜇1 at 𝑡 = 0 as long as the gradient flow are equivalent at 𝑡 = 0. We use the flow

interchange method in discrete time via minimizing movement scheme (see [14, Definition A.1]) to

make the argument rigorous by avoiding differentiating S(𝜇𝜖 ) in time.

Definition 2.5.4 (Minimizing movement scheme). Suppose that G is proper, lower semi-continuous,

and 𝜔-convex along generalized geodesics. Define the proximal operator 𝐽𝜏 by

𝐽𝜏 𝜇 = argmin𝜈∈𝑃2 (R𝑑)

1
2𝜏

𝑊 2

2 (𝜇, 𝜈) + G(𝜈),

40

and define the minimizing movement scheme 𝐽𝑛

𝜏 𝜇 by

𝐽𝑛
𝜏 𝜇 = 𝐽𝜏 ◦ 𝐽𝜏 ◦ · · · ◦ 𝐽𝜏
(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)
(cid:32)(cid:32)(cid:32)(cid: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)
𝑛 times

𝜇.

Remark 2.5.5 (Minimizing movement scheme). Given 𝜇 ∈ P2(R𝑑), let 𝐽𝑛
the minimizing movement scheme of F𝜖,𝑘 with time step 𝜏 and initial data 𝐽0

𝜏,𝜖 𝜇 = 𝜇.

𝜏,𝜖 𝜇 denote the 𝑛th step of

Definition 2.5.6 (Heat flow semigroup). Given 𝜇 ∈ P2(R𝑑) and ℎ ≥ 0, we will let 𝑆ℎ 𝜇 denote the
strongly continuous gradient flow of S with the initial data 𝜇 at time ℎ. Moreover, 𝑆ℎ is the heat

flow semigroup operator.

Note that in the following proofs, we use that for 𝜇 ∈ P2(R𝑑), 𝜁𝜖 ∗ 𝑆ℎ (𝜇) = 𝑆ℎ (𝜁𝜖 ∗ 𝜇) as in [14].

Lemma 2.5.7 (derivatives along the heat semigroup). Suppose Assumptions 2.1.3, 2.1.4, 2.1.5,
2.1.7 hold. Let 𝜇 ∈ P2(R𝑑). We have,

=

∫

=

𝜏,𝜖 𝜇))

W𝜖 (𝐽𝑛

𝜏,𝜖 𝜇) − W𝜖 (𝑆ℎ (𝐽𝑛
ℎ
𝜏,𝜖 𝜇) − V𝜖 (𝑆ℎ (𝐽𝑛
ℎ
𝜏,𝜖 𝜇) − V𝑘 (𝑆ℎ (𝐽𝑛
ℎ
𝜏,𝜖 𝜇) − E𝜖 (𝑆ℎ (𝐽𝑛
ℎ

V𝜖 (𝐽𝑛

V𝑘 (𝐽𝑛

E𝜖 (𝐽𝑛

𝜏,𝜖 𝜇))

𝜏,𝜖 𝜇))

𝜏,𝜖 𝜇))

lim sup
ℎ→0+

lim sup
ℎ→0+

lim sup
ℎ→0+

lim sup
ℎ→0+

∫

R𝑑

⟨𝜁𝜖 ∗ ∇𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇, ∇𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇⟩ 𝑑L𝑑,

⟨∇𝑉, ∇𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇⟩ 𝑑L𝑑,

R𝑑
∫

R𝑑

= −

Δ𝑉𝑘 𝑑𝐽𝑛

𝜏,𝜖 𝜇,

(Δ𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) 𝑑L𝑑.

= −

∫

R𝑑

1
𝑎

41

Proof. Using the fact that 𝑊, 𝜁𝜖 are both even functions combined with the definition of convolution,

W𝜖 (𝐽𝑛

𝜏,𝜖 𝜇) − W𝜖 (𝑆ℎ (𝐽𝑛
ℎ

𝜏,𝜖 𝜇))

=

=

=

=

∫

∫

1
2ℎ
∫

1
2ℎ
1
2ℎ
∫

1
2ℎ
1
2ℎ

−

1
2ℎ

+

−

1
2ℎ

+

1
2ℎ

𝜁𝜖 ∗ 𝑊 ∗ 𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇 𝑑𝐽𝑛

𝜁𝜖 ∗ 𝑊 ∗ 𝜁𝜖 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇) 𝑑𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇)

∫

𝜏,𝜖 𝜇 −

1
2ℎ
𝜏,𝜖 𝜇) 𝑑L𝑑

(𝜁𝜖 ∗ 𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)(𝜁𝜖 ∗ 𝐽𝑛

∫

(𝜁𝜖 ∗ 𝑊 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇)) (𝜁𝜖 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇)) 𝑑L𝑑

(𝜁𝜖 ∗ 𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)(𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) − (𝜁𝜖 ∗ 𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) (𝜁𝜖 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇)) 𝑑L𝑑

∫

∫

(𝜁𝜖 ∗ 𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)(𝜁𝜖 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇))L𝑑

(𝜁𝜖 ∗ 𝑊 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇)) (𝜁𝜖 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇)) 𝑑L𝑑

(𝜁𝜖 ∗ 𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) (cid:0)(𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) − (𝜁𝜖 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇))(cid:1) 𝑑L𝑑

∫

(𝜁𝜖 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇)) (cid:0)(𝜁𝜖 ∗ 𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) − (𝜁𝜖 ∗ 𝑊 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇))(cid:1) 𝑑L𝑑

=: 𝐽1 + 𝐽2

where we add and subtract a term in the third equality. As 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇) satisfies the heat equation

classically, then by the fundamental theorem of calculus

∫

𝐽1 = −

(𝜁𝜖 ∗ 𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)

1
2ℎ

∫ ℎ

0

Δ𝑆𝑡 (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) 𝑑𝑡 𝑑L𝑑.

To get 𝐽2 to look similar to 𝐽1, we use that both 𝜁𝜖 ∗ 𝑊, 𝜁𝜖 are even functions,

𝐽2 =

=

=

∫

∫

∫

1
2ℎ
1
2ℎ
1
2ℎ
∫

(𝜁𝜖 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇)) (cid:0)𝜁𝜖 ∗ 𝑊 ∗ (𝐽𝑛

𝜏,𝜖 𝜇 − 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇))(cid:1) 𝑑L𝑑

(𝜁𝜖 ∗ 𝑊 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇)) (cid:0)𝜁𝜖 ∗ (𝐽𝑛

𝜏,𝜖 𝜇 − 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇))(cid:1) 𝑑L𝑑

(𝜁𝜖 ∗ 𝑊 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇)) (cid:0)(𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇 − 𝑆ℎ (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇))(cid:1) 𝑑L𝑑

= −

(𝜁𝜖 ∗ 𝑊 ∗ 𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇))

1
2ℎ

∫ ℎ

0

Δ𝑆𝑡 (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) 𝑑𝑡 𝑑L𝑑.

From here we want to say that either item 1 or item 2 of Assumption 2.1.7 is enough to justify

passing the limit in ℎ. Let us recall that 𝑆𝑡 (𝜇) = 𝐾𝑡 ∗ 𝜇 where 𝐾𝑡 is the heat kernel. Therefore by

42

Young’s convolution inequality,

∥𝑆𝑡 (𝜇)∥ 𝐿 𝑝 (R𝑑) ≤ ∥𝐾𝑡 ∥ 𝐿1 (R𝑑) ∥𝜇∥ 𝐿 𝑝 (R𝑑) = ∥𝜇∥ 𝐿 𝑝 (𝑅𝑑) ≤ 𝐶.

Let us first start by using item 1. By integration by parts, Hölder’s inequality, and Jensen’s inequality

∫

𝐽1 ≤

(𝜁𝜖 ∗ ∇𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)

1
2ℎ

∫ ℎ

∇𝑆𝑡 (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) 𝑑𝑡 𝑑L𝑑

∥𝜁𝜖 ∗ ∇𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇∥ 𝐿2 (R𝑑)

∥𝜁𝜖 ∗ ∇𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇∥ 𝐿2 (R𝑑)

∥𝜁𝜖 ∗ ∇𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇∥ 𝐿2 (R𝑑)𝐶𝜏,𝑛,𝜖

∥𝜁𝜖 ∗ ∇𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇∥ 𝐿2 (R𝑑)𝐶𝜏,𝑛,𝜖

∫ ℎ

0
∫ ℎ

0
(cid:18)∫ 1
ℎ
(cid:18)∫ 1
ℎ
0
(cid:18)∫ 1
ℎ
(cid:18)∫ 1
ℎ

|∇𝑆𝑡 (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)|2 𝑑𝑡 𝑑L𝑑

(cid:19) 1/2

|Δ𝑆𝑡 (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)||𝑆𝑡 (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)| 𝑑𝑡 𝑑L𝑑

(cid:19) 1/2

(cid:19) 1/2

|𝑆𝑡 (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)| 𝑑𝑡 𝑑L𝑑

|𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇| 𝑑𝑡 𝑑L𝑑

(cid:19) 1/2

∫ ℎ

0
∫ ℎ

0

∥𝜁𝜖 ∗ ∇𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇∥ 𝐿2 (R𝑑)𝐶𝜏,𝑛,𝜖

(cid:18)∫

|𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇| 𝑑L𝑑

(cid:19) 1/2

≤

≤

≤

≤

≤

1
2

1
2

1
2

1
2

1
2

≤ 𝐶𝜏,𝑛,𝜖

where contractivity of the semigroup was used, ∥Δ𝑆𝑡 (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) ∥ 𝐿∞ (R𝑑) ≤ 𝐶𝜏,𝑛,𝜖 for all 𝑡, and item

1 of Assumption 2.1.7. The calculation for 𝐽2 is similar. Since we have a bound independent of ℎ,

we may pass the limit.

Now let us try to get a bound of again 𝐽1, 𝐽2 independent of ℎ by using item 2 of Assumption

2.1.7. We use integration by parts twice, Hölder’s inequality, and contractivity of the heat semigroup

so that

𝐽1 ≤

≤

≤

∫

1
2
1
2

|𝜁𝜖 ∗ Δ𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇|

1
2ℎ

∫ ℎ

0

|𝑆𝑡 (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)| 𝑑𝑡 𝑑L𝑑

∥𝜁𝜖 ∗ 𝐷2𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇∥ 𝐿2 (R𝑑) ∥𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇∥ 𝐿2 (R𝑑)

𝐶𝜏,𝑛,𝜖 ∥𝐽𝑛

𝜏,𝜖 𝜇∥ 𝐿2 (R𝑑) ∥𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇∥ 𝐿2 (R𝑑)

which is finite and independent of ℎ. The calculation for 𝐽2 is similar. Since we have a bound

43

independent of ℎ, we may pass the limit. Taking lim sup gives

W𝜖 (𝐽𝑛

𝜏,𝜖 𝜇) − W𝜖 (𝑆ℎ (𝐽𝑛
ℎ

𝜏,𝜖 𝜇))

lim sup
ℎ→0+

∫

(𝜁𝜖 ∗ 𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)Δ(𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) 𝑑𝑡 𝑑L𝑑

⟨𝜁𝜖 ∗ ∇𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇, ∇𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇⟩ 𝑑L𝑑.

= −

∫

=

We can justify getting the same result for V𝜖 by using the same techniques used for W𝜖 . The results

of V𝑘 , E𝜖 carry over from [14, Lemma 4.4].

The next lemma is one small piece in the 𝐻1 bound proof.

Lemma 2.5.8 (Mollified nth step of minimizing movement scheme, [14] Lemma 4.5). For

𝐽𝑛
𝜏,𝜖 ∈ 𝐷 (E𝜖 ),

−

∫

R𝑑

1
𝑎

Δ(𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)(𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) 𝑑L𝑑 ≥ 𝐶𝑎 ∥∇𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇∥2

𝐿2 (R𝑑) − 𝐶′

𝑎 ∥𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇∥2

𝐿2 (R𝑑)

.

We seek the control the second moment of 𝜇𝜖,𝑘 so that it is independent of 𝜖 by controlling the

second moment along a curve at any time by the second moment along the same curve at the initial

time.

Proposition 2.5.9 (𝑀2 bound for 𝐴𝐶2 curves, [14] Proposition A.3). Suppose
𝜇 ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)). Then for all 𝑡 ∈ [0, 𝑇],

𝑀2(𝜇(𝑡)) ≤ (1 + 𝑡𝑒𝑡)

(cid:18)

𝑀2(𝜇(0)) +

∫ 𝑇

0

|𝜇′|2(𝑟) 𝑑𝑟

(cid:19)

.

Proof. It suffices to show for any 𝜌 ∈ P2(R𝑑),

𝑊 2

2 (𝜇(𝑡), 𝜌) ≤ (1 + 𝑡𝑒𝑡)

(cid:18)
𝑊 2

2 (𝜇(0), 𝜌) +

∫ 𝑇

0

(cid:19)

|𝜇′|(𝑟) 𝑑𝑟

for all 𝑡 ∈ [0, 𝑇] by taking 𝜌 = 𝛿0. Define H (𝜇) = − 1
2

2 (𝜇, 𝜌). By [1, Proposition 9.3.12] H is
(−1)-convex and lower semi-continuous and by [1, Definition 1.2.1, Corollary 2.4.10] the local

𝑊 2

slope |𝜕H |(𝜇) is a strong upper gradient for H . Thus,

|H (𝜇(𝑡)) − H (𝜇(0)) ≤

|𝜕H |(𝜇(𝑠))|𝜇′|(𝑠) 𝑑𝑠.

∫ 𝑡

0

44

Using the definition of local slope and triangle inequality,

|𝜕H |(𝜇) = lim sup

𝜈→𝜇

𝑊 2

2 (𝜈, 𝜌) − 𝑊 2
2𝑊2(𝜇, 𝜈)

2 (𝜇, 𝜌)

= lim sup
𝜈→𝜇

(𝑊2(𝜈, 𝜌) − 𝑊2(𝜇, 𝜌)) (𝑊2(𝜈, 𝜌) + 𝑊2(𝜇, 𝜌))
2𝑊2(𝜇, 𝜈)

≤ lim sup
𝜈→𝜇

𝑊2(𝜈, 𝜇) (𝑊2(𝜈, 𝜌) + 𝑊2(𝜇, 𝜌))
2𝑊2(𝜇, 𝜈)

= 𝑊2(𝜇, 𝜌).

Combining the previous two estimates,

1
2

𝑊 2

2 (𝜇(𝑡), 𝜌) −

1
2

𝑊 2

2 (𝜇(0), 𝜌) ≤ |H (𝜇(𝑡)) − H (𝜇(0))|

≤

≤

∫ 𝑡

0
∫ 𝑡
1
2

0

𝑊2(𝜇(𝑠), 𝜌)|𝜇′|(𝑠) 𝑑𝑠

𝑊 2

2 (𝜇(𝑡), 𝜌) 𝑑𝑠 +

1
2

∫ 𝑇

0

|𝜇′|2(𝑠) 𝑑𝑠.

Applying Gröwall’s inequality gives us the result.

Proposition 2.5.10 (𝐻1 bound on convolved gradient flow). Let 𝑉, 𝑊 satisfy Assumptions 2.1.3,
2.1.4, and Assumption 2.1.7. For all 𝑇 > 0 and 𝜖 > 0, suppose that 𝜇𝜖 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) is a

gradient flow of F𝜖,𝑘 . Then,

∫ 𝑇

0

∥∇𝜁𝜖 ∗ 𝜇𝜖 (𝑠)∥2

𝐿2 (R𝑑)

𝑑𝑠 ≤ 𝐶 (cid:0)S(𝜇𝜖 (0)) + 𝑀2(𝜇𝜖 (0)) + F𝜖,𝑘 (𝜇𝜖 (0)) + 1(cid:1)

where 𝐶 = 𝐶 (𝑎, 𝑇, 𝑉, 𝑉𝑘 , 𝑊).

Proof. By definition of minimizing movement scheme for any 𝜇 ∈ 𝐷 (F𝜖,𝑘 ),

F𝜖,𝑘 (𝐽𝑛

𝜏,𝜖 𝜇) − F𝜖,𝑘 (𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇)) ≤

1
2ℎ

(cid:16)
𝑊 2

2 (𝑆ℎ (𝐽𝑛

𝜏,𝜖 𝜇), 𝐽𝑛−1

𝜏,𝜖 𝜇) − 𝑊 2

2 (𝐽𝑛

𝜏,𝜖 𝜇, 𝐽𝑛−1

𝜏,𝜖 𝜇)

(cid:17)

.

By the EVI condition in Theorem 2.3.13, we get

F𝜖,𝑘 (𝐽𝑛

𝜏,𝜖 𝜇) − F𝜖,𝑘 (𝑆ℎ (𝐽𝑛
ℎ

𝜏,𝜖 𝜇))

lim sup
ℎ→0+

≤

≤

45

𝜏,𝜖 𝜇), 𝐽𝑛−1

𝜏,𝜖 𝜇)|ℎ=0

𝑊 2

𝑑+
1
2 (𝑆ℎ (𝐽𝑛
𝑑ℎ
2𝜏
𝜏,𝜖 𝜇) − S(𝐽𝑛
S(𝐽𝑛−1
𝜏

𝜏,𝜖 𝜇)

.

Using the derivatives along the heat semigroup from Lemma 2.5.7,

lim sup
ℎ→0+
∫

=

R𝑑

F𝜖,𝑘 (𝐽𝑛

𝜏,𝜖 𝜇) − F𝜖,𝑘 (𝑆ℎ (𝐽𝑛
ℎ

𝜏,𝜖 𝜇))

⟨𝜁𝜖 ∗ ∇𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇, ∇𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇⟩ 𝑑L𝑑 +

−

∫

R𝑑

1
𝑎

(Δ𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇)(𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) 𝑑L𝑑.

∫

R𝑑

⟨∇𝑉, ∇𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇⟩ 𝑑L𝑑 −

∫

R𝑑

Δ𝑉𝑘 𝑑𝐽𝑛

𝜏,𝜖 𝜇

Bounding the difference quotient of the heat entropy from below,

S(𝐽0

𝜏,𝜖 𝜇) − S(𝐽𝑛
𝜏

𝜏,𝜖 𝜇)

𝑛
∑︁

𝑖=1
∫

R𝑑

=

≥

S(𝐽𝑖−1

𝜏,𝜖 𝜇) − S(𝐽𝑖
𝜏

𝜏,𝜖 𝜇)

⟨𝜁𝜖 ∗ ∇𝑊 ∗ 𝐽𝑛

𝜏,𝜖 𝜇⟩ 𝑑L𝑑 +

⟨∇𝑉, ∇𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇⟩ 𝑑L𝑑

𝜏,𝜖 𝜇, ∇𝜁𝜖 ∗ 𝐽𝑛
∫

1
𝑎

R𝑑

Δ𝑉𝑘 𝑑𝐽𝑛

𝜏,𝜖 𝜇 −

(Δ𝜁𝜖 ∗ 𝐽𝑛

−

∫

R𝑑

R𝑑
𝜏,𝜖 𝜇) (𝜁𝜖 ∗ 𝐽𝑛

𝜏,𝜖 𝜇) 𝑑L𝑑.

∫

Choose 𝜏 = 𝑇/𝑛 and let 𝜇𝜏,𝜖 (𝑡) be the piecewise constant interpolation of the minimizing movement

scheme 𝐽𝑛

𝜏,𝜖 . Therefore,

S(𝜇𝜏,𝜖 (0)) − S(𝜇𝜏,𝜖 (𝑇))

∫ 𝑇

∫

≥

0
∫ 𝑇

R𝑑
∫

+

0

R𝑑

−1
𝑎

Δ(𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠))(𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠)) + ⟨∇𝑉, ∇(𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠))⟩ 𝑑L𝑑 𝑑𝑠

⟨𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜏,𝜖 (𝑠), ∇𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠)⟩ 𝑑L𝑑 𝑑𝑠 −

∫ 𝑇

∫

0

R𝑑

Δ𝑉𝑘 𝑑𝜇𝜏,𝜖 (𝑠) 𝑑𝑠.

The last term can easily be bounded from below by 𝑇 ∥Δ𝑉𝑘 ∥ 𝐿∞. By Lemma 2.5.8,

∫

R𝑑

−1
𝑎

Δ(𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠))(𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠)) 𝑑L𝑑 ≥ 𝐶𝑎 ∥∇𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) ∥2

𝐿2 (R𝑑) − 𝐶′

𝑎 ∥𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) ∥2

𝐿2 (R𝑑)

It follows by Lemma 2.5.1 that

∫ 𝑇

∫

0

R𝑑

−1
𝑎

Δ(𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠))(𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠)) 𝑑L𝑑 𝑑𝑠 ≥ 𝐶𝑎

∫ 𝑇

∥∇𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) ∥2

𝑑𝑠

𝐿2 (R𝑑)

0
𝑎𝑇 (F𝜖,𝑘 (𝜇𝜖 (0)) + 2𝐶).

− 𝐶′

From here we want to say that either item 1 or item 2 of Assumption 2.1.7 is enough to get a

lower bound on the other terms. If we have item 1 of Assumption 2.1.7, then we can use Cauchy

46

inequality,

∫ 𝑇

∫

0

R𝑑

⟨∇𝑉, ∇(𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠))⟩ 𝑑L𝑑 𝑑𝑠 ≥

≥

−𝐶𝑎
2
−𝐶𝑎
2

∫ 𝑇

0
∫ 𝑇

0

∥∇𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) ∥2

𝐿2 (R𝑑)

𝑑𝑠 − 𝐶′

𝑎𝑇 ∥∇𝑉 ∥ 𝐿2 (R𝑑)

∥∇𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) ∥2

𝐿2 (R𝑑)

𝑑𝑠 − 𝐶′

𝑎𝑇𝐶

and

∫ 𝑇

∫

0

R𝑑

⟨𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜏,𝜖 (𝑠), ∇𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠)⟩ 𝑑L𝑑 𝑑𝑠 ≥

−𝐶𝑎
2

− 𝐶′
𝑎

≥

−𝐶𝑎
2

∫ 𝑇

0
∫ 𝑇

0
∫ 𝑇

0

∥∇𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) ∥2

𝐿2 (R𝑑)

𝑑𝑠

∥𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜏,𝜖 (𝑠) ∥ 𝐿2 (R𝑑) 𝑑𝑠

∥∇𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) ∥2

𝐿2 (R𝑑)

𝑑𝑠 − 𝐶′

𝑎𝑇𝐶.

Notice that last inequality is because of Lemma 2.5.1 allows us to apply Assumption 2.1.7 (and

2.4.10). To use item 2 instead, we first use integration by parts and then Young’s inequality,

∫ 𝑇

∫

0

R𝑑

⟨∇𝑉, ∇(𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠))⟩ 𝑑L𝑑 𝑑𝑠 ≥

≥

∫ 𝑇

−1
2
0
−𝑇𝐶𝑎
2

∥𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) ∥2

𝐿2 (R𝑑)

𝑑𝑠 − 𝑇 ∥Δ𝑉 ∥ 𝐿2 (R𝑑)

(F𝜖,𝑘 (𝜇𝜖 (0)) + 2𝐶) − 𝑇𝐶

and

∫ 𝑇

∫

0

R𝑑

⟨𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜏,𝜖 (𝑠), ∇(𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠))⟩ 𝑑L𝑑 𝑑𝑠 ≥

∫ 𝑇

−1
2
0
∫ 𝑇

−

0
−𝑇𝐶𝑎
2

≥

∥𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) ∥2

𝐿2 (R𝑑)

𝑑𝑠

∥𝜁𝜖 ∗ Δ𝑊 ∗ 𝜇𝜏,𝜖 (𝑠) ∥ 𝐿2 (R𝑑) 𝑑𝑠

(F𝜖,𝑘 (𝜇𝜖 (0)) + 2𝐶) − 𝑇𝐶.

In either case,

S(𝜇𝜏,𝜖 (0)) − S(𝜇𝜏,𝜖 (𝑇)) ≥

−𝐶
2

∫ 𝑇

0

∥∇𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) ∥2

𝐿2 (R𝑑)

𝑑𝑠 − 𝐶 (F𝜖,𝑘 (𝜇𝜖 (0)) + 1)

where 𝐶 = 𝐶 (𝑎, 𝑇). Using 𝜇𝜏,𝜖 (𝑡) → 𝜇𝜖 (𝑡) narrowly for all 𝑡 ≥ 0 ([14, Theorem A.2]), then for

𝑓 ∈ 𝐿2(R𝑑) and 𝑠 ∈ [0, 𝑇], we have that ∇𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) → ∇𝜁𝜖 ∗ 𝜇𝜖 (𝑠) weakly in 𝐿2(R𝑑) and

𝑠 ∈ [0, 𝑇]. That is as 𝑛 → ∞,

∫

R𝑑

𝑓 ∇𝜁𝜖 ∗ 𝜇𝜏,𝜖 (𝑠) =

∫

R𝑑

(∇𝜁𝜖 ∗ 𝑓 )𝜇𝜏,𝜖 (𝑠) →

∫

R𝑑

(∇𝜁𝜖 ∗ 𝑓 )𝜇𝜖 (𝑠) =

∫

R𝑑

𝑓 ∇𝜁𝜖 ∗ 𝜇𝜖 (𝑠).

47

By lower semi-continuity of 𝐿2(R𝑑) norm with respect to weak convergence and Fatou’s lemma,

taking 𝑛 → ∞

lim sup
𝑛→∞

S(𝜇𝜏,𝜖 (0)) − S(𝜇𝜏,𝜖 (𝑇)) ≥

𝐶

2

∫ 𝑇

0

∥∇𝜁𝜖 ∗ 𝜇𝜖 (𝑠) ∥2

𝐿2 (R𝑑)

𝑑𝑠 − 𝐶 (sup
𝜖 >0

F𝜖 (𝜇𝜖 (0)) + 1)

On the left-hand side we use the initial data from the minimizing movement scheme S(𝜇𝜏,𝜖 (0)) =

S(𝜇𝜖 (0)) for all 𝜏 > 0, the lower semi-continuity of the entropy with respect to narrow conver-

gence lim sup𝑛→∞ −S(𝜇𝜏,𝜖 (𝑇)) ≤ −S(𝜇𝜖 (𝑇)), a Carleman-type bound (from [9, Proposition 3.8])
−S(𝜇𝜖 (𝑇)) ≤ (2𝜋)1/2 + 𝑀2(𝜇𝜖 (𝑇)), and Proposition 2.5.9

S(𝜇𝜏,𝜖 (0)) − S(𝜇𝜏,𝜖 (𝑇)) ≤ S(𝜇𝜖 (0)) + (2𝜋)1/2 + (1 + 𝑇 𝑒𝑇 ) (𝑀2(𝜇𝜖 (0)) + F𝜖,𝑘 (𝜇𝜖 (0))).

lim sup
𝑛→∞

Thus,

𝐶

2

∫ 𝑇

0

∥∇𝜁𝜖 ∗ 𝜇𝜖 (𝑠)∥2

𝐿2 (R𝑑)

𝑑𝑠 − 𝐶 (F𝜖,𝑘 (𝜇𝜖 (0)) + 1) ≤ S(𝜇𝜖 (0)) + (2𝜋)1/2

+ (1 + 𝑇 𝑒𝑇 ) (𝑀2(𝜇𝜖 (0)) + F𝜖,𝑘 (𝜇𝜖 (0))).

The results follow.

2.6 Convergence of Gradient Flow

2.6.1 Γ-convergence of the energies

We establish one of the key hypotheses of Serfaty’s Theorem 2.3.16.

Proposition 2.6.1 (Γ-convergence of energies). Suppose Assumptions 2.1.3, 2.1.4 holds. Fix 𝑘 ∈ N.

Then F𝜖,𝑘 Γ-converges to F𝑘 as 𝜖 → 0.

Proof. As 𝜇𝜖 → 𝜇 narrowly as 𝜖 → 0, then 𝜁𝜖 ∗ 𝜇𝜖 → 𝜇 narrowly as 𝜖 → 0 by Lemma 2.3.3. Since

V is lower semi-continuous with respect to narrow convergence,

Similarly, we get that

lim inf
𝜖→0

V𝜖 (𝜇𝜖 ) = lim inf
𝜖→0

V (𝜁𝜖 ∗ 𝜇𝜖 ) ≥ V (𝜇).

lim inf
𝜖→0

W𝜖 (𝜇𝜖 ) = lim inf
𝜖→0

W (𝜁𝜖 ∗ 𝜇𝜖 ) ≥ W (𝜇),

lim inf
𝜖→0

E𝜖 (𝜇𝜖 ) = lim inf
𝜖→0

E (𝜁𝜖 ∗ 𝜇𝜖 ) ≥ E (𝜇).

48

It remains to show that

𝐸𝜖 (𝜇) ≤ 𝐸 (𝜇)

lim sup
𝜖→0

for 𝐸 = V, W as the case for E is documented in [14, Theorem 5.1]. Let us look at when 𝐸 = W.

The inequality is trivially true for W (𝜇) = ∞, so we can assume it is finite. We might as well

assume that there exists a 𝐶 > 0 such that

∫

W (𝜇) =

𝑊 ∗ 𝜇 𝑑𝜇 ≤ 𝐶.

Thus,

∫

𝜁𝜖 ∗ 𝑊 ∗ 𝜇 𝑑𝜇 ≤

∬

𝑊 ∗ 𝜇(𝑥 − 𝑦) 𝑑𝜇(𝑥) 𝜁𝜖 (𝑦) 𝑑𝑦 ≤ 𝐶

where the 𝐶 may of changed. Then,

∬

W𝜖 (𝜇) ≤

𝜁𝜖 ∗ 𝑊 ∗ 𝜇(𝑥 − 𝑦) 𝑑𝜇(𝑥) 𝜁𝜖 (𝑦) 𝑑𝑦 ≤ 𝐶

Therefore we can take the lim sup as 𝜖 → 0 and interchange the lim sup and the integral to get the

result. Similarly, it holds for when 𝐸 = V.

2.6.2 LSC of the local slope of the energies

Lemma 2.6.2 (Upgraded convergence). Suppose Assumptions 2.1.1, 2.1.2, 2.1.3, 2.1.4, 2.1.5 hold.

Fix 𝑘 ∈ N. Consider any sequence 𝜇𝜖 in P (R𝑑) and 𝜇𝑘 ∈ P (R𝑑) such that 𝜇𝜖 narrowly converges

to 𝜇𝑘 and

F𝜖,𝑘 (𝜇𝜖 ),

sup
𝜖 >0

lim inf
𝜖→0

∥∇𝜁𝜖 ∗ 𝜇𝜖 ∥ 𝐿2 (R𝑑)

are all finite. Then, 𝜇𝑘 ∈ 𝐿2(R𝑑) and there exists a subsequence (denoted 𝜇𝜖 ) along which we have,

∥𝜁𝜖 ∗ 𝜇𝜖 ∥𝐻1 (R𝑑) < ∞

sup
𝜖 >0

and 𝜁𝜖 ∗ 𝜇𝜖 converges to 𝜇𝑘 in 𝐿2

𝑙𝑜𝑐 (R𝑑).

49

Proof. By Proposition 2.6.6 and (2.3),

F𝑘 (𝜇𝑘 ) ≤ sup
𝜖 >0

F𝜖,𝑘 (𝜇𝜖 ) < ∞.

Using the ideas of the proof of Lemma 2.5.1, we get that 𝜇𝑘 ∈ 𝐿2(R𝑑). Combining the results of

Lemma 2.5.1 and lim inf𝜖→0 ∥∇𝜁𝜖 ∗ 𝜇𝜖 ∥ 𝐿2 (R𝑑) < ∞, then up to a subsequence we have

∥𝜁𝜖 ∗ 𝜇𝜖 ∥𝐻1 (R𝑑) < ∞.

sup
𝜖 >0

By Rellich-Kondrachov, up to another subsequence, 𝜁𝜖 ∗ 𝜇𝜖 converges in 𝐿2

𝑙𝑜𝑐 (R𝑑). By Lemma
2.3.3, 𝜁𝜖 ∗ 𝜇𝜖 narrowly converges to 𝜇𝑘 . Uniqueness of limits gives us the convergence result in
𝑙𝑜𝑐 (R𝑑).
𝐿2

We require the weak limit of the subdifferentials to achieve to Γ-convergence (lower semi-

continuity) of the local slopes.

Lemma 2.6.3 (Weak limit of subdifferentials, [14] Lemma 5.5). Suppose Assumptions 2.1.1, 2.1.2,

2.1.3, 2.1.4, 2.1.5 hold. Fix 𝑘 ∈ N. Consider any sequence 𝜇𝜖 in P (R𝑑) and 𝜇𝑘 ∈ P (R𝑑) such that

𝜇𝜖 narrowly converges to 𝜇𝑘 and

F𝜖,𝑘 (𝜇𝜖 ),

sup
𝜖 >0

lim inf
𝜖→0

∥∇𝜁𝜖 ∗ 𝜇𝜖 ∥ 𝐿2 (R𝑑)

are all finite. For all 𝜖 > 0 and 𝑓 ∈ 𝐶∞

𝑐 (R𝑑), define,

𝐿𝜖 ( 𝑓 ) =

∫

R𝑑

𝑓 ∇𝜁𝜖 ∗

(cid:19)

(cid:18) 𝜁𝜖 ∗ 𝜇𝜖
𝑎

𝑑𝜇𝜖 ,

𝐿( 𝑓 ) = −

∫

R𝑑

𝜇2
𝑘
2

(cid:19)

(cid:18) 𝑓
𝑎

∇

𝑑𝑥 +

∫

R𝑑

𝑓 𝜇2

𝑘 ∇

(cid:19)

(cid:18) 1
𝑎

𝑑𝑥.

There exists a subsequence, denoted by 𝜖, such that for any 𝑓 ∈ 𝐶∞

𝑐 (R𝑑), we have

𝐿𝜖 ( 𝑓 ) = 𝐿 ( 𝑓 ).

lim
𝜖→0

Furthermore, 𝐿 is a bounded linear operator on 𝐿2(𝜇𝑘 ).

Here we establish another key hypothesis of Serfaty’s Theorem.

50

Proposition 2.6.4 (lower semi-continuity of local slopes). Suppose Assumptions 2.1.3, 2.1.4 hold.
Fix 𝑘 ∈ N. Let 𝜇𝜖 ∈ P2(R𝑑) such that

F𝜖,𝑘 (𝜇𝜖 ),

sup
𝜖 >0

lim inf
𝜖→0

∥∇𝜁𝜖 ∗ 𝜇𝜖 ∥ 𝐿2 (R𝑑),

lim inf
𝜖→0

∫ (cid:12)
(cid:12)
(cid:12)
(cid:12)

∇𝜁𝜖 ∗

(cid:18) 𝜁𝜖 ∗ 𝜇𝜖
𝑎

2

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑑𝜇𝜖

are all finite. Suppose ∃ 𝜇𝑘 ∈ P (R𝑑) such that 𝜇𝜖 converges to 𝜇𝑘 . Then 𝜇2

𝑘 ∈ 𝑊 1,1(R𝑑) and

∃ 𝜂𝑘 ∈ 𝐿2(𝜇𝑘 ) where

𝜂𝑘 𝜇𝑘 =

(cid:17) 2(cid:19)

𝑎

2

∇

(cid:18) (cid:16) 𝜇𝑘
𝑎

+ 𝜇𝑘 (∇𝑊 ∗ 𝜇𝑘 ) + 𝜇𝑘 ∇(𝑉 + 𝑉𝑘 )

and

lim inf
𝜖→0

∫ (cid:12)
(cid:12)
(cid:12)
(cid:12)

∇𝜁𝜖 ∗

(cid:19)

(cid:18) 𝜁𝜖 ∗ 𝜇𝜖
𝑎

+ ∇(𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊) ∗ 𝜇𝜖 + ∇(𝜁𝜖 ∗ 𝑉) + ∇𝑉𝑘

∫

𝑑𝜇𝜖 ≥

(cid:12)
2
(cid:12)
(cid:12)
(cid:12)

|𝜂𝑘 |2 𝑑𝜇𝑘 .

Proof. We may choose a subsequence, denoted 𝜇𝜖 , so that

lim
𝜖→0

|𝜕F𝜖,𝑘 |(𝜇𝜖 ) = lim inf
𝜖→0

|𝜕F𝜖,𝑘 |(𝜇𝜖 ).

By [1, Theorem 5.4.4(ii)] (or [9, Proposition B.2(ii)]), it is now sufficient to show there exists

𝜂𝑘 ∈ 𝐿2(𝜇) satisfying the hypothesis above and up to another subsequence,

∫

(cid:18)

𝑓

∇𝜁𝜖 ∗

(cid:19)

(cid:18) 𝜁𝜖 ∗ 𝜇𝜖
𝑎

lim
𝜖→0

+ ∇(𝜁𝜖 ∗ 𝜁𝜖 ∗ 𝑊) ∗ 𝜇𝜖 + ∇(𝜁𝜖 ∗ 𝑉) + ∇𝑉𝑘

(cid:19)

∫

𝑑𝜇𝜖 =

𝑓 𝜂𝑘 𝑑𝜇𝑘

for all 𝑓 ∈ 𝐶∞

𝑐 (R𝑑). Given the continuity of ∇𝑉, ∇𝑉𝑘 ,

∫

R𝑑

lim
𝜖→0

𝑓 ∇(𝜁𝜖 ∗ 𝑉) 𝑑𝜇𝜖 = lim
𝜖→0
∫

∫

𝑓 ∇𝑉 𝑑𝜇𝜖 =

lim
𝜖→0

R𝑑

R𝑑

∫

R𝑑

∇𝑉 (𝜁𝜖 ∗ ( 𝑓 𝜇𝜖 )) 𝑑𝑥 =

∫

R𝑑

𝑓 ∇𝑉 𝑑𝜇𝑘

𝑓 ∇𝑉𝑘 𝑑𝜇𝑘 .

By using 𝐿𝜖 ( 𝑓 ), 𝐿( 𝑓 ) from Lemma 2.6.3 as well as the lemma itself, we may apply the Riesz

Representation Theorem on 𝐿2(𝜇𝑘 ), so that there exists ˜𝜂𝑘 ∈ 𝐿2(𝜇𝑘 ) such that

∫

R𝑑

lim
𝜖→0

𝑓 ∇𝜁𝜖 ∗

(cid:19)

(cid:18) 𝜁𝜖 ∗ 𝜇𝜖
𝑎

𝑑𝜇𝜖 = −

∫

R𝑑

𝜇2
𝑘
2

(cid:19)

(cid:18) 𝑓
𝑎

∇

𝑑𝑥 +

∫

R𝑑

𝑓 𝜇2

𝑘 ∇

(cid:19)

(cid:18) 1
𝑎

𝑑𝑥 =

∫

R𝑑

𝑓 ˜𝜂𝑘 𝑑𝜇𝑘 .

By rearranging we get,

−

∫

R𝑑

𝜇2
𝑘
2

(cid:19)

(cid:18) 𝑓
𝑎

∇

𝑑𝑥 =

∫

R𝑑

(cid:16)

𝑓
𝑎

˜𝜂𝑘 𝜇𝑘 𝑎 − 𝑎𝜇2

𝑘 ∇(1/𝑎)

(cid:17)

𝑑𝑥.

51

We find that 𝜇2

𝑘 ∈ 𝑊 1,1(R𝑑) and its weak derivative is

(cid:33)

(cid:32) 𝜇2
𝑘
2

∇

= ˜𝜂𝑘 𝜇𝑘 𝑎 − 𝑎𝜇2

𝑘 ∇(1/𝑎).

By the chain rule for 𝑊 1,1(R𝑑) functions, we obtain

˜𝜂𝑘 𝜇𝑘 =

𝑎

2

∇

(cid:18) (cid:16) 𝜇𝑘
𝑎

(cid:17) 2(cid:19)

.

We have to check the aggregation portion and verify that 𝜂𝑘 is 𝐿2(𝜇𝑘 ). First, we check that

𝜂𝑘 = ˜𝜂𝑘 + ∇𝑉 + ∇𝑊 ∗ 𝜇𝑘 + ∇𝑉𝑘 is 𝐿2(𝜇𝑘 ). We get this easily by Assumption 2.1.3, 2.1.4 and 2.1.5.

Namely, that ∥∇𝑊 ∗ 𝜇𝑘 ∥ 𝐿2 (𝜇) ≤ 𝐶, ∥∇𝑉 ∥ 𝐿2 (𝜇𝑘) ≤ 𝐶, and ∥∇𝑉𝑘 ∥ 𝐿2 (𝜇𝑘) ≤ 𝐶. Second, we require
that for 𝑓 ∈ 𝐶∞

𝑐 (R𝑑),

∫

𝑓 (𝜁𝜖 ∗ 𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜖 ) 𝑑𝜇𝜖 →

∫

𝑓 (∇𝑊 ∗ 𝜇𝑘 ) 𝑑𝜇𝑘

as 𝜖 → 0. Let us look at the difference and write

∫

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

𝑓 (𝜁𝜖 ∗ 𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜖 ) 𝑑𝜇𝜖 −

∫

𝑓 (∇𝑊 ∗ 𝜇𝑘 ) 𝑑𝜇𝑘

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

≤ ∥ 𝑓 ∥ 𝐿∞

∫

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

(𝜁𝜖 ∗ 𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜖 )𝜇𝜖 − (∇𝑊 ∗ 𝜇𝑘 )𝜇𝑘 𝑑𝑥

Thus it is sufficient to look at

(cid:12)
∫
(cid:12)
(cid:12)
(cid:12)

≤

(cid:12)
∫
(cid:12)
(cid:12)
(cid:12)

+

(𝜁𝜖 ∗ 𝜁𝜖 ∗ ∇𝑊 ∗ 𝜇𝜖 ) 𝑑𝜇𝜖 −

∫

(∇𝑊 ∗ 𝜇𝑘 ) 𝑑𝜇𝑘

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

(∇𝑊 ∗ 𝜁𝜖 ∗ 𝜇𝜖 ) 𝑑𝜁𝜖 ∗ 𝜇𝜖 −

∫

(∇𝑊 ∗ 𝜇𝑘 ) 𝑑𝜁𝜖 ∗ 𝜇𝜖

(cid:12)
∫
(cid:12)
(cid:12)
(cid:12)

(∇𝑊 ∗ 𝜇𝑘 ) 𝑑𝜁𝜖 ∗ 𝜇𝜖 −

∫

(∇𝑊 ∗ 𝜇𝑘 ) 𝑑𝜇𝑘

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

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

.

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

=: 𝐽1 + 𝐽2.

We have 𝐽2 → 0 as 𝜖 → 0 as 𝜁𝜖 ∗ 𝜇𝜖 converges narrowly to 𝜇𝑘 along with 𝑓 ∈ 𝐶∞

𝑐 (R𝑑) and ∇𝑊 ∗ 𝜇𝑘

is continuous. For 𝐽1, we apply Assumption 2.1.3 (item 4),

𝐽1 ≤ 𝐶𝑊2((𝜁𝜖 ∗ 𝜇𝜖 )1supp( 𝑓 ), 𝜇𝑘 1supp( 𝑓 )) → 0

as 𝜁𝜖 ∗ 𝜇𝜖 → 𝜇𝑘 narrowly and supp( 𝑓 ) is compact (namely bounded). This gives us the result. Note

that due to the minimality of the subdifferential of F𝜖,𝑘 and F𝑘 and properties of convergence of

varying measure ([9, Proposition B.2(ii)]), we get the lower semi-continuity of the local slopes.

52

2.6.3 Convergence as 𝜖 limit

One would expect that gradient flows of F𝜖,𝑘 to convergence to gradient flows of F𝑘 . Unfortunately,

due to the triviality of having 𝑎(𝑥) be log-concave on all of R𝑑, we lack the regularity to define some

notion of gradient flows of F𝑘 . However, we are able to identify to limit and have it “almost” satisfy

the conditions of gradient flow of F𝑘 .

Definition 2.6.5 (“almost” curves of maximal slope of F𝑘 ). A curve 𝜇𝑘 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) is

an “almost” curve of maximal slope of F𝑘 if it satisfies,

1
2

∫ 𝑡

0

|𝜇′

𝑘 |2(𝑟) 𝑑𝑟 +

1
2

∫ 𝑡

∫

0

R𝑑

|𝜂𝑘 (𝑟)|2 𝑑𝜇𝑘 (𝑟) 𝑑𝑟 ≤ F𝑘 (𝜇𝑘 (0)) − F𝑘 (𝜇𝑘 (𝑡))

for all 𝑡 ∈ [0, 𝑇]. Here 𝜇2

𝑘 (𝑡) ∈ 𝑊 1,1(R𝑑) and 𝜂𝑘 (𝑡) ∈ 𝐿2(𝜇𝑘 (𝑡)) satisfies

𝜂𝑘 𝜇𝑘 =

(cid:17) 2(cid:19)

𝑎

2

∇

(cid:18) (cid:16) 𝜇𝑘
𝑎

+ 𝜇𝑘 (∇𝑊 ∗ 𝜇𝑘 ) + 𝜇𝑘 ∇(𝑉 + 𝑉𝑘 )

for almost every 𝑡 ∈ [0, 𝑇].

It is worth noting that if 𝑎(𝑥) was log-concave on all R𝑑, then ∫

R𝑑 |𝜂𝑘 (𝑟)|2 𝑑𝜇𝑘 (𝑟) would be a

strong upper gradient of F𝑘 so that an almost curve of maximal slope would actually be a curve of

maximal slope.

Proposition 2.6.6 (Convergence as 𝜖 → 0). Suppose Assumptions 2.1.3, 2.1.4, 2.1.5, 2.1.7 hold. Fix
𝑇 > 0 and 𝑘 ∈ N. For 𝜖 > 0, let 𝜇𝜖,𝑘 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) be a gradient flow of F𝜖,𝑘 satisfying,

S(𝜇𝜖,𝑘 (0)) < ∞,

sup
𝜖 >0

𝑀2(𝜇𝜖,𝑘 (0)) < ∞.

sup
𝜖 >0

Suppose there exists 𝜇𝑘 (0) ∈ 𝐷 (F ) ∩ P2(R𝑑) that is an “almost” curve of maximal slope of F𝑘 in
the sense of Definition 2.6.5, and a subsequence 𝜖 (𝑘)
𝑛

, depending on 𝑘, such that

lim
𝑛→∞

𝑊1(𝜇

𝑛 ,𝑘 (𝑡), 𝜇𝑘 (𝑡)) = 0
𝜖 (𝑘 )

uniformly for 𝑡 ∈ [0, 𝑇].

Proof. By Theorem 2.3.13, 𝜇𝜖,𝑘 is a curve of maximal slope of F𝜖,𝑘 so that for all 0 ≤ 𝑡 ≤ 𝑇,

1
2

∫ 𝑡

0

|𝜇′

𝜖,𝑘 |2(𝑟) 𝑑𝑟 +

1
2

∫ 𝑡

0

|𝜕F𝜖,𝑘 |2(𝜇𝜖,𝑘 (𝑟)) 𝑑𝑟 ≤ F𝜖,𝑘 (𝜇𝜖,𝑘 (0)) − F𝜖,𝑘 (𝜇𝜖,𝑘 (𝑡)).

53

We check the hypotheses of the weak Serfaty framework, Theorem 2.3.16, to apply the Theorem.

Proposition 2.6.1 gives us the required Γ-convergence. Proposition 2.4.17 gives an explicit

characterization of the local slope, |𝜕F𝜖,𝑘 | and therefore, an explicit characterization of 𝜂𝜖,𝑘 (𝑟) ∈

𝐿2(𝜇𝜖,𝑘 (𝑟)). The hypotheses of this proposition ensure the initial data is well prepared. We may now
apply Theorem 2.3.16. There exists 𝜇𝑘 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) and a subsequence 𝜖 (𝑘)
on 𝑘, so that

, depending

𝑛

lim
𝑛→∞

𝑊1(𝜇

𝑛 ,𝑘 (𝑡), 𝜇𝑘 (𝑡)) = 0
𝜖 (𝑘 )

and for all 𝑡 ∈ [0, 𝑇],

1
2

∫ 𝑡

0

|𝜇′

𝑘 |2(𝑟) 𝑑𝑟 +

1
2

∫ 𝑡

0

lim inf
𝑛→∞

∫

R𝑑

|𝜂

𝜖 (𝑘 )
𝑛

(𝑟)|2 𝑑𝜇

𝑛 ,𝑘 (𝑟) 𝑑𝑟 ≤ F𝑘 (𝜇𝑘 (0)) − F𝑘 (𝜇𝑘 (𝑡)).
𝜖 (𝑘 )

To conclude we must show the 𝜇𝑘 is an “almost” curve of maximal slope (Definition 2.6.5). To do

so, we seek to apply Proposition 2.6.4. As the right-hand side above is finite, then the left-hand side

is finite for almost every 𝑟 ∈ [0, 𝑇]. In particular, for almost every 𝑟 ∈ [0, 𝑇],

Given that

lim inf
𝑛→∞

∫

R𝑑

|𝜂

𝜖 (𝑘 )
𝑛

(𝑟)|2 𝑑𝜇

𝑛 ,𝑘 (𝑟) 𝑑𝑟 < ∞.
𝜖 (𝑘 )

F𝜖 (𝑘 )

𝑛 ,𝑘 (𝜇

𝑛 ,𝑘 (0)) < ∞,
𝜖 (𝑘 )

sup
𝑛∈N

and the hypotheses that the heat entropy and second moment is finite at the initial data we apply the

𝐻1 bound (Proposition 2.5.10) so that,

lim inf
𝑛→∞

∫ 𝑇

0

∥∇𝜁

𝜖 (𝑘 )
𝑛

∗ 𝜇

𝑛 ,𝑘 (𝑟) ∥2
𝜖 (𝑘 )

𝐿2 (R𝑑)

𝑑𝑟 < ∞.

Fatou’s lemma gives the integrand must be finite for almost every 𝑟 ∈ [0, 𝑇]. As the energy decreases

along its gradient flow in time,

F𝜖 (𝑘 )

𝑛 ,𝑘 (𝜇

𝑛 ,𝑘 (𝑡)) < ∞.
𝜖 (𝑘 )

sup
𝑛∈N

Now we may apply Proposition 2.6.4 and obtain the result.

54

2.7 Convergence of the Confining Limit

Now after taking the limit in 𝜖, we take the limit in 𝑘.

Proposition 2.7.1 (Γ-convergence of confining energy, [14] Theorem 6.1). Suppose Assumptions

2.1.3, 2.1.4, 2.1.5, 2.1.7 hold. Then V𝑘 , F𝑘 Γ-converge to VΩ, F respectively as 𝑘 → ∞. In
particular, lim𝑘→∞ V𝑘 (𝜇) = VΩ(𝜇) for any 𝜇 ∈ P2(R𝑑).

Proof. Notice that the Γ-convergence of F𝑘 to F follows from the Γ-convergence of V𝑘 to VΩ. We

may assume that lim inf 𝑘→∞ V𝑘 (𝜇𝑘 ) < ∞, so up to a subsequence, sup𝑘∈N V𝑘 (𝜇𝑘 ) < ∞. To show
(2.3), it suffices to show that supp 𝜇 ⊆ Ω, since V𝑘 (𝜇𝑘 ) is nonnegative and VΩ(𝜇) would be zero.

By contradiction, suppose that supp 𝜇 ⊈ Ω, so that there exists 𝑥 ∈ Ω

𝑐

and an open ball 𝐵 containg

𝑥 so that 𝐵 ⊂⊂ Ω

𝑐

and 𝜇(𝐵) > 0. By equivalent definitions of weak convergence of sequence

measures (Portmanteau Theorem), 𝜇𝑘 → 𝜇 narrowly implies lim inf 𝑘→∞ 𝜇𝑘 (𝐵) ≥ 𝜇(𝐵) > 0. So

up to another subsequence, we may assume that there exists 𝛿 > 0 so that 𝜇𝑘 (𝐵) ≥ 𝛿 for all 𝑘 ∈ N.

By Assumption 2.1.5,

lim inf
𝑘→∞

∫

R𝑑

𝑉𝑘 𝑑𝜇𝑘 ≥ lim inf
𝑘→∞

≥ lim inf
𝑘→∞

∫

𝐵

(cid:18)

𝑉𝑘 𝑑𝜇𝑘

(cid:19)

𝑉𝑘 (𝑥)

𝜇𝑘 (𝐵)

inf
𝑥∈𝐵
(cid:18)

inf
𝑥∈𝐵

(cid:19)

𝑉𝑘 (𝑥)

≥ 𝛿 lim inf
𝑘→∞

= ∞,

a contradiction. To show the Γ − lim sup convergence, we use Assumption 2.1.5, so that

lim sup
𝑘→∞

V𝑘 (𝜇) = lim sup
𝑘→∞

∫

∫

𝑉𝑘 𝑑𝜇 ≤

𝑉Ω 𝑑𝜇 = VΩ(𝜇).

This ends the proof.

We take the limit in the confining variable so that the “almost” gradient flows of F𝑘 converge to

the gradient flows of F .

55

Proposition 2.7.2 (Convergence as 𝑘 → ∞). Suppose Assumptions 2.1.3, 2.1.4, 2.1.5, 2.1.7 hold.
Fix 𝑇 > 0. For 𝑘 ∈ N, let 𝜇𝑘 ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)) be an “almost” curve of maximal slope of
F𝑘 , in the sense of Definition 2.6.5, and suppose there exists 𝜇(0) ∈ 𝐷 (F ) ∩ P2(R𝑑) such that
sup𝑘∈N 𝑀2(𝜇𝑘 (0)) < ∞ and as 𝑘 → ∞

𝜇𝑘 (0) → 𝜇(0) narrowly, F𝑘 (𝜇𝑘 (0)) → F (𝜇(0)).

Then,

𝑊1(𝜇𝑘 (𝑡), 𝜇(𝑡)) = 0, uniformly for 𝑡 ∈ [0, 𝑇],

lim
𝑘→∞

where 𝜇 ∈ 𝐴𝐶2([0, 𝑇]; P2(R𝑑)) is the unique gradient flow of F with initial condition 𝜇(0).

Proof. Given that the proof here is similar to the proof of [14, Proposition 6.2], we will only

provided the necessary updates. Recall that Proposition 2.7.1 gives us Γ-convergence of F𝑘 to F . So
we can apply Theorem 2.3.16. There exists 𝜇 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) so that up to a subsequence,

𝑊1(𝜇𝑘 (𝑡), 𝜇(𝑡)) = 0, uniformly for 𝑡 ∈ [0, 𝑇],

lim
𝑘→∞

holds and for all 𝑡 ∈ [0, 𝑇],

1
2

∫ 𝑡

0

|𝜇′|2(𝑟) 𝑑𝑟 +

1
2

∫ 𝑡

0

lim inf
𝑘→∞

∫

R𝑑

|𝜂𝑘 (𝑟)|2 𝑑𝜇𝑘 (𝑟) 𝑑𝑟 ≤ F (𝜇(0)) − F (𝜇(𝑡)).

By definition of “almost” curve of maximal slope, F𝑘 (𝜇𝑘 (𝑡)) ≤ F𝑘 (𝜇𝑘 (0)) for all 𝑘 ∈ N, 𝑡 ∈ [0, 𝑇].

As the initial data is well prepared,

sup
𝑡∈[0,𝑇],𝑘 ∈N

F𝑘 (𝜇𝑘 (𝑡)) ≤ sup
𝑘∈N

F𝑘 (𝜇𝑘 (0)) < ∞.

As we have 𝑊1 convergence of the density, then the density converges narrowly for all 𝑡 ∈ [0, 𝑇].

By Γ-convergence,

sup
𝑡∈[0,𝑇]

F (𝜇(𝑡)) ≤ lim inf
𝑘→∞

F𝑘 (𝜇𝑘 (𝑡)) ≤

sup
𝑡∈[0,𝑇],𝑘 ∈N

F𝑘 (𝜇𝑘 (𝑡)) < ∞.

It suffices to show that for almost every 𝑡 ∈ [0, 𝑇], we have

(2.9)

lim inf
𝑘→∞

∫

𝑅𝑑

|𝜂𝑘 (𝑡)|2 𝑑𝜇𝑘 (𝑡) ≥

∫

R𝑑

|𝜂(𝑡)|2 𝑑𝜇(𝑡)

56

for 𝜇 and 𝜂 satisfying on Ω,

𝜂𝜇 =

(cid:17) 2(cid:19)

𝑎

2

∇

(cid:18) (cid:16) 𝜇
𝑎

+ 𝜇(∇𝑊 ∗ 𝜇) + 𝜇∇𝑉,

(cid:19) 2

(cid:18) 𝜇(𝑡)
𝑎

∈ 𝑊 1,1(Ω).

Since that F (𝜇(0)) − F (𝜇(𝑡)) is finite, then the left-hand side above is finite and up to a subsequence

in 𝑘, we may assume sup𝑘∈N ∥𝜂𝑘 (𝑡)∥ 𝐿2 (𝜇𝑘 (𝑡)) < ∞. Given that we can bound the drift and the
aggregation from below (say by 2𝐶), then

∥𝜇𝑘 (𝑡)∥ 𝐿2 (R𝑑)
2∥𝑎∥𝑙∞

∫

|𝜇𝑘 (𝑡)|2
2𝑎

sup
𝑘∈N

≤ sup
𝑘∈N

= sup
𝑘∈N
Moreover, as sup𝑡∈[0,𝑇] F (𝜇(𝑡)) < ∞, supp 𝜇(𝑡) ⊆ Ω. By Assumptions 2.1.3, 2.1.4, we have
∇𝑉, ∇𝑊 ∗ 𝜇𝑘 ∈ 𝐿2(𝜇𝑘 ). Therefore,

E (𝜇𝑘 (𝑡)) ≤ sup
𝑘 ∈N

F𝑘 (𝜇𝑘 (𝑡)) + 2𝐶 < ∞.

R𝑑

∫ (cid:12)
(cid:12)
(cid:12)
(cid:12)

𝑎

2

∇

(cid:18) (cid:16) 𝜇𝑘
𝑎

sup
𝑘

(cid:17) 2(cid:19)

+ ∇𝑉𝑘 𝜇𝑘

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

∥𝜂𝑘 − ∇𝑉 − ∇𝑊 ∗ 𝜇𝑘 ∥ 𝐿1 (𝜇𝑘)

∥𝜂𝑘 − ∇𝑉 − ∇𝑊 ∗ 𝜇𝑘 ∥ 𝐿2 (𝜇𝑘)

(cid:16)

∥𝜂𝑘 ∥ 𝐿2 (𝜇𝑘) + ∥∇𝑊 ∗ 𝜇𝑘 ∥ 𝐿2 (𝜇𝑘) + ∥∇𝑉 ∥ 𝐿2 (𝜇𝑘)

(cid:17)

= sup
𝑘

≤ sup
𝑘

≤ sup
𝑘

< ∞.

By [1, Theorem 5.4.4(ii)] (or [9, Proposition B.2(ii)]), to get the lower semi-continuity of the 𝜂, as

in (2.9), it suffices to show weak convergence with varying measure,

∫

R𝑑

𝑓

(cid:18) 𝑎

2

∇

(cid:18) (cid:16) 𝜇𝑘
𝑎

(cid:17) 2(cid:19)

lim
𝑘→∞

+ 𝜇𝑘 (∇𝑊 ∗ 𝜇𝑘 ) + 𝜇𝑘 ∇(𝑉 + 𝑉𝑘 )

∫

=

Ω

𝑓

(cid:18) 𝑎

2

∇

(cid:18) (cid:16) 𝜇
𝑎

(cid:17) 2(cid:19)

+ 𝜇(∇𝑊 ∗ 𝜇) + 𝜇∇𝑉

(cid:19)

(cid:19)

for all 𝑓 ∈ 𝐶∞

𝑐 (R𝑑) where we recall that 𝜇 = 0 a.e. on Ω𝑐.

By Assumptions 2.1.3, 2.1.4, we have 𝑉, 𝑊 ∗𝜇𝑘 ∈ 𝐶1(R𝑑). Moreover, 𝑓 ∇𝑉, 𝑓 ∇𝑊 ∗𝜇𝑘 ∈ 𝐶𝑏 (R𝑑).

By narrow convergence,

lim inf
𝑘→∞

∫

R𝑑

𝑓 ∇𝑉 𝑑𝜇𝑘 =

∫

Ω

𝑓 ∇𝑉 𝑑𝜇,

lim inf
𝑘→∞

∫

R𝑑

𝑓 ∇𝑊 ∗ 𝜇𝑘 𝑑𝜇𝑘 =

∫

Ω

𝑓 ∇𝑊 ∗ 𝜇 𝑑𝜇

for a.e 𝑡 ∈ [0, 𝑇]. Thus is suffices to show for all 𝑓 ∈ 𝐶∞

𝑐 (R𝑑),
∫

=

Ω

𝑓

(cid:18) 𝑎

2

∇

(cid:18) (cid:16) 𝜇
𝑎

(cid:17) 2(cid:19)(cid:19)

.

∫

R𝑑

𝑓

(cid:18) 𝑎

2

∇

(cid:18) (cid:16) 𝜇𝑘
𝑎

(cid:17) 2(cid:19)

lim
𝑘→∞

(cid:19)

+ 𝜇𝑘 ∇𝑉𝑘

57

What remains is to first show it is true for 𝑓 ∈ 𝐶∞

𝑐 (Ω) then generalized to 𝑓 ∈ 𝐶∞

𝑐 (R𝑑). We define

the operator

𝐿( 𝑓 ) :=

∫

Ω

(cid:17) 2

𝑓

𝑎

2

∇

(cid:16) 𝜇
𝑎

= lim inf
𝑘→∞

∫

Ω

𝑓

(cid:18) 𝑎

2

∇

(cid:16) 𝜇𝑘
𝑎

(cid:17) 2

+ ∇𝑉𝑘 𝜇𝑘

(cid:19)

,

show that it is bounded, and apply the Riesz Representation Theorem. To extend the results to

𝑓 ∈ 𝐶∞

𝑐 (R𝑑), a cut off function is used. This is exactly the same as [14, Proposition 6.2].

2.8 Proofs of Main Results

Proof of Theorem 2.2.1. Let 𝜇 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑)) be the unique gradient flow of F with the
initial condition 𝜇(0). By Proposition 2.6.1, for all 𝑘 ∈ N,

F𝜖,𝑘 (𝜇(0)) = F𝑘 (𝜇(0)).

lim
𝜖→0

By Proposition 2.6.6, there exists an “almost” curve of maximal slope 𝜇𝑘 ∈ 𝐴𝐶2( [0, 𝑇]; P2(R𝑑))
and a subsequence {𝜖 (𝑘)

𝑗=1, depending on 𝑘, such that
}∞

𝑗

lim
𝑗→∞

𝑊1(𝜇

𝜖 (𝑘 )
𝑗

,𝑘 (𝑡), 𝜇𝑘 (𝑡)) = 0

uniformly for 𝑡 ∈ [0, 𝑇]. In particular, for each 𝑘 ∈ N, there exists 𝜖𝑘 > 0 so that lim𝑘→∞ 𝜖𝑘 = 0 and

𝑊1(𝜇𝜖𝑘,𝑘 (𝑡), 𝜇𝑘 (𝑡)) < 1
𝑘

,

for all 𝑡 ∈ [0, 𝑇]. By Proposition 2.7.1,

So by Proposition 2.7.2,

F𝑘 (𝜇(0)) = F (𝜇(0)).

lim
𝑘→∞

𝑊1(𝜇𝑘 (𝑡), 𝜇(𝑡)) = 0,

lim
𝑘→∞

uniformly for 𝑡 ∈ [0, 𝑇]. Fix 𝛿 > 0. Choose 𝐾𝛿 > 0 such that, for all 𝑘 ≥ 𝐾𝛿, 𝑊1(𝜇𝑘 (𝑡), 𝜇(𝑡)) < 𝛿/2

for all 𝑡 ∈ [0, 𝑇]. Then, for all 𝑘 ≥ max{2/𝛿, 𝐾𝛿},

𝑊1(𝜇𝜖𝑘,𝑘 (𝑡), 𝜇(𝑡)) ≤ 𝑊1(𝜇𝑘 (𝑡), 𝜇(𝑡)) + 𝑊1(𝜇𝜖𝑘,𝑘 (𝑡), 𝜇𝑘 (𝑡)) ≤

𝛿

2

+

1
𝑘

≤ 𝛿,

for all 𝑡 ∈ [0, 𝑇].

58

Proof of Theorem 2.2.2. Let 𝜇𝜖,𝑘 (𝑡) be the gradient flow of F𝜖,𝑘 with initial data 𝜇(0). By Theorem

2.2.1, as 𝑘 → ∞, 𝜖 = 𝜖 (𝑘) → 0,

𝑊1(𝜇𝜖,𝑘 (𝑡), 𝜇(𝑡)) = 0

lim
𝑘→∞

uniformly for 𝑡 ∈ [0, 𝑇], where 𝜇(𝑡) is the gradient flow of F with initial data 𝜇(0). Recall that F𝜖,𝑘 is

lower semi-continuous and 𝜔-convex along generalized geodesics with 𝜆𝜔,𝜖 = −𝜖 −𝑑−2∥𝐷2𝜁/𝑎∥ 𝐿∞ +

8𝐶. Note that 𝜆𝜔,𝜖 is nonpositive for sufficiently small 𝜖. The empirical measure, 𝜇𝑁

𝜖,𝑘 (𝑡) as defined

in the hypothesis, is the unique gradient flow of F𝜖,𝑘 with initial data 𝜇𝑁

𝜖,𝑘 (0). By Theorem 2.2.1, it

suffices to show that as 𝑘 → ∞, 𝜖 = 𝜖 (𝑘) → 0, 𝑁 = 𝑁 (𝜖) → ∞,

𝑊1(𝜇𝑁

𝜖,𝑘 (𝑡), 𝜇𝜖,𝑘 (𝑡)) = 0

lim
𝑘→∞

uniformly for 𝑡 ∈ [0, 𝑇]. As both 𝜇𝜖,𝑘 (𝑡), 𝜇𝑁

𝜖,𝑘 (𝑡) are both gradient flows of the 𝜔-convex energy
functional F𝜖,𝑘 , we may apply [13, Theorem 3.11(iii)]. That is, given that 𝜇𝜖,𝑘 (𝑡) has initial data

𝜇(0),

𝑊 2

2 (𝜇𝑁

𝜖,𝑘 (𝑡), 𝜇𝜖,𝑘 (𝑡)) ≤ 𝐹−2𝑡,𝜖 (𝑊 2

2 (𝜇𝑁

𝜖,𝑘 (0), 𝜇(0))).

By hypothesis as 𝑘 → ∞,

𝑊1(𝜇𝑁

𝜖,𝑘 (𝑡), 𝜇𝜖,𝑘 (𝑡)) ≤ 𝑊2(𝜇𝑁

𝜖,𝑘 (𝑡), 𝜇𝜖,𝑘 (𝑡)) ≤

√︃

𝐹−2𝑡,𝜖 (𝑊 2
2

(𝜇𝑁

𝜖,𝑘 (0), 𝜇(0))) → 0

uniformly in 𝑡 ∈ [0, 𝑇].

Proof of Corollary 2.2.3. By Proposition 2.4.19,

𝑊1

lim
𝑡→∞

(cid:0)𝜇(𝑡), 𝑎1

(cid:1) ≤ lim
𝑡→∞

Ω

𝑊2

(cid:0)𝜇(𝑡), 𝑎1

(cid:1) = 0.

Ω

The result follows from Theorem 2.2.2.

Proof of Theorem 2.2.4. We would like to use Theorem 2.3.17 to conclude the results. By Lemma

2.3.15, up to a subsequence, we get that 𝜇𝜖 → 𝜇 narrowly (as convergence in distance implies

convergence narrowly) and the Γ-convergence of the metric derivatives. By Proposition 2.6.1, the

Γ-convergence of the energy holds and lim𝜖→0 G𝜖 (𝜇𝜖 (0)) = G(𝜇(0)). Finally, by Proposition 2.6.4,

the Γ-convergence (or lower semi-continuity) of the local slopes hold. Now we may apply Theorem

2.3.17 and gain the results immediately.

59

2.9 Applications

2.9.1 Keller-Segel Chemotaxis Model

Next we look at the general Keller-Segel Model for chemotaxis in [3],

𝜕𝑡𝑢 = ∇ · (𝜙(𝑢, 𝑣)∇𝑢 − 𝜓(𝑢, 𝑣)∇𝑣) + 𝑓 (𝑢, 𝑣)

𝜏𝜕𝑡𝑣 = 𝑑Δ𝑣 + 𝑔(𝑢, 𝑣) − ℎ(𝑢, 𝑣)𝑣.





We have 𝑢 represents the cell density on Ω ⊂ R𝑑, 𝑣 represents the concentration of the chemical signal,

𝜙 represents the diffusivity of the cells, and 𝜓 represents the chemotactic sensitivity. Chemotaxis is

when cells movements are affected by chemicals in their environment. In this model the aggregation

term and the diffusion term compete. When aggregation wins, it can be studied in mathematics as

blow-up in finite time; however, this is not realistic from a biological viewpoint. Moreover with a

more general diffusion, the system has solutions based on the inequalities that involve the diffusion

exponent and the dimension ([5], [15]). Here we choose

𝜙(𝑢, 𝑣) = 𝜓(𝑢, 𝑣) = 𝑔(𝑢, 𝑣) = 𝑢,

𝑑 = 1,

𝜏 = 𝑓 (𝑢, 𝑣) = 0,

ℎ(𝑢, 𝑣) = 𝛼 ≥ 0.

Most choices above are relatively common as seen in table 1 of [3]. With that we obtain that

𝑣 = B𝛼,𝑑 ∗ 𝑢 ([8]) where B𝛼,𝑑 is the Bessel kernel

B𝛼,𝑑 (𝑥) =

∫ ∞

0

1
(4𝜋𝑡)𝑑/2

𝑒 − | 𝑥 |2

4𝑡 −𝛼𝑡 𝑑𝑡.

Notice that B0,𝑑 = −N (see [3]), where we get the original PDE that we are considering with 𝑎(x)

removed (or 𝑎(𝑥) ≡ 1). Note that for potential future work choosing 𝜏 = 1 implies that we get the

heat kernel for 𝛼 = 0 and the integrand of the Bessel kernel for 𝛼 > 0. The Newtonian potential

satisfies Assumption 2.1.3 by [13, Proposition 4.4], however, we need to show that it satisfies the

additional assumptions.

Remark 2.9.1 (The Newtonian potential satisfies Assumption 2.1.7). As ∥𝜇∥ 𝐿 𝑝 (R𝑑) ≤ 𝐶 for 𝑝 = 1, 2,

by interpolation it is true for all 𝑝 ∈ [1, 2]. Using Young’s convolution inequality, as long as

∥∇𝑊 ∥ 𝐿 𝑝 (R𝑑\𝐵𝑅) ≤ 𝐶𝑅 for some 𝑝 ∈ [1, 2], then we get item 1. For the Newtonian potential, we

60

only have this for 𝑑 ≥ 3 (see Lemma A.0.4). However, the Newtonian potential does satisfy item 2

for all 𝑑 (see [13] equation 61).

2.9.2 Bessel Kernel

Naturally, we will now show that the Bessel kernel satisfies Assumption 2.1.3 and Assumption

2.1.7.

Proposition 2.9.2. The Bessel kernel, −B𝛼,𝑑, with 𝑑 ≥ 3 satisfies Assumption 2.1.3. Furthermore,

it satisfies Assumption 2.1.7 item 1 for 𝑑 ≥ 3 and item 2 for all 𝑑.

Proof. Given that most of the assumptions involve a norm or modulus, it suffices to show B𝛼,𝑑

satisfies the above assumptions (barring the lower semi-continuity). Furthermore 𝛼 = 0 is already

shown as B0,𝑑 = −N , so we will assume 𝛼 > 0. We first start with the items of Assumption 2.1.3.

1. Since we have the chain of inequalities B−

𝛼,𝑑 ≤ B−

𝛼,𝑑 + B+

𝛼,𝑑 = B𝛼,𝑑 ≤ B0,𝑑 ≤ −N ≤ N −, we

get |B−

𝛼,𝑑 ∗ 𝜇| ≤ |N − ∗ 𝜇| ≤ 𝐶.

2. It is sufficient to show that |∇B𝛼,𝑑 ∗ 𝜇| ≤ 𝐶 as 𝜈 is a probability measure. By computation

or [8, Lemma 2.4], |∇B𝛼,𝑑 (𝑥)| ≤ 𝐶𝑑 |𝑥|1−𝑑𝑔𝛼 (|𝑥|) where 𝑔𝛼 (|𝑥|) is a positive radial function
exponentially decreasing from 1 to 0 as |𝑥| → ∞. Thus, ∃𝑐 such that |∇B𝛼,𝑑 | ≤ 𝑐 on R𝑑\𝐵1
where 𝐵1 is the unit ball centered at the origin. By computation, ∥∇B𝛼,𝑑 ∥ 𝐿 𝑝 (𝐵1) ≤ 𝑐 for
𝑝 < 𝑑

𝑑−1 (see Lemma A.0.4). Using Hölder’s inequality,

|∇B𝛼,𝑑 ∗ 𝜇| ≤ ∥∇B𝛼,𝑑 ∥ 𝐿 𝑝′ (𝐵1) ∥𝜇∥ 𝐿 𝑝 (R𝑑) + 𝑐∥𝜇∥ 𝐿1 (R𝑑) ≤ 𝑐.

3. This results from Proposition 2.1 of [10].

4. Using the ideas of [13, Proposition 4.4], it suffices to show that ∥𝐷2B𝛼,𝑑 ∗ 𝜇∥ 𝐿2 (R𝑑) ≤

𝐶 ∥𝜇∥ 𝐿2 (R𝑑).

Define 𝑣 = B𝛼,𝑑 ∗ 𝜇 so that 𝑣 satisfies −Δ𝑣 + 𝛼𝑣 = 𝜇. By interchanging derivatives via

61

integration by parts twice (or see [20, Theorem 9.9, Corollary 9.10]),

∫

𝐵𝑅

|𝐷2𝑣|2 =

=

∫

𝐵𝑅

∫

(Δ𝑣)2

(𝛼𝑣 − 𝜇)2

𝐵𝑅
∫

≤ 2

𝛼2𝑣2 + 2

∫

𝐵𝑅

𝜇2.

𝐵𝑅

Using Young’s convolution inequality,

𝛼2∥B𝛼,𝑑 ∗ 𝜇∥2

𝐿2 (𝐵𝑅) ≤ ∥B𝛼,𝑑 ∥2

𝐿1 (𝐵𝑅)

𝛼2∥𝜇∥2

𝐿2 (𝐵𝑅)

≤

1
𝛼2

𝛼2∥𝜇∥2

𝐿2 (𝐵𝑅)

= ∥𝜇∥2

𝐿2 (𝐵𝑅)

.

Taking 𝑅 → ∞ and square roots we get ∥𝐷2B𝛼,𝑑 ∗ 𝜇∥ 𝐿2 (R𝑑) ≤ 2∥𝜇∥ 𝐿2 (R𝑑).

5. Define 𝑔(𝑥, 𝑡) =

−1
(4𝜋𝑡)𝑑/2

𝑒−|𝑥|2/(4𝑡), 𝑓 (𝑥, 𝑡) = 𝑒−𝛼𝑡𝑔(𝑥, 𝑡). Then,

𝐹 (𝑥) :=

∫ ∞

0

𝑓 (𝑥, 𝑡) 𝑑𝑡 = −B𝛼,𝑑 (𝑥).

As 𝑔 is lower semi-continuous in 𝑥, ∀𝜖 > 0 ∃𝛿 > 0 such that 𝑔(𝑥0, 𝑡) < 𝑔(𝑥, 𝑡) + 𝜖 for all
𝑥 ∈ 𝐵𝛿 (𝑥0). So, 𝑓 (𝑥0, 𝑡) < 𝑓 (𝑥, 𝑡) + 𝜖 𝑒−𝛼𝑡 holds. So, 𝐹 (𝑥0) < 𝐹 (𝑥) + 𝜖/𝛼 holds for 𝛼 > 0.
Thus, 𝐹 (𝑥) = −B𝛼,𝑑 (𝑥) is lower semi-continuous in 𝑥.

Now we address the items in Assumption 2.1.7.

(i) As with the Newtonian potential, using Young convolution inequality and 𝐿 𝑝 bounds (Lemma

A.0.4) gives the result.

(ii) This is proven in item 4 of this proof.

All of the items are complete and this finishes the proof.

Corollary 2.9.3. The Bessel kernel, −B𝛼,𝑑, is convex for 𝑑 = 1.

Proof. Taking 𝑔(𝑥, 𝑡), 𝑓 (𝑥, 𝑡), 𝐹 (𝑥) in item 5 of the previous proof. We have that 𝑔(𝑥, 𝑡) is convex

in 𝑥 and so therefore 𝐹 (𝑥) = −B𝛼,1(𝑥) is convex in 𝑥 by linearity of the integral.

62

2.10 Numerical Simulations

We now implement the particle method discussed earlier. Here we have simulations in dimension

𝑑 = 1. We explore different targets, aggregation kernels, and initial conditions. We also calculate the

𝐿1 error for the convergence rate in 𝑁 for a fixed final time. We start with some of the basic details.

We define the domain Ω = [−1, 1] where we define a confining potential

𝑉𝑘 (𝑥) =

𝑘

2 (𝑥 + 1)2

𝑘

2 (𝑥 − 1)2

0

if 𝑥 < −1,

if 𝑥 > 1,

otherwise.






The confinement strength is controlled by the value of 𝑘 ∈ N. Most simulation we set 𝑘 = 100,

which is a medium strength confinement. Medium strength confinement is preferred here because

the strong the confinement, the lower the convergence rate in 𝑁 is (see [14]). We choose to use the

Gaussian mollifier 𝜁𝜖 (𝑥) = exp(−𝑥2/2𝜖 2)/

√

2𝜋𝜖 2 where we define 𝜖 = 4/𝑁 0.99. The relationship

between 𝜖 and 𝑁 is better than the expected qualitative results previously mentioned. From Theorem

2.2.2, we define the empirical measure

𝜇𝑁
𝜖,𝑘 (𝑡) =

𝑁
∑︁

𝑖=1

𝛿𝑋 𝑖

𝜖 ,𝑘 (𝑡)𝑚𝑖, 𝑚𝑖 ≥ 0,

𝑁
∑︁

𝑚𝑖 = 1.

𝑖=1
𝜖,𝑘 (0) = (cid:205)𝑁
𝑖=1

Moreover, we define the approximate initial condition as 𝜇𝑁

𝛿𝑋 𝑖
0, 𝜖

𝑚𝑖. The particles are

uniformly spread out in the domain Ω and the mass 𝑚𝑖 is computed from the initial condition 𝜇(0).
In most cases, we choose the initial condition as the Barenblatt profile 𝜓𝜏 (𝑥) = 𝜏−1/3
12

+
with 𝜏 = 0.0625. This corresponds to a more general profile in [9]. The function is chosen because

34/3 − |𝑥|2
𝜏2/3

(cid:16)

(cid:17)

the Barenblatt profile is a solution to the homogeneous (non-weighted) PME. We typically stick

with three weights: uniform 𝑎(𝑥) = 1/2, log-cave 𝑎(𝑥) = 2/𝜋/(1 + |𝑥|2), and piecewise 𝑎(𝑥) = 2/3

for 𝑥 ∈ [−0.75, −0.25) ∪ [0.25, 0.75) and 𝑎(𝑥) = 1/3 otherwise. To visualize 𝜇𝑁

𝜖,𝑘 and compute 𝐿1

errors of 𝜇, we use 𝜁𝜖 ∗ 𝜇𝑁

𝜖,𝑘 instead.

2.10.1 Demonstrating the Particle Method

In most simulations, we choose 𝑁 = 100. We start by demonstrating the particle method

in the case when we only have diffusion (𝑉 = 𝑊 = 0) in Figure 2.1. We have three different

63

weights: uniform 𝑎(𝑥) = 1/2, log-cave 𝑎(𝑥) = 2/𝜋/(1 + |𝑥|2), and piecewise 𝑎(𝑥) = 2/3 for

𝑥 ∈ [−0.75, −0.25) ∪ [0.25, 0.75) and 𝑎(𝑥) = 1/3 otherwise. This echos the simulations in [14].

(a) Uniform weight

(b) Log-concave weight

(c) Piecewise weight

Figure 2.1 Density of diffusion energy with different targets evolves in time.

Figure 2.2 Particles location evolution in time of Figure 2.1.

In Figure 2.3, we look at the diffusion (with uniform weight) and aggregation energies. Here the

attractive Newtonian potential is used for the kernel of the aggregation. Thus, the aggression wants

to bring the particles together while the diffusion wants to spread the particles out. In Figure 2.3a

and 2.4a, the coefficients for both the aggregation and diffusion is one. We see that the diffusion wins

and particles separate. In Figure 2.3b and 2.4b, the coefficient for the aggregation is increased while

keeping the coefficient for the diffusion one. Very quickly the particles come together, however, the

diffusion is strong enough to keep the the particles from moving to the origin. The particles are

located between −0.25 and 0.25. In Figure 2.3c and 2.4c, The aggregation strength falls between

the previous two cases and the diffusion is dramatically decreased. As expected, we see the particles

come together and meet at the origin.

64

(a)

(b)

(c)

Figure 2.3 Density of aggregation and diffusion energy with uniform target evolution in time.

(a)

(b)

(c)

Figure 2.4 Particles location evolution in time of Figure 2.3.

In the previous figures, the initial data considered is the Barenblatt profile. An obvious question

rises that would another initial data be sufficient to get convergence to the steady-state solution.

We would still expect convergence with another initial data. Indeed, Figure 2.5 shows this. Figure

2.5a, a cosine function is used as the initial condition. In particular, (cos(10𝑥) + 2)/(ℎ𝐶) where

𝐶 = 2(sin(10)/10 + 2) is the normalization term and ℎ = |Ω|/𝑁. In Figure 2.5b, the piecewise

function 𝜇0 = 2/3 for 𝑥 ∈ [−0.75, −0.25) ∪ [0.25, 0.75) and 𝜇0 = 1/3 otherwise, is the initial

condition. The particle evolution in time in Figure 2.6 is different than the previous figures.

The particles are still spreading out, however, there is a noticeable partitioning of the particles

corresponding to the number of peaks of the initial data. This is most likely because the particles

initially near the peaks have more mass than the surrounding particles.

65

(a)

(b)

Figure 2.5 Density of diffusion and aggregation energy with different initial data evolution in time.

Figure 2.6 Particles location evolution in time of Figure 2.5.

We can also use different initial data when the energy in consideration is the sum of the diffusion,

aggregation, and drift. We see the evolution of the density of the energy in Figure 2.7.

66

(a) Cosine initial data

(b) Piecewise initial data

Figure 2.7 Density of all energies (diffusion, aggregation, drift) with different initial data evolution
in time.

2.10.2 Convergence rate in 𝑁

In Figure 2.8, we examine the 𝐿1 error of the density (for 𝑁 between 10 and 113) and the density

at 𝑁 = 226. In this case, we are only focusing on the diffusion energy with different weights. We

get the expected results that the convergence rate is faster for the smoother weights (uniform, and

log-concave) and slower for the discontinuous weight (piecewise).

(a) Uniform weight

(b) Log-concave weight

(c) Piecewise weight

Figure 2.8 𝐿1 errors of the density of the diffusion energy for three different targets demonstrating
convergence rate in 𝑁.

Now we do the same experiment now including the aggregation and drift energies in Figure 2.9.

The addition of the energies slightly slowed down the convergence rate in each scenario.

67

(a) Uniform weight

(b) Log-concave weight

(c) Piecewise weight

Figure 2.9 𝐿1 errors of the density of the diffusion, aggregation, and drift energies for different
targets demonstrating convergence rate in 𝑁.

In Figure 2.10, we see if changing the initial data degrades the convergence rate in 𝑁. Indeed,

we see a convergence rate below quadratic with the change of initial data, compared to quadratic

convergence with the Barenblatt profile initial condition and more than quadratic convergence with

only the diffusion energy.

(a) Cosine initial data

(b) Piecewise initial data

Figure 2.10 𝐿1 errors of the density of the diffusion, aggregation, and drift energies for different
initial data demonstrating convergence rate in 𝑁.

2.10.3 Computational Complexity

Based on the ODE (velocity law) of Theorem 2.2.2, we would expect that for a general mollifier

the computation complexity for the drift is 𝑂 (𝑁), the diffusion is 𝑂 (𝑁 2), and the aggregation is

𝑂 (𝑁 3). However, using properties of a specific mollifier, we can do better. Recall that we use a

Gaussian mollifier in these numerical simulations. Using the Fourier transform, one can show that

68

the convolution of two Gaussians is Gaussian. Namely, 𝜑𝜖 := 𝜁𝜖 ∗ 𝜁𝜖 is Gaussian. In particular, the

mean of the new Gaussian is the sum of the means of the Gaussians and the variance of the new

Gaussian is the sum of the variances of the Gaussians.

Thus, we can reduce the aggregation to 𝑂 (𝑁 2),

(cid:164)𝑋𝑖
𝜖,𝑘 (𝑡) = −

𝑁
∑︁

𝑚 𝑗 ∫

∇𝜁𝜖 (𝑋𝑖

𝜖,𝑘 − 𝑧)𝜁𝜖 (𝑋 𝑗

𝜖,𝑘 − 𝑧)

1
𝑎(𝑧)

𝑑𝑧 − ∇𝜁𝜖 ∗ 𝑉 (𝑋𝑖

𝜖,𝑘 ) − ∇𝑉𝑘 (𝑋𝑖

𝜖,𝑘 )

𝑅𝑑

𝑗=1
𝑁
∑︁

𝑗=1

−

𝑚 𝑗 ∇𝜑𝜖 ∗ 𝑊 (𝑋𝑖

𝜖,𝑘 − 𝑋 𝑗

𝜖,𝑘 ).

This in turn reduces the computation complexity of the ODE from 𝑂 (𝑁 3) to 𝑂 (𝑁 2). This can be

seen in Figure 2.11a. In specific scenarios, we can reduce the complexity ever further. If 𝑎(𝑧) is a

scalar (such as the uniform weight), then we can reduce the diffusion to 𝑂 (𝑁),

(cid:164)𝑋𝑖
𝜖,𝑘 (𝑡) = −

−

𝑁
∑︁

𝑗=1
𝑁
∑︁

𝑗=1

𝑚 𝑗 1
𝑎

∇𝜑𝜖 (𝑋𝑖

𝜖,𝑘 − 𝑋 𝑗

𝜖,𝑘 ) − ∇𝜁𝜖 ∗ 𝑉 (𝑋𝑖

𝜖,𝑘 ) − ∇𝑉𝑘 (𝑋𝑖

𝜖,𝑘 )

𝑚 𝑗 ∇𝜑𝜖 ∗ 𝑊 (𝑋𝑖

𝜖,𝑘 − 𝑋 𝑗

𝜖,𝑘 ).

Thus, having the diffusion energy with the uniform weight and the drift (with no aggregation, 𝑊 = 0),

we have a 𝑂 (𝑁) complexity of the ODE. We can observe this in Figure 2.11b.

(a) The complexity of the ODE of the diffusion,
aggregation, and drift.

(b) The complexity of the ODE of the diffusion
(with the uniform weight) and drift.

Figure 2.11 The computational complexity of the ODE (velocity law).

69

CHAPTER 3

INCOMPRESSIBLE LIMIT OF INHOMOGENEOUS POROUS MEDIUM EQUATIONS

3.1

Introduction

The focus of our work is the 𝑚 → ∞ limit (called the incompressible limit or stiff pressure limit)

of the inhomogeneous porous medium equation with reaction,

(3.1)

𝜕𝑡𝑢𝑚 = ∇ ·

(cid:18) 𝑢𝑚
𝑎(𝑥, 𝑡)

(cid:19)

+

∇𝑝𝑚

𝑢𝑚
𝑎(𝑥, 𝑡)

Φ(𝑥, 𝑡, 𝑝𝑚) on R𝑑 × (0, ∞),

where the pressure 𝑝𝑚 is given in terms of the density 𝑢𝑚 by the power law,

𝑝𝑚 =

𝑚
𝑚 − 1

(cid:18) 𝑢𝑚
𝑏(𝑥, 𝑡)

(cid:19) 𝑚−1

, 𝑚 ≥ 2.

It is sometimes useful to rewrite (3.1) as

(3.2)

𝜕𝑡𝑢𝑚 = ∇ ·

(cid:18) 𝑏
𝑎

∇

(cid:16) 𝑢𝑚
𝑏

(cid:17) 𝑚(cid:19)

+

𝑢𝑚
𝑎

Φ(𝑥, 𝑡, 𝑝𝑚).

The coefficients, 𝑎 and 𝑏, which are assumed to be bounded from above and strictly away from zero,

represent heterogeneity in the underlying medium and in the cellular packing density, respectively

[26]. It is also assumed that the growth term Φ is strictly decreasing in 𝑝 and that there exists

𝑝 𝑀 > 0 with Φ(𝑥, 𝑡, 𝑝 𝑀) = 0, which corresponds to a ceiling on the maximum pressure that the

medium can support.

The aim of our work is to study the limit 𝑚 → ∞ of (3.1). In our first result we establish that the

density and pressure converge to the pair (𝑢∞, 𝑝∞), which is the (unique) weak solution of

(3.3)

𝜕𝑡𝑢∞ = ∇ ·

(cid:18) 𝑏
𝑎

∇𝑝∞

(cid:19)

+

𝑢∞
𝑎

Φ(𝑥, 𝑡, 𝑝∞)

𝑝∞(𝑥, 𝑡) ∈ 𝑃∞(𝑢∞(𝑥, 𝑡), 𝑏(𝑥, 𝑡)) almost everywhere ,

where we use the notation 𝑃∞(𝑢, 𝑏) for the Hele-Shaw graph: for any 𝑢, 𝑏 ∈ [0, ∞),

𝑃∞(𝑢, 𝑏) =

0,

0 ≤ 𝑢 < 𝑏,

[0, ∞),

𝑢 = 𝑏.





70

In our next two main results we provide more detail on the behavior of the limiting density and

pressure. The heuristics for this can be seen by examining the equation that the pressure satisfies,

(3.4)

𝜕𝑡 𝑝𝑚 −

|∇𝑝𝑚 |2
𝑎

= (𝑚 − 1)

(cid:18)

𝑝𝑚
𝑎

∇𝑝𝑚 · ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

+ Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚) − 𝑎𝜕𝑡 log(𝑏)

(cid:19)

,

which is obtained by multiplying (3.2) by 𝑚
𝑏
some calculations, it is easier to evaluate the normalized density 𝑣𝑚 = 𝑢𝑚

(cid:0) 𝑢𝑚
𝑏

(cid:1) 𝑚−2 and performing standard manipulations. For

𝑏 , which solves

(3.5)

𝜕𝑡𝑣𝑚 + 𝑣𝑚𝜕𝑡 log(𝑏) =

(cid:18)

1
𝑎

∇𝑣𝑚

𝑚 · ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

+ Δ𝑣𝑚

𝑚 + 𝑣𝑚Φ(𝑥, 𝑡, 𝑝𝑚)

(cid:19)

.

It is natural to guess that, in the 𝑚 → ∞ limit of (3.4), the limit of the term on the right-hand side of

(3.4) should be zero:

(3.6)

(cid:18)

𝑝∞
𝑎

∇𝑝∞ · ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

+ Δ𝑝∞ + Φ(𝑥, 𝑡, 𝑝∞) − 𝑎𝜕𝑡 log(𝑏)

(cid:19)

= 0.

This is the so-called complementarity condition. In Theorem 3.3.3, we prove that (𝑢∞, 𝑝∞) does

indeed satisfy (3.6) in the sense of distributions. This means that, for each time 𝑡, there are two

regions of interest: the region where 𝑝∞ is zero, and the region where 𝑝∞ is positive (and therefore

the term in the parentheses of (3.6) is identically zero). Thus, it is natural to attempt to characterize

the evolution of the boundary between these two regions.

Examining (3.4) suggests that, on this boundary, the limit as 𝑚 → ∞ of the left-hand side of

(3.4) should be identically zero:

𝜕𝑡 𝑝∞
|∇𝑝∞|

=

|∇𝑝∞|
𝑎

.

The left-hand side of the previous line is exactly the normal velocity of the zero level set of 𝑝∞.
Thus, the guess is that this normal velocity is exactly |∇𝑝∞|

. It turns out that this guess is correct,

𝑎

in the absence of an external density; this follows from Proposition 3.3.4, our third main result, in

which we characterize the normal velocity of 𝜕{𝑥 : 𝑝∞(𝑥, 𝑡) = 0} in the sense of comparison with

barriers [21].

3.2 Notation and Assumptions

Throughout, we use the notation 𝑄𝑇 = R𝑑 × (0, 𝑇) and 𝑄 = R𝑑 × (0, ∞). We will also use the

notation 𝑢+ and 𝑢− to denote the positive and negative part of 𝑢, respectively. Throughout, we use 𝐶

71

to denote any positive constant that is independent of 𝑚 but may depend on 𝑑 and the constants Λ,

𝜆, and 𝑝 𝑀 in Assumption 3.2.1. Note that the constant 𝐶 will only depend on 𝑑 in the AB-estimate,

Proposition 3.7.3.

Assumption 3.2.1. Suppose that 𝑎, 𝑏 ∈ 𝐶3(R𝑑 × (0, ∞)) and there exists Λ > 0 such that

1/Λ ≤ 𝑎, 𝑏 ≤ Λ for all (𝑥, 𝑡) ∈ 𝑄𝑇 . Suppose Φ ∈ 𝐶2(R𝑑 × (0, ∞) × [0, 𝑝 𝑀]) satisfies 𝜕𝑝Φ < −𝜆

and Φ(𝑥, 𝑡, 𝑝 𝑀) = 0, for some 𝑝 𝑀 > 0 and 𝜆 > 0.

Example 1 (Example of Φ). One choice of Φ is where the pressure is separate from the space and

time such as Φ(𝑥, 𝑡, 𝑝) = 𝑔( 𝑝)ℎ(𝑥, 𝑡). Here 𝑔 is decreasing, 𝑔( 𝑝 𝑀) = 0, and ℎ is a positive function

that is bounded. A standard choice, as in [23, Fig 1], is to choose a linear function 𝑔( 𝑝) = 𝐶 ( 𝑝 𝑀 − 𝑝),

where 𝐶 > 0.

Assumption 3.2.2 (Initial Data). For some 𝑢0 ∈ 𝐿1(R𝑑), suppose that the initial data 𝑢0

𝑚 satisfies,

𝑚 ≥ 0,
𝑢0




𝑚) ⊂ Ω0 for Ω0 ⊂ R𝑑 compact. There exists a constant 𝐶 > 0 such that

𝑚 − 𝑢0∥ 𝐿1 (R𝑑) → 0 as 𝑚 → ∞,

𝑚 ∥ 𝐿1 (R𝑑) ≤ 𝐶, 𝑖 = 1, . . . , 𝑑,

∥𝜕𝑥𝑖 𝑢0

≤ 𝑝 𝑀,

(cid:16) 𝑢0
𝑚
𝑏

𝑚
𝑚−1

∥𝑢0

(cid:17) 𝑚−1

(3.7)

and supp(𝑢0

(3.8)

∥∇𝑝0

𝑚 ∥ 𝐿2 (R𝑑) + ∥Δ𝑝0

𝑚 ∥ 𝐿2 (R𝑑) + ∥𝜕𝑡 𝑝0

𝑚 ∥ 𝐿1 (R𝑑) ≤ 𝐶.

Assumptions 3.2.1 and 3.2.2 are similar to standard assumptions in the literature, such as in

[12, 16, 17, 21, 23].

Assumption 3.2.3 (Construction of Supersolution). Suppose that either,

1. 𝑎(𝑥, 𝑡) = 𝑎(|𝑥|, 𝑡), 𝑏(𝑥, 𝑡) = 𝑏(|𝑥|, 𝑡) (𝑎, 𝑏 are radial in space),

2. For all 𝑡 ∈ [0, 𝑇], there exists 𝑅(𝑡) > 0, such that if |𝑥| ≥ 𝑅, then we can find an 0 < 𝜖 ≤

𝑑− 1
2
Λ2

such that

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

∇

(cid:18) 𝑏
𝑎

(cid:19)(cid:12)
(cid:12)
(cid:12)
(cid:12)

≤

𝜖
|𝑥|

.

We make precise our notion of weak solution, as in [26, Definition 5.4].

72

Definition 3.2.4 (Notion of weak solution to (3.1)). A non-negative 𝑢 ∈ 𝐿1(𝑄𝑇 ) is a weak solution

of (3.1) with initial data 𝑢0 ∈ 𝐿1(R𝑑) if

∈ 𝐿2(0, 𝑇; 𝐻1(R𝑑))

(cid:1) 𝑚

1. (cid:0) 𝑢
𝑏
2. for 𝑝 = 𝑚
𝑚−1

(cid:0) 𝑢
𝑏

(cid:1) 𝑚−1, 𝑢 satisfies

∬

𝑄𝑇

𝑢𝜕𝑡 𝜁 −

(cid:17) 𝑚

𝑏
𝑎

∇

(cid:16) 𝑢
𝑏

· ∇𝜁 +

𝑢
𝑎

Φ(𝑥, 𝑡, 𝑝)𝜁 = −

∫

R𝑑

𝑢0(𝑥)𝜁 (𝑥, 0),

for any 𝜁 ∈ 𝐶1(𝑄𝑇 ) that vanishes for 𝑡 = 𝑇.

Existence can be achieved by approximating the weak solution with smooth functions that solve

the PME with strictly positive initial data such as in [26, Theorem 5.14] or [26, Section 9.3].

Definition 3.2.5 (Solution to liming problem). Let 𝑇 > 0. We say

(𝑢∞, 𝑝∞) ∈ 𝐶 ([0, 𝑇); 𝐿1(R𝑑)) × 𝐿1( [0, 𝑇); 𝐿1(R𝑑)) ∩ 𝐿2((0, 𝑇); 𝐻1(R𝑑))

is a weak subsolution (resp. supersolution) of (3.3) if for all test functions 𝜁 ∈ 𝐶1

𝑐 (R𝑑 × [0, 𝑇)) we

have

∬

−

𝑄𝑇

𝑢∞𝜕𝑡 𝜁 +

(cid:17) 𝑚

𝑏
𝑎

∇

(cid:16) 𝑢∞
𝑏

· ∇𝜁 −

𝑢∞
𝑎

Φ(𝑥, 𝑡, 𝑝∞)𝜁 ≤

∫

R𝑑

𝑢0(𝑥)𝜁 (𝑥, 0)

(resp. ≥),

and if 𝑝∞(𝑥, 𝑡) ∈ 𝑃∞(𝑢∞(𝑥, 𝑡), 𝑏(𝑥, 𝑡)) and 0 ≤ 𝑢∞(𝑥, 𝑡) ≤ 𝑏(𝑥, 𝑡) hold almost everywhere in 𝑄𝑇 .

We say (𝑢∞, 𝑝∞) is a weak solution of (3.3) if it is both a weak subsolution and a weak supersolution.

3.3 Main Results

Theorem 3.3.1 (Convergence to limiting problem). Suppose Assumptions 3.2.1 and 3.2.2 hold and

fix 𝑇 > 0. Then, up to a subsequence, the density 𝑢𝑚 and the pressure 𝑝𝑚 solution pair of (3.1)

converge strongly in 𝐿1(𝑄𝑇 ) as 𝑚 → ∞ to 𝑢∞, 𝑝∞ respectively, which satisfy

𝑢∞, 𝑝∞ ∈ 𝐵𝑉 (𝑄𝑇 ),

𝑢∞ ∈ 𝐶 𝑠 ([0, ∞); 𝐻−1(R𝑑)) ∩ 𝐶 ( [0, ∞); 𝐿1(R𝑑)) ∀𝑠 < 1
2

, 𝑝∞ ∈ 𝐿2(0, 𝑇; 𝐻1(R𝑑)),

and solve (3.3).

0 ≤ 𝑝∞ ≤ 𝑝 𝑀,

73

Theorem 3.3.2 (Uniqueness of solutions to (3.3)). Suppose Assumptions 3.2.1, 3.2.2, and 3.2.3

hold. Let 𝑇 > 0. Then, there exists a unique pair

(𝑢∞, 𝑝∞) ∈ 𝐶 ([0, 𝑇); 𝐿1(R𝑑)) × 𝐿1((0, 𝑇); 𝐿1(R𝑑)) ∩ 𝐿2((0, 𝑇); 𝐻1(R𝑑))

such that 𝑢(·, 𝑡) is compactly supported for each 0 < 𝑡 < 𝑇, which solves (3.3) in the sense of

Definition 3.2.5.

Theorem 3.3.3 (Complementarity condition). Suppose Assumptions 3.2.1, 3.2.2, and 3.2.3 hold.

Then (3.6) holds in the sense of distributions. More precisely, for any 𝜁 ∈ 𝐶1(𝑄𝑇 ) that vanishes for

𝑡 = 𝑇,

0 =

∬

𝑄𝑇

𝜁

𝑝∞
𝑎

∇𝑝∞ · ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

− |∇𝑝∞|2

𝜁
𝑎

− ∇𝑝∞ · ∇

(cid:19)

(cid:18) 𝜁
𝑎

𝑝∞

∬

+

𝑄𝑇

𝑝∞
𝑎

𝜁Φ(𝑥, 𝑡, 𝑝∞) − 𝑝∞𝜁 𝜕𝑡 log(𝑏).

Let D be a ball in R𝑑. For a time interval [𝑡1, 𝑡2] ⊂ [0, ∞), consider a function (that represent
the pressure) 𝜁 ∈ 𝐶𝑐 ( ¯𝐷 × [𝑡1, 𝑡2]) such that the initial density 𝑢1(𝑥) satisfies 𝑢1(𝑥) = 𝑏(𝑥, 𝑡1) in

{𝜁 (𝑡1) > 0}. For all 𝑥 ∉ {𝜁 (𝑡1) > 0}, we define 𝑡 (𝑥) as the last time that 𝜁 (𝑥, 𝑡) = 0 (with 𝑡 (𝑥) = 𝑡2

is 𝜁 (𝑥, 𝑡2 = 0)) and define the external density

𝑢𝐸
𝜁 (𝑥, 𝑡) = 𝑢1(𝑥) exp

(cid:18)∫ 𝑡

𝑡1

Φ(𝑥, 𝑠, 𝜁 (𝑥, 𝑠))
𝑎(𝑥, 𝑠)

(cid:19)

𝑑𝑠

for all 𝑡 < 𝑡 (𝑥). We assume that the external density satisfies,

𝑢𝐸
𝜁 (𝑥, 𝑡) < 𝑏(𝑥, 𝑡) in {𝜁 = 0}.

The external density solves 𝜕𝑡𝑢 = 𝑢

𝑎 Φ(𝑥, 𝑡, 𝜁) in {𝜁 = 0}. The density in 𝐷 × (𝑡1, 𝑡2) is defined by

𝑢𝜁 (𝑥, 𝑡) = 𝑏(𝑥, 𝑡) 𝜒{𝜁 >0} (𝑥) + 𝑢𝐸

𝜁 (𝑥, 𝑡) (1 − 𝜒{𝜁 >0} (𝑥))

=





𝑏(𝑥, 𝑡), in {𝜁 > 0}

𝑢𝐸
𝜁 (𝑥, 𝑡), in {𝜁 = 0}.

74

Proposition 3.3.4 (Velocity Law). The external density, 𝑢𝐸

∞, is the limit of the density from

outside the saturated region {𝑥 : 𝑝∞(𝑥, 𝑡) > 0}. The normal velocity, 𝑉∞, of the free boundary

𝜕{𝑥 : 𝑝∞(𝑥, 𝑡) > 0} satisfies in a viscosity sense

(cid:18)

1 −

(cid:19)

𝑢𝐸
∞
𝑏

𝑉∞ =

|∇𝑝∞|
𝑎

.

3.4 Estimates

This section is devoted to estimates for solutions of (3.1); these estimates will allow us to take

the incompressible limit and obtain our first main result, Theorem 3.3.1.

Remark 3.4.1. We often manipulate the equation pointwise and/or differentiate the equation. These

manipulations are justified by approximating the solution by the solution with initial data 𝑢0

𝑚 + 𝜖

(that solution is uniformly positive and thus smooth), establishing the desired estimate, and then

taking the limit 𝜖 → 0. See, for example, [26, Section 9.3].

3.4.1 𝐿∞ bound and compact support

We begin by using the comparison principle for (3.1) to establish that the solutions (𝑢𝑚, 𝑝𝑚) of

(3.1) are uniformly bounded and have a (uniformly) finite speed of propagation. For the latter, we

construct appropriate supersolutions to (3.1); our construction is similar to [12, Lemma 3.1].

Lemma 3.4.2 (𝐿∞ bounds and compact support). Suppose that Assumptions 3.2.1, 3.2.2, and 3.2.3

hold and let (𝑢𝑚, 𝑝𝑚) solve (3.1). Then, there exits a constant 𝐶 > 0 such that,

0 ≤ 𝑝𝑚 ≤ 𝑝 𝑀,

0 ≤ 𝑢𝑚 (𝑥, 𝑡) ≤ 𝐶𝑏(𝑥, 𝑡),

0 ≤ 𝑣𝑚 ≤ 𝐶 a.e. 𝑄𝑇 ,

𝑠𝑢 𝑝 𝑝(𝑢𝑚 (·, 𝑡)) ⊂ 𝐵𝐶𝑡

a.e. 0 ≤ 𝑡 ≤ 𝑇 .

Proof. By assumption on the initial data and the comparison principle, 0 ≤ 𝑝𝑚 ≤ 𝑚
𝑚−1

(cid:17) 𝑚−1

(cid:16) 𝑢0
𝑚
𝑏

≤

𝑝 𝑀 a.e. in 𝑄𝑇 . By definition of the density,

0 ≤

𝑢𝑚
𝑏

(cid:18) 𝑚 − 1
𝑚

≤

𝑝 𝑀

(cid:19) 1/(𝑚−1)

→ 1,

as 𝑚 → ∞. Thus, 𝑢𝑚 ≤ 𝐶𝑏 a.e. in 𝑄𝑇 . Moreover, 𝑣𝑚 ≤ 𝐶 a.e. in 𝑄𝑇 . Now, let 𝑍 (𝑥, 𝑡) be as in

Lemma B.0.2 (if Assumption 3.2.3(i) holds) or as in Lemma B.0.3 (if Assumption 3.2.3(ii) holds).

75

In either case, let the constant 𝛼 be chosen large enough, depending on the initial data, to ensure

𝑍 (𝑥, 0) ≥ 𝑢0

𝑚 (𝑥) on R𝑑. The comparison principle for (3.1) thus ensures 𝑢𝑚 (𝑥, 𝑡) ≤ 𝑍 (𝑥, 𝑡) for all

𝑡 > 0. By construction, 𝑍 (𝑥, 𝑡) is compactly supported in 𝑥; therefore, so are 𝑢𝑚 and 𝑝𝑚.

Now that we have compact support and 𝐿∞ bounds, we immediately get 𝐿1 bounds.

Corollary 3.4.3 (𝐿1 bounds for 𝑢𝑚, 𝑝𝑚). Suppose Assumptions 3.2.1 and 3.2.2 hold and let (𝑢𝑚, 𝑝𝑚)

solve (3.1). There exists a constant 𝐶 > 0 such that for 𝑡 ∈ [0, 𝑇] and 𝑚 ≥ 2,

∥𝑢𝑚 (𝑡)∥ 𝐿1 (R𝑑) + ∥𝑣𝑚 (𝑡) ∥ 𝐿1 (R𝑑) + ∥ 𝑝𝑚 (𝑡) ∥ 𝐿1 (R𝑑) ≤ 𝐶.

3.4.2 Derivative bounds

In the next two lemmas, we establish integral bounds for the time and spatial derivatives of the

pressure and density. The techniques are similar to those of [23]; however, more care has to be

taken, especially in the proof of the time derivative bounds, due to the coefficients’ dependence on

space and time.

Lemma 3.4.4 (𝐿1 and 𝐿2 bound for ∇𝑝𝑚). Suppose Assumptions 3.2.1 and 3.2.2 hold, and let

(𝑢𝑚, 𝑝𝑚) solve (3.1). There exists a constant 𝐶 > 0 such that for 𝑚 > 3,

∥∇𝑝𝑚 ∥ 𝐿1 (𝑄𝑇 ) + ∥∇𝑝𝑚 ∥ 𝐿2 (𝑄𝑇 ) ≤ 𝐶.

Proof. First we point out the identity,

|∇𝑝𝑚 |2 + 𝑝𝑚Δ𝑝𝑚 = ∇ · ( 𝑝𝑚∇𝑝𝑚) = Δ( 𝑝2

𝑚).

Rearranging the equation for the pressure (3.4), and using this identity, yields,

|∇𝑝𝑚 |2 =

𝑚 − 1
2(𝑚 − 2)

Δ( 𝑝2

𝑚) −

𝑎𝜕𝑡 𝑝𝑚
𝑚 − 2

+

𝑚 − 1
𝑚 − 2

(cid:18)

𝑝𝑚

Φ(𝑥, 𝑡, 𝑝𝑚) − 𝑎𝜕𝑡 log(𝑏) + ∇𝑝𝑚 · ∇ log

(cid:19)(cid:19)

.

(cid:18) 𝑏
𝑎

We integrate over 𝑄𝑇 and note that, since 𝑝𝑚 is compactly supported in 𝑥, the integral of the

Laplacian term vanishes upon integrating by parts. Thus we find,

∬

𝑄𝑇

|∇𝑝𝑚 |2 ≤

≤

∥𝜕𝑡𝑎∥ 𝐿∞
𝑚 − 2
(cid:18)
𝑚 − 1
𝑚 − 2

𝐶 +

∬

𝑄𝑇

𝑝𝑚 +

𝑚 − 1
𝑚 − 2

𝐶

∬

𝐶
𝑚 − 1

(cid:19) ∬

𝑄𝑇

𝑝𝑚 +

76

𝑝𝑚 +

𝑄𝑇
(cid:18) 𝑚 − 1
𝑚 − 2

𝐶

∬

𝐶

𝑚 − 1
𝑚 − 2
(cid:19) 2 ∬

𝑄𝑇

𝑝2
𝑚 +

𝑝𝑚 |∇𝑝𝑚 |

1
2

∬

𝑄𝑇

|∇𝑝𝑚 |2,

𝑄𝑇

where we’ve also used Young’s inequality. Therefore,

∥∇𝑝𝑚 ∥ 𝐿2 (𝑄𝑇 ) ≤ 𝐶 ∥ 𝑝𝑚 ∥ 𝐿1 (𝑄𝑇 ) + 𝐶 ∥ 𝑝𝑚 ∥ 𝐿2 (𝑄𝑇 ).

According to Lemma 3.4.2, we have 𝑝𝑚 ∈ 𝐿1(𝑄𝑇 ) ∩ 𝐿∞(𝑄𝑇 ). The 𝐿2 bound follows. Using

the compact support of 𝑝𝑚 (Lemma 3.4.2) along with Hölder’s inequality yields the 𝐿1 bound as

well.

Corollary 3.4.5. Suppose that Assumptions 3.2.1, and 3.2.2 hold. For 𝑇 > 0, there exists a constant

𝐶 > 0 such that for 𝑚 > 3,

∥∇𝑣𝑚

𝑚 ∥ 𝐿1 (𝑄𝑇 ) + ∥∇𝑣𝑚

𝑚 ∥ 𝐿2 (𝑄𝑇 ) ≤ 𝐶.

Proof. By Lemma 3.4.2,

|∇𝑣𝑚

𝑚 | = 𝑣𝑚 |∇𝑝𝑚 | ≤ 𝐶 |∇𝑝𝑚 |.

Integrate in 𝑄𝑇 and apply Lemma 3.4.4 to get the result.

Using the previous corollary, we establish the 𝐿1 bound for ∇𝑢𝑚.

Lemma 3.4.6 (𝐿1 bound for ∇𝑢𝑚). Suppose Assumptions 3.2.1 and 3.2.2 hold and let (𝑢𝑚, 𝑝𝑚)

solve (3.1). For 𝑇 > 0, there exists a constant 𝐶 > 0 such that for 𝑚 > 3,

∥𝜕𝑥𝑖 𝑢𝑚 ∥ 𝐿1 (𝑄𝑇 ) + ∥𝜕𝑥𝑖 𝑣𝑚 ∥ 𝐿1 (𝑄𝑇 ) ≤ 𝐶.

Proof. Let 𝜆 = min𝑝∈[0,𝑝 𝑀 ] |𝜕𝑝Φ(𝑥, 𝑡, 𝑝)| > 0. We differentiate (3.5) with respect to 𝑥𝑖, multiply by
sgn(𝜕𝑥𝑖 𝑣𝑚) = sgn(𝜕𝑥𝑖 𝑝𝑚) = sgn(𝜕𝑥𝑖 𝑣𝑚

𝑚) and using Kato’s inequality,

𝜕𝑡 |𝜕𝑥𝑖 𝑣𝑚 | + 𝑣𝑚𝜕𝑥𝑖 𝜕𝑡 log(𝑏)sgn(𝜕𝑥𝑖 𝑣𝑚) + |𝜕𝑥𝑖 𝑣𝑚 |𝜕𝑡 log(𝑏)

≤ sgn(𝜕𝑥𝑖 𝑣𝑚)𝜕𝑥𝑖 (1/𝑎)

(cid:18)

∇𝑣𝑚

𝑚 · ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

+ Δ𝑣𝑚

𝑚 + 𝑣𝑚Φ(𝑥, 𝑡, 𝑝𝑚)

(cid:19)

+ (1/𝑎)(sgn(𝜕𝑥𝑖 𝑣𝑚)∇𝑣𝑚

𝑚 · 𝜕𝑥𝑖 ∇ log(𝑏/𝑎) + ∇|𝜕𝑥𝑖 𝑣𝑚

𝑚 | · ∇ log(𝑏/𝑎) + Δ|𝜕𝑥𝑖 𝑣𝑚

𝑚 | + |𝜕𝑥𝑖 𝑣𝑚 |Φ(𝑥, 𝑡, 𝑝𝑚)

+ 𝑣𝑚 (Φ𝑥𝑖 sgn(𝜕𝑥𝑖 𝑣𝑚) + Φ𝑝 |𝜕𝑥𝑖 𝑝𝑚 |)).

77

Integrating in space, using integration by parts, and rearranging,

𝑑
𝑑𝑡

∫

R𝑑

|𝜕𝑥𝑖 𝑣𝑚 | ≤ 𝐶

∫

R𝑑

𝑣𝑚 + 𝐶

∫

R𝑑

|∇𝑣𝑚

𝑚 | + 𝐶

∫

R𝑑

|𝜕𝑥𝑖 𝑣𝑚 | − 𝜆

∫

R𝑑

𝑣𝑚
𝑎

|𝜕𝑥𝑖 𝑝𝑚 |.

By Gr onwall’s inequality and Corollary 3.4.5,

∥𝜕𝑥𝑖 𝑣𝑚 ∥ 𝐿1 (R𝑑) + 𝜆

∬

𝑄𝑇

𝑣𝑚
𝑎

|𝜕𝑥𝑖 𝑝𝑚 | ≤ 𝑒𝑡𝐶 (∥𝜕𝑥𝑖 𝑣0

𝑚 ∥ 𝐿1 (R𝑑) + 𝐶 ∥𝑣𝑚 ∥ 𝐿1 (𝑄𝑇 )

+ 𝐶 ∥∇𝑣𝑚

𝑚 ∥ 𝐿1 (𝑄𝑇 ))

≤ 𝐶.

Notice that each term on the left-hand side is bounded by this constant. Then,

∥𝜕𝑥𝑖 𝑣𝑚 ∥ 𝐿1 (R𝑑) ≤ 𝐶,

𝜆
∥𝑎∥ 𝐿∞

∬

𝑄𝑇

𝑣𝑚 |𝜕𝑥𝑖 𝑝𝑚 | ≤ 𝐶.

Using the fact that

𝜕𝑥𝑖 𝑢𝑚 = 𝑏𝜕𝑥𝑖 𝑣𝑚 + 𝑢𝑚𝜕𝑥𝑖 log(𝑏),

we achieve the 𝐿1 bound for the density.

We continue with 𝐿1 bounds for 𝜕𝑡𝑢𝑚, 𝜕𝑡 𝑝𝑚.

Lemma 3.4.7 (𝐿1 bounds for 𝜕𝑡𝑢𝑚, 𝜕𝑡 𝑝𝑚). Suppose Assumptions 3.2.1 and 3.2.2 hold and let

(𝑢𝑚, 𝑝𝑚) solve (3.1). For 𝑇 > 0, there exists a constant 𝐶 > 0 such that for 𝑚 > 3,

∥𝜕𝑡𝑢𝑚 ∥ 𝐿1 (𝑄𝑇 ) + ∥𝜕𝑡𝑣𝑚 ∥ 𝐿1 (𝑄𝑇 ) + ∥𝜕𝑡 𝑝𝑚 ∥ 𝐿1 (𝑄𝑇 ) ≤ 𝐶.

Proof. Let 𝜆 = min𝑝∈[0,𝑝 𝑀 ] |𝜕𝑝Φ(𝑥, 𝑡, 𝑝)| > 0. We rearrange (3.5) so that

𝑏𝜕𝑡𝑣𝑚 + 𝑣𝑚𝑏𝜕𝑡 log(𝑏) = ∇ ·

(cid:19)

∇𝑣𝑚
𝑚

(cid:18) 𝑏
𝑎

+ 𝑣𝑚

𝑏
𝑎

Φ(𝑥, 𝑡, 𝑝𝑚).

We differentiate with respect to 𝑡, multiply by sgn(𝜕𝑡𝑣𝑚) = sgn(𝜕𝑡 𝑝𝑚) = sgn(𝜕𝑡𝑣𝑚

𝑚) and using Kato’s

inequality,

𝜕𝑡 (𝑏|𝜕𝑡𝑣𝑚 |) + |𝜕𝑡𝑣𝑚 |𝑏𝜕𝑡 log(𝑏) + 𝑣𝑚𝜕𝑡 𝑏𝜕𝑡 log(𝑏)sgn(𝜕𝑡𝑣𝑚) + 𝑣𝑚𝑏𝜕𝑡 𝜕𝑡 log(𝑏)sgn(𝜕𝑡𝑣𝑚) ≤

≤ |𝜕𝑡∇ (𝑏/𝑎) ||∇𝑣𝑚

𝑚 | + ∇ · (cid:0)(𝑏/𝑎)∇|𝜕𝑡𝑣𝑚

𝑚 |(cid:1) + |𝜕𝑡𝑣𝑚 |(𝑏/𝑎)Φ(𝑥, 𝑡, 𝑝𝑚)

+ 𝑣𝑚𝜕𝑡 (𝑏/𝑎)Φ(𝑥, 𝑡, 𝑝𝑚)sgn(𝜕𝑡𝑣𝑚) + 𝑣𝑚 (𝑏/𝑎)Φ𝑡sgn(𝜕𝑡𝑣𝑚) + 𝑣𝑚 (𝑏/𝑎)Φ𝑝 |𝜕𝑡 𝑝𝑚 |.

78

Integrating in space, using integration by parts, and rearranging,

𝑑
𝑑𝑡

∫

R𝑑

𝑏|𝜕𝑡𝑣𝑚 | ≤ 𝐶

∫

R𝑑

𝑣𝑚 + 𝐶

∫

R𝑑

|∇𝑣𝑚

𝑚 | + 𝐶

∫

R𝑑

𝑏|𝜕𝑡𝑣𝑚 | − 𝜆

∫

R𝑑

𝑣𝑚

𝑏
𝑎

|𝜕𝑡 𝑝𝑚 |.

By Gr onwall’s inequality and Corollary 3.4.5,

∥𝑏𝜕𝑡𝑣𝑚 ∥ 𝐿1 (R𝑑) + 𝜆

∬

𝑄𝑇

𝑏
𝑎

𝑣𝑚 |𝜕𝑡 𝑝𝑚 | ≤ 𝑒𝑡𝐶 (𝐶 ∥𝜕𝑡𝑣0

𝑚 ∥ 𝐿1 (R𝑑) + 𝐶 ∥𝑣𝑚 ∥ 𝐿1 (𝑄𝑇 )

+ 𝐶 ∥∇𝑣𝑚

𝑚 ∥ 𝐿1 (𝑄𝑇 ))

≤ 𝐶.

Notice that each term on the left-hand side is bounded by this constant. Then,

∥𝜕𝑡𝑣𝑚 ∥ 𝐿1 (R𝑑) ≤ 𝐶,

∬

𝑄𝑇

𝑣𝑚 |𝜕𝑡 𝑝𝑚 | ≤ 𝐶.

Bounding the pressure by using the estimates above,

∥𝜕𝑡 𝑝𝑚 ∥ 𝐿1 (𝑄𝑇 ) ≤

∬

𝑚𝑣𝑚−2

𝑚 |𝜕𝑡𝑣𝑚 | +

∬

𝑄𝑇 ∩{𝑣𝑚≥ 1
2 }

2𝑣𝑚 |𝜕𝑡 𝑝𝑚 |

𝑄𝑇 ∩{𝑣𝑚≤ 1
2 }
(cid:19) 𝑚−2
(cid:18) 1
2

𝐶 + 𝐶

≤ 𝑚

≤ 𝐶,

where we use the fact that 𝑚

(cid:17) 𝑚−2

(cid:16) 1
2

the fact that

≤ 2 for all 𝑚 ≥ 2 and 𝜆 is absorbed into the constant. Using

𝜕𝑡𝑢𝑚 = 𝑏𝜕𝑡𝑣𝑚 + 𝑢𝑚𝜕𝑡 log(𝑏),

we achieve the 𝐿1 bound for the density.

3.5 Proof of Convergence to Limiting Problem

Before we establish uniqueness of the limit in a bounded region, we let 𝑢∞ be any subsequential

limit of the density with the same initial condition when establishing the regularity and initial data

of 𝑢∞ below.

Lemma 3.5.1. Suppose Assumptions 3.2.1 and 3.2.2 hold. Let 𝑢∞ be any subsequential limit

of the density with the same initial condition. The sequence {𝑢𝑚} is relatively compact in

𝐶 𝑠 ([0, 𝑇); 𝐻−1(R𝑑)) for 𝑠 ∈ (0, 1/2). Furthermore, 𝑢∞ ∈ 𝐶 𝑠 ( [0, ∞); 𝐻−1(R𝑑)) for 𝑠 ∈ (0, 1/2).

79

Proof. Suppose Assumptions 3.2.1 and 3.2.2 hold. From Lemma 3.4.2, 𝑢𝑚 ∈ 𝐿∞(0, 𝑇; 𝐿1(R𝑑)) ∩

𝐿∞(0, 𝑇; 𝐿∞(R𝑑)) and by interpolation have 𝑢𝑚 ∈ 𝐿∞(0, 𝑇; 𝐿2(R𝑑)). Taking the test function

𝜑 ∈ 𝐻1

0 and using (3.2),

𝜕𝑡𝑢𝑚 𝜑 = −

∫

𝑢𝑚
𝑎

∇𝜑 · ∇𝑝𝑚 +

∫

R𝑑

Φ(𝑥, 𝑡, 𝑝𝑚)

∥∇𝜑∥ 𝐿2 (R𝑑) ∥∇𝑝𝑚 (𝑡) ∥ 𝐿2 (R𝑑) +

≤

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

𝑏
𝑎

R𝑑
(cid:13)
(cid:13)
(cid:13)
(cid:13)𝐿∞

𝑢𝑚
𝑎
(cid:13)
Φ
(cid:13)
(cid:13)
𝑎
(cid:13)

𝜑

(cid:13)
(cid:13)
(cid:13)
(cid:13)𝐿∞

∥𝜑∥ 𝐿2 (R𝑑) ∥𝑢𝑚 (𝑡) ∥ 𝐿2 (R𝑑).

∫

R𝑑

Thus,

∥𝜕𝑡𝑢𝑚 ∥2

𝐿2 (0,𝑇;𝐻 −1 (R𝑑)) ≤ ∥∇𝜑∥2

𝐿2 (R𝑑) ∥∇𝑝𝑚 ∥2

𝐿2 (𝑄𝑇 ) + 𝐶 ∥𝑢𝑚 ∥2

𝐿∞ (0,𝑇;𝐿2 (R𝑑)) ∥𝜑∥2

𝐿2 (R𝑑)

.

Since 𝐻−1(R𝑑) is compactly embedded in 𝐿2(R𝑑), by Lions-Aubin (for reference see [16, Proposition

1.2.5]) we have that {𝑢𝑚} is relatively compact in 𝐶 𝑠 ( [0, 𝑇); 𝐻−1(R𝑑)) for 𝑠 ∈ (0, 1/2). By

compactness, the result on 𝑢∞ follows.

3.5.1 Time Continuity

Lemma 3.5.2. Suppose Assumptions 3.2.1 and 3.2.2 hold. Let 𝑢∞ be any subsequential limit of the

density with the same initial condition. Then the limiting density is continuous in time. In particular,

𝑢∞ ∈ 𝐶 ([0, ∞); 𝐿1(R𝑑)).

Proof. For times 0 < 𝑡1 < 𝑡2 ≤ 𝑇, given that 𝑢∞ solves the limiting PDE and using integration by

parts,

∫

R𝑑

|𝑢∞(𝑡2) − 𝑢∞(𝑡1)| =

=

∫ 𝑡2

∫

𝑡1
∫ 𝑡2

R𝑑

∫

𝑡1

R𝑑

∇ ·

𝑢∞
𝑎

(cid:18) 𝑏
𝑎

∇𝑝∞

(cid:19)

+

𝑢∞
𝑎

Φ(𝑥, 𝑡, 𝑝∞)

Φ(𝑥, 𝑡, 𝑝∞)

≤ 𝐶 (𝑡2 − 𝑡1).

Thus, 𝑢∞ ∈ 𝐶 ([0, ∞); 𝐿1(R𝑑)).

80

3.5.2

Initial Condition

Lemma 3.5.3. Suppose Assumptions 3.2.1 and 3.2.2 hold. Let 𝑢∞ be any subsequential limit of the

density with the same initial condition. Then the limiting density at 𝑡 = 0 coincides with the the

limiting initial condition in 𝐿1. That is, 𝑢∞(0) = 𝑢0 in 𝐿1(R𝑑)

Proof. For time 0 < 𝑡 ≤ 𝑇, given that 𝑢𝑚 is a solution to the PME,

∫

R𝑑

𝑢𝑚 (𝑡) −

∫

R𝑑

𝑢0

𝑚 =

∫ 𝑡

∫

0

R𝑑

∇ ·

(cid:18) 𝑏
𝑎

∇

(cid:16) 𝑢𝑚
𝑏

(cid:17) 𝑚(cid:19)

+

𝑢𝑚
𝑎

Φ(𝑥, 𝑡, 𝑝𝑚).

Letting 𝑚 → ∞,

∫

R𝑑

𝑢∞(𝑡) −

∫

R𝑑

𝑢0
∞ =

∫ 𝑡

∫

0

R𝑑

∇ ·

(cid:18) 𝑏
𝑎

∇𝑝∞

(cid:19)

+

𝑢∞
𝑎

Φ(𝑥, 𝑡, 𝑝∞).

Using integration by parts and letting 𝑡 → 0,

∫

R𝑑

𝑢∞(0) −

∫

R𝑑

𝑢0 = 0.

Thus, 𝑢∞(0) = 𝑢0 in 𝐿1(R𝑑).

3.5.3 Proof of convergence to limiting problem

Proof of Theorem 3.3.1. From previous estimates Lemmas 3.4.2, 3.4.4, 3.4.7, we have that 𝑢𝑚, 𝑝𝑚

are bounded in 𝑊 1,1(𝑄𝑇 ). By Rellich-Kondrachov, 𝑢𝑚, 𝑝𝑚 converge (up to a subsequence) strongly

in 𝐿1(𝑄𝑇 ).

Integrating (3.2) against a test function 𝜓 ∈ 𝐶1(𝑄𝑇 ) that vanishes for 𝑡 = 𝑇,

∬

𝑄𝑇

𝑢𝑚𝜕𝑡𝜓 +

(cid:17) 𝑚

(cid:16) 𝑢𝑚
𝑏

∇ ·

(cid:19)

+

∇𝜓

(cid:18) 𝑏
𝑎

𝑢𝑚
𝑎

Φ(𝑥, 𝑡, 𝑝𝑚)𝜓 = −

∫

R𝑑

𝑚𝜓(𝑥, 0).
𝑢0

As 𝑢𝑚, 𝑝𝑚 converges strongly in 𝐿1(𝑄𝑇 ), we obtain that

𝜕𝑡𝑢∞ = ∇ ·

(cid:18) 𝑏
𝑎

∇𝑝∞

(cid:19)

+

𝑢∞
𝑎

Φ(𝑥, 𝑡, 𝑝∞)

in D′(𝑄).

Using the definition of 𝑝𝑚 and rearranging

𝑢𝑚
𝑏

(cid:18) 𝑚 − 1
𝑚

=

𝑝𝑚

(cid:19) 1/(𝑚−1)

⇒

𝑢𝑚
𝑏

𝑝𝑚 =

(cid:18) 𝑚 − 1
𝑚

(cid:19) 1/(𝑚−1)

𝑝𝑚/(𝑚−1)
𝑚

.

81

Up to a subsequence, we can pass to the limit a.e. so that 𝑢∞ 𝑝∞/𝑏 = 𝑝∞ or equivalently,

(cid:16)

1 −

(cid:17)

𝑢∞
𝑏

𝑝∞ = 0

This gets us 𝑝∞ ∈ 𝑃∞(𝑢∞, 𝑏). We also obtain almost everywhere in 𝑄𝑇 that 0 ≤ 𝑢∞ ≤ 𝑏, 0 ≤ 𝑝∞ ≤

𝑝 𝑀, and 𝑢∞, 𝑝∞ ∈ 𝐵𝑉 (𝑄𝑇 ) for all 𝑇 > 0.

3.6 Uniqueness of the Limit

To establish uniqueness of solutions to (3.3), we follow the Hilbert’s duality method, which was

used for the homogeneous version of (3.3) in [23].

For the remainder of this section, let us fix two non-negative densities 𝑢1, 𝑢2 with corresponding

pressures 𝑝1, 𝑝2, solving (3.3). Let Ω be a bounded domain containing the supports of both solutions

for all time 𝑡 ∈ [0, 𝑇] and Ω𝑇 = Ω × (0, 𝑇). For ease of notation, we will abbreviate Φ(𝑥, 𝑡, 𝑝) as

Φ( 𝑝).

First we shall prove that the densities must agree. To this end, we use the definition of weak

solution to find,

(3.9)

∬

0 =

Ω𝑇

(𝑢1 − 𝑢2)𝜕𝑡𝜓 + ( 𝑝1 − 𝑝2)∇ ·

(cid:19)

+

∇𝜓

(cid:18) 𝑏
𝑎

𝜓
𝑎

(𝑢1Φ( 𝑝1) − 𝑢2Φ( 𝑝2)).

We shall denote 𝑍 := 𝑢1 − 𝑢2 + 𝑝1 − 𝑝2 and define

𝐴 :=

𝑢1 − 𝑢2
𝑍

, 𝐵 :=

𝑝1 − 𝑝2
𝑍

, 𝐶 := (−𝑢2)

Φ( 𝑝1) − Φ( 𝑝2)
𝑝1 − 𝑝2

.

We define 𝐴 = 0 when 𝑢1 = 𝑢2 (even when 𝑝1 = 𝑝2) and similarly set 𝐵 = 0 when 𝑝1 = 𝑝2 (even

when 𝑢1 = 𝑢2), which yields the bounds,

(3.10)

0 ≤ 𝐴 ≤ 1,

0 ≤ 𝐵 ≤ 1,

0 ≤ 𝐶 ≤ 𝜈.

With this in hand, we see that (3.9) may be rewritten as,

(3.11)

∬

0 =

(cid:18)

𝑍

Ω𝑇

𝐴𝜕𝑡𝜓 + 𝐵∇ ·

(cid:19)

∇𝜓

(cid:18) 𝑏
𝑎

+ 𝐴Φ( 𝑝1)

𝜓
𝑎

− 𝐶𝐵

(cid:19)

.

𝜓
𝑎

82

If, given any smooth 𝐺, we could find 𝜓 solving the dual problem,

𝐴𝜕𝑡𝜓 + 𝐵∇ ·

(cid:17)

(cid:16) 𝑏
𝑎 ∇𝜓

+ 𝐴Φ( 𝑝1)

𝜓

𝑎 − 𝐶𝐵 𝜓

𝑎 = 𝐴𝐺 in Ω𝑇 ,

𝜓 = 0

in 𝜕Ω × (0, 𝑇), 𝜓(·, 𝑇) = 0 in Ω,





then, by taking 𝜓 as the test function in (3.11), we would obtain

∬

0 =

Ω𝑇

(𝑢1 − 𝑢2 + 𝑝1 − 𝑝2) 𝐴𝐺 =

∬

Ω𝑇

(𝑢1 − 𝑢2)𝐺.

From this we have uniqueness for the density, as the smooth function 𝐺 in the previous line is

arbitrary.

However, since we may not be able to solve the dual problem due to the degeneracy of the

coefficients 𝐴, 𝐵, 𝐶, we proceed by an approximation argument. Let {𝐴𝑛}, {𝐵𝑛}, {𝐶𝑛}, {Φ1,𝑛} be

sequences of smooth bounded functions such that

∥ 𝐴 − 𝐴𝑛 ∥ 𝐿2 (Ω𝑇 ) < 𝐾/𝑛,

1/𝑛 < 𝐴𝑛 ≤ 1,

∥𝐵 − 𝐵𝑛 ∥ 𝐿2 (Ω𝑇 ) < 𝐾/𝑛,

1/𝑛 < 𝐵𝑛 ≤ 1,

∥𝐶 − 𝐶𝑛 ∥ 𝐿2 (Ω𝑇 ) < 𝐾/𝑛,

0 ≤ 𝐶𝑛 < 𝐾,

∥𝜕𝑡𝐶𝑛 ∥ 𝐿1 (Ω𝑇 ) ≤ 𝐾

∥Φ( 𝑝1) − Φ1,𝑛 ∥ 𝐿2 (Ω𝑇 ) < 𝐾/𝑛,

|Φ1,𝑛| < 𝐾,

∥∇Φ1,𝑛 ∥ 𝐿2 (Ω𝑇 ) ≤ 𝐾






where 𝐾 is a positive constant. Fix a smooth function 𝐺. Standard theory for parabolic PDEs yields

that there exists a unique solution 𝜓𝑛 to the regularized dual problem

(RDP)

𝜕𝑡𝜓𝑛 + 𝐵𝑛
𝐴𝑛

∇ ·

(cid:16) 𝑏
𝑎 ∇𝜓𝑛

(cid:17)

+ Φ1,𝑛

𝜓𝑛
𝑎 − 𝐶𝑛

𝐵𝑛
𝐴𝑛

𝜓𝑛
𝑎 = 𝐺 in Ω𝑇 ,

𝜓𝑛 = 0 in 𝜕Ω × (0, 𝑇), 𝜓𝑛 (·, 𝑇) = 0 in Ω.





We shall establish some estimates on the solution 𝜓𝑛:

Lemma 3.6.1. There are constants 𝜅𝑖 = 𝜅𝑖 (𝑎, 𝑏, 𝑇, 𝐺, |Ω|) for 𝑖 = 1, 2, 3 such that

∥𝜓𝑛 ∥ 𝐿∞ (Ω𝑇 ) ≤ 𝜅1,
(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)

(cid:18) 𝐵𝑛
𝐴𝑛

(cid:18) 𝑏
𝑎

(cid:19) 1/2 (cid:18)

∇ ·

sup
0≤𝑡≤𝑇
(cid:19)

∇𝜓𝑛

−

∥∇𝜓𝑛 (𝑡) ∥ 𝐿2 (Ω) ≤ 𝜅2

𝐶𝑛
𝑎

𝜓𝑛

(cid:19)(cid:13)
(cid:13)
(cid:13)
(cid:13)
(cid:13)𝐿2 (Ω𝑇 )

≤ 𝜅3.

83

Proof. We will follow the proof of [23, Lemma 3.1], which the first bound is obtained in the same

way (also see [12, Lemma 4.1]). Multiplying (RDP) by ∇ ·

(cid:16) 𝑏
𝑎 ∇𝜓𝑛

(cid:17)

− 𝐶𝑛

𝑎 𝜓𝑛,

∫

Ω

𝑏(𝑡)
2𝑎(𝑡)

|∇𝜓𝑛 (𝑡)|2 +

(cid:19)

−

∇𝜓𝑛

|∇𝜓𝑛|2 −

𝜓𝑛

𝐶𝑛
𝑎
∫ 𝑇

𝑡

(cid:12)
2
(cid:12)
(cid:12)
(cid:12)
∫

𝜕𝑡

Ω
Φ1,𝑛
𝑎

𝑏
𝑎

(cid:19)

· ∇𝜓𝑛 +

∇ ·

(cid:18) 𝑏
𝑎

𝜕𝑡

𝜓𝑛∇

(cid:18) 𝑏
𝑎
(cid:19) 1
2
(cid:18) Φ1,𝑛
𝑎

∫ 𝑇

∫

𝐵𝑛
𝐴𝑛
Ω
∫ 𝑇
∫

𝑡

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

= −

𝑡
∫ 𝑇

∫

+

+

𝑡
∫ 𝑇

Ω
∫

𝑡

Ω

Ω
(cid:32) 𝑏
𝑎

(cid:18) 𝑏
𝑎

∇𝐺 · ∇𝜓𝑛 +

(cid:19)

.

𝐺𝜓𝑛

𝐶𝑛
𝑎

(cid:18) 𝐶𝑛
𝑎

(cid:19) 1
2

𝜓2

𝑛 −

∫

Ω

|∇𝜓𝑛|2 + Φ1,𝑛𝐶𝑛

(𝑡)

(cid:19)

𝜓2
𝑛

(cid:19) 2(cid:33)

(cid:18) 𝐶𝑛
2𝑎
(cid:18) 𝜓𝑛
𝑎

Bounding the right-hand side,

∫

Ω

|∇𝜓𝑛 (𝑡)|2 +

∫ 𝑇

∫

𝑡

Ω

𝐵𝑛
𝐴𝑛

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

∇ ·

(cid:19)

−

∇𝜓𝑛

(cid:18) 𝑏
𝑎

𝐶𝑛
𝑎

𝜓𝑛

2

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

(cid:18)

≤ 𝐾

1 + 𝑡 +

∫ 𝑇

∫

𝑡

Ω

(cid:19)

,

|∇𝜓𝑛|2

where 𝐾 is independent of 𝑛 but contains various constants. From here, we use Grönwall’s inequality

to get the second bound. The third bound is obtained when using the equation above and the second

bound.

With these estimates in hand, we proceed with the uniqueness proof. Combining (3.11) and

(RDP),

∬

Ω𝑇

where

(𝑢1 − 𝑢2)𝐺 = (𝑢1 − 𝑢2)

(cid:18)

𝜕𝑡𝜓𝑛 +

∇ ·

∇𝜓𝑛

+ Φ1,𝑛

− 𝐶𝑛

(cid:19)

(cid:18) 𝑏
𝑎
(cid:19)

𝐵𝑛
𝐴𝑛
(cid:18) 𝑏
𝑎

𝜓𝑛
𝑎
𝜓𝑛
𝑎

(cid:19)

𝜓𝑛
𝑎

𝐵𝑛
𝐴𝑛
(cid:19)
𝜓𝑛
𝑎

𝐴𝜕𝑡𝜓𝑛 + 𝐵∇ ·

∇𝜓𝑛

+ 𝐴Φ( 𝑝1)

− 𝐶𝐵

(cid:18)

− 𝑍

= 𝐼1

𝑛 − 𝐼2

𝑛 − 𝐼3

𝑛,
𝑛 + 𝐼4

𝐼1
𝑛 =

𝐼2
𝑛 =

𝐼3
𝑛 =

𝐼4
𝑛 =

∬

Ω𝑇

∬

Ω𝑇

∬

Ω𝑇

∬

Ω𝑇

(cid:18)

( 𝐴 − 𝐴𝑛)

𝑍

𝐵𝑛
𝐴𝑛

𝑍 (𝐵 − 𝐵𝑛)

(cid:18)

∇ ·

∇ ·

(cid:18) 𝑏
𝑎

(cid:18) 𝑏
𝑎

∇𝜓𝑛

(cid:19)

−

(cid:19)

𝜓𝑛

𝐶𝑛
𝑎

∇𝜓𝑛

(cid:19)

−

(cid:19)

𝜓𝑛

𝐶𝑛
𝑎

𝑢1 − 𝑢2
𝑎
(cid:18) 𝐶
𝑎

𝑍 𝐵

(Φ( 𝑝1) − Φ1,𝑛)𝜓𝑛

(cid:19)

𝐶𝑛
𝑎

−

𝜓𝑛.

84

Our goal is to show that 𝐼𝑖

𝑛 → 0 as 𝑛 → 0 for 𝑖 = 1, 2, 3, 4. Now we bound the integrals above by

using Hölder’s inequality, bounds of the coefficients (namely ( 𝐴𝑛/𝐵𝑛)1/2, (𝐵𝑛/𝐴𝑛)1/2 ≤ 𝑛1/2), and

the convergence of the coefficients,

|𝐼1

𝑛 | ≤ 𝐾

|𝐼2

𝑛 | ≤ 𝐾

∬

Ω𝑇

∬

Ω𝑇

𝐵𝑛
𝐴𝑛

| 𝐴 − 𝐴𝑛|

|𝐵 − 𝐵𝑛|

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

∇ ·

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

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

≤

𝐾
𝑛1/2

,

(cid:19)

−

∇𝜓𝑛

(cid:18) 𝑏
𝑎

𝐶𝑛
𝑎

𝜓𝑛

(cid:19)

−

∇𝜓𝑛

𝐶𝑛
𝑎

𝜓𝑛

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

∇ ·

(cid:18) 𝑏
𝑎

≤ 𝐾 ∥( 𝐴𝑛/𝐵𝑛)1/2(𝐵 − 𝐵𝑛) ∥ 𝐿2 (Ω𝑇 )

,

𝐾
𝑛1/2
∬

≤

|𝐼3

𝑛 | ≤

Ω𝑇
∬

≤ 𝐾

|𝑢1 − 𝑢2|
𝑎

|Φ( 𝑝1) − Φ1,𝑛||𝜓𝑛|

|𝑢1 − 𝑢2||Φ( 𝑝1) − Φ1,𝑛|

Ω𝑇

≤ 𝐾 ∥Φ( 𝑝1) − Φ1,𝑛 ∥ 𝐿2 (Ω𝑇 )

≤

𝐾
𝑛

|𝐼4

𝑛 | ≤ 𝐾

,

∬

Ω𝑇

|𝐶𝑛 − 𝐶 ||𝜓𝑛|

≤ 𝐾 ∥𝐶𝑛 − 𝐶 ∥ 𝐿2 (Ω𝑇 )

≤

𝐾
𝑛

,

where 𝐾 = 𝐾 (𝑎, 𝑏, 𝑇, 𝐺, |Ω𝑇 |).

For uniqueness of the pressure, using uniqueness of the density (𝑢1 = 𝑢2) and defining

𝜓 := 𝑝1 − 𝑝2,

∬

0 =

Ω𝑇
∬

= −

(𝑢1 − 𝑢2)𝜕𝑡𝜓 + ( 𝑝1 − 𝑝2)∇ ·

𝑏
𝑎

|∇( 𝑝1 − 𝑝2)|2 +

∬

Ω𝑇

Ω𝑇

∇𝜓

(cid:18) 𝑏
𝑎
𝑝1 − 𝑝2
𝑎

(cid:19)

+

𝜓
𝑎

(𝑢1Φ( 𝑝1) − 𝑢2Φ( 𝑝2))

(𝑢1Φ( 𝑝1) − 𝑢1Φ( 𝑝2)).

Recalling that Φ strictly decreasing in 𝑝 implies sgn( 𝑝1 − 𝑝2) = −sgn(Φ( 𝑝1) − Φ( 𝑝2)), the density

85

is non-negative, and 𝑎, 𝑏 are strictly positive,

|∇( 𝑝1 − 𝑝2)|2

( 𝑝1 − 𝑝2) (Φ( 𝑝1) − Φ( 𝑝2))

𝑏
𝑎
𝑢1
𝑎

∬

0 ≤

Ω𝑇

∬

Ω𝑇

=

≤ 0.

Given that the pressures have the same initial condition, we achieve the uniqueness of the pressure.

Now that we have uniqueness for the limiting solution, we get a comparison principle.

Corollary 3.6.2 (Comparison principle). Suppose Assumptions 3.2.1, 3.2.2, and 3.2.3 hold. Let

(𝑢∞, 𝑝∞) be the limit solution of (3.1) in 𝑄𝑇 . Let (𝑢1, 𝑝1) be a weak solution of (3.1) in Ω × [𝑡1, 𝑡2].

If 𝑝∞ ≤ 𝑝1 on 𝜕Ω × [𝑡1, 𝑡2] and 𝑢∞ ≤ 𝑢1 when 𝑡 = 𝑡1, then 𝑝∞ ≤ 𝑝1 and 𝑢∞ ≤ 𝑢1 in Ω × [𝑡1, 𝑡2].

3.7 Complementarity Condition

By Lemma 3.4.2, let Ω be a compact domain containing the support of 𝑝𝑚 for almost every time

𝑡 ∈ [0, 𝑇] and Ω𝑇 = Ω × (0, 𝑇). As shown in [23], the complementarity condition is equivalent

to strong 𝐿2 convergence of the pressure gradient. We use ideas in [18]. In particular, an 𝐿3

AB-estimate is obtained to get space compactness, instead of the classical 𝐿∞ AB-estimate. The 𝐿3

bound of the pressure gradient gives enough compactness needed to pass the limit. Indeed, to get

the 𝐿3 AB-estimate we first require the 𝐿3 bound for the pressure gradient.

3.7.1 𝐿3 bound for ∇𝑝𝑚

Proposition 3.7.1 (𝐿3 bound for ∇𝑝𝑚). Suppose Assumptions 3.2.1, 3.2.2, and 3.2.3 hold. For

𝑇 > 0 and 𝑚 > 4, there exists 𝐶 > 0, independent of 𝑚, such that

∬

Ω𝑇

𝑝𝑚 (Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2 ≤ 𝐶 (𝑇, 𝑎, 𝑏, 𝑝 𝑀, Φ)

and

∬

Ω𝑇

|∇𝑝𝑚 |3 ≤ 𝐶 (𝑇, 𝑎, 𝑏, 𝑝 𝑀, Φ).

86

Proof. By integration by parts, Young’s inequality, and Lemma 3.4.4,

∬

Ω𝑇

|∇𝑝𝑚 |3 = −2

∬

≤

∬

Ω𝑇

𝑝𝑚Δ𝑝𝑚 |∇𝑝𝑚 |

𝑚 |Δ𝑝𝑚 |2 +
𝑝2

∬

Ω𝑇

|∇𝑝𝑚 |2

𝑝𝑚 |Δ𝑝𝑚 |2 + 𝐶.

Ω𝑇

∬

≤ 𝑝 𝑀

Ω𝑇

It is sufficient to show that the right hand-side is bounded. By triangle inequality, it is enough to
show that ∬

𝑝𝑚 (Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2 is controlled. Multiplying the pressure equation (3.4) by

Ω𝑇

−(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚)), and integrating in space and time,

∫ 𝑇

0

𝑑
𝑑𝑡

∫

Ω

|∇𝑝𝑚 |2
2

+ (𝑚 − 1)

+ 𝐼1 + 𝐼2 + 𝐼3 + 𝐼4 = −

∬

Ω𝑇

∬

𝑝𝑚
𝑎
Ω𝑇
|∇𝑝𝑚 |2
𝑎

Φ(𝑥, 𝑡, 𝑝𝑚)

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2

where

𝐼1 = −(𝑚 − 1)

∬

Ω𝑇

𝑝𝑚 (Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))𝜕𝑡 log(𝑏)

∬

Ω𝑇

𝑝𝑚
𝑎

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))

(cid:18)

∇𝑝𝑚 · ∇ log

(cid:19)(cid:19)

(cid:18) 𝑏
𝑎

𝐼2 = (𝑚 − 1)

𝐼3 = −

∬

Ω𝑇

𝜕𝑡 𝑝𝑚Φ(𝑥, 𝑡, 𝑝𝑚)

𝐼4 =

∬

Ω𝑇

Δ𝑝𝑚

|∇𝑝𝑚 |2
𝑎

.

Integrating by parts,

𝐼4 =

∬

Ω𝑇

𝑝𝑚

(cid:18) Δ|∇𝑝𝑚 |2
𝑎

+ |∇𝑝𝑚 |2Δ

(cid:19)

(cid:18) 1
𝑎

+ 2∇

(cid:19)

(cid:18) 1
𝑎

(cid:19)

· ∇|∇𝑝𝑚 |2

= 𝐼4,1 + 𝐼4,2 + 𝐼4,3.

87

Using integration by parts

𝐼4,1 = 2

= 2

∬

Ω𝑇

∬

Ω𝑇

− 2

𝑝𝑚
𝑎

𝑝𝑚
𝑎

∑︁

𝑖, 𝑗

∑︁

𝑖, 𝑗

(𝜕2

𝑖, 𝑗 𝑝𝑚)2 + 2

(𝜕2

𝑖, 𝑗 𝑝𝑚)2 − 2

∬

𝑝𝑚Δ𝑝𝑚∇𝑝𝑚 · ∇

∬

= 2

Ω𝑇

Ω𝑇
𝑝𝑚
𝑎

∑︁

𝑖, 𝑗

(𝜕2

𝑖, 𝑗 𝑝𝑚)2 − 2

∬

Ω𝑇

∬

Ω𝑇
(cid:19)

(cid:18) 1
𝑎
∬

Ω𝑇

𝑝𝑚
𝑎

𝑝𝑚
𝑎

𝑝𝑚
𝑎

∇𝑝𝑚∇Δ𝑝𝑚

|Δ𝑝𝑚 |2 − 2

∬

Ω𝑇

Δ𝑝𝑚
𝑎

|∇𝑝𝑚 |2

|Δ𝑝𝑚 |2 − 2𝐼4 − 2

∬

Ω𝑇

𝑝𝑚Δ𝑝𝑚∇𝑝𝑚 · ∇

(cid:19)

(cid:18) 1
𝑎

.

Thus combining the 𝐼4 terms,

𝐼4 = −

2
3

∬

Ω𝑇

𝑝𝑚
𝑎

|Δ𝑝𝑚 |2 +

2
3

∬

Ω𝑇

𝑝𝑚
𝑎

∑︁

𝑖, 𝑗

(𝜕2

𝑖, 𝑗 𝑝𝑚)2 −

2
3

∬

Ω𝑇

𝑝𝑚Δ𝑝𝑚∇𝑝𝑚 · ∇

(cid:19)

(cid:18) 1
𝑎

+

1
3

𝐼4,2 +

1
3

𝐼4,3.

So,

∫ 𝑇

0

𝑑
𝑑𝑡

∫

Ω

|∇𝑝𝑚 |2
2

+ 𝐼1 + 𝐼2 + 𝐼3 −

∬

Ω𝑇

𝑝𝑚
𝑎

|Δ𝑝𝑚 |2 +

+ (𝑚 − 1)

2
3

∬

Ω𝑇

𝑝𝑚
𝑎

∬

≤ −

Ω𝑇

|∇𝑝𝑚 |2
𝑎

Φ(𝑥, 𝑡, 𝑝𝑚) −

1
3

𝐼4,2 −

Lemmas 3.4.2 and 3.4.4 imply

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2

𝑝𝑚
𝑎

(𝜕2

𝑖, 𝑗 𝑝𝑚)2

∬

∑︁

Ω𝑇

𝑖, 𝑗

𝐼4,3 +

2
3

2
3

1
3

∬

Ω𝑇

𝑝𝑚Δ𝑝𝑚∇𝑝𝑚 · ∇

(cid:19)

(cid:18) 1
𝑎

.

∬

−

Ω𝑇

|∇𝑝𝑚 |2
𝑎

Φ(𝑥, 𝑡, 𝑝𝑚) −

1
3

𝐼4,2 ≤

(cid:18)

∥

𝜙
𝑎

∥ 𝐿∞ + 𝐶 ∥Δ(1/𝑎) ∥ 𝐿∞ 𝑝 𝑀

(cid:19) ∬

Ω𝑇

|∇𝑝𝑚 |2 ≤ 𝐶.

Young’s inequality and Lemma 3.4.4 imply,

2
3

∬

Ω𝑇

𝑝𝑚Δ𝑝𝑚∇𝑝𝑚 · ∇

(cid:19)

(cid:18) 1
𝑎

≤

≤

1
3
1
3

∬

Ω𝑇

∬

Ω𝑇

𝑝𝑚
𝑎
𝑝𝑚
𝑎

|Δ𝑝𝑚 |2 + 𝐶

∬

Ω𝑇

𝑝𝑚
𝑎

|∇𝑝𝑚 |2 |∇𝑎|2
𝑎2

|Δ𝑝𝑚 |2 + 𝐶.

Calculating we have,

𝐼4,3 = −4

∬

Ω𝑇

𝑝𝑚
𝑎

∑︁

𝑖, 𝑗

(𝜕2

𝑖, 𝑗 𝑝𝑚)

∇𝑝𝑚 · ∇𝑎
𝑎

88

and

−1
3

𝐼4,3 ≤

≤

1
3

1
3

∬

Ω𝑇

∬

Ω𝑇

𝑝𝑚
𝑎

𝑝𝑚
𝑎

∑︁

𝑖, 𝑗

∑︁

𝑖, 𝑗

(𝜕2

𝑖, 𝑗 𝑝𝑚)2 + 𝐶

∬

Ω𝑇

𝑝𝑚
𝑎

|∇𝑝𝑚 |2 |∇𝑎|2
𝑎2

(𝜕2

𝑖, 𝑗 𝑝𝑚)2 + 𝐶.

Combining like terms,

∫ 𝑇

0

𝑑
𝑑𝑡

∫

Ω

|∇𝑝𝑚 |2
2

+ (𝑚 − 1)

∬

Ω𝑇

𝑝𝑚
𝑎

∬

Ω𝑇

𝑝𝑚
𝑎

|Δ𝑝𝑚 |2 +

1
3

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2

∬

∑︁

Ω𝑇

𝑖, 𝑗

𝑝𝑚
𝑎

(𝜕2

𝑖, 𝑗 𝑝𝑚)2 ≤ 𝐶.

+ 𝐼1 + 𝐼2 + 𝐼3 −

Now using Young’s inequality in each of −𝐼1 and −𝐼2, then adding, and finally using Assumption

3.2.1 and Lemma 3.4.4 yields,

−𝐼1 − 𝐼2 ≤

(𝑚 − 1)
2

∬

Ω𝑇

∬

+ (𝑚 − 1)𝐶

≤

(𝑚 − 1)
2

Ω𝑇

∬

Ω𝑇

𝑝𝑚
𝑎
𝑝𝑚
𝑎
𝑝𝑚
𝑎

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2

𝑎2|𝜕𝑡 log(𝑏)|2 + (𝑚 − 1)𝐶

∬

Ω𝑇

𝑝𝑚
𝑎

|∇𝑝𝑚 |2|∇ log(𝑏/𝑎)|2

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2 + (𝑚 − 1)𝐶.

Thus,

where

∫ 𝑇

0

𝑑
𝑑𝑡

∫

Ω

|∇𝑝𝑚 |2
2

+ 𝐼3 + 𝐼5 ≤ (𝑚 − 1)𝐶,

𝐼5 =

(𝑚 − 1)
2

∬

Ω𝑇

𝑝𝑚
𝑎

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2 −

∬

Ω𝑇

𝑝𝑚
𝑎

|Δ𝑝𝑚 |2 +

1
3

∬

Ω𝑇

𝑝𝑚
𝑎

∑︁

𝑖, 𝑗

(𝜕2

𝑖, 𝑗 𝑝𝑚)2.

Using the fact that Φ(𝑥, 𝑡, 𝑝) ≥ 0 yields,

𝐼5 ≥

=

(𝑚 − 3)
2
(𝑚 − 3)
2

∬

Ω𝑇

∬

Ω𝑇

𝑝𝑚
𝑎
𝑝𝑚
𝑎

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2 +

∬

Ω𝑇

𝑝𝑚
𝑎

(Δ𝑝𝑚)2 −

∬

Ω𝑇

𝑝𝑚
𝑎

|Δ𝑝𝑚 |2

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2.

Define Ψ(𝑥, 𝑡, 𝑝𝑚) = ∫ 𝑝𝑚

0

Φ(𝑥, 𝑡, 𝑞) 𝑑𝑞. Then, 𝜕𝑡Ψ = Φ(𝑥, 𝑡, 𝑝𝑚)𝜕𝑡 𝑝𝑚 + ∫ 𝑝𝑚

0

Φ𝑡 (𝑥, 𝑡, 𝑞) 𝑑𝑞. Given

that

∫ 𝑝𝑚

0

Φ𝑡 (𝑥, 𝑡, 𝑞) 𝑑𝑞 ≤ ∥Φ𝑡 ∥ 𝐿∞ 𝑝 𝑀,

89

then

Therefore,

𝐼3 ≥ −

∫ 𝑇

0

𝑑
𝑑𝑡

∫

Ω

Ψ − 𝐶.

∫ 𝑇

0

𝑑
𝑑𝑡

∫

Ω

|∇𝑝𝑚 |2
2

∫

−

Ω(𝑇)

Ψ(𝑥, 𝑇, 𝑝𝑚) +

(𝑚 − 3)
2

∬

Ω𝑇

𝑝𝑚
𝑎

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2 ≤ (𝑚 − 1)𝐶.

Moreover by using previous bounds and Assumption 3.2.2,

(𝑚 − 3)
2

∬

Ω𝑇

𝑝𝑚
𝑎

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2 ≤ (𝑚 − 1)𝐶.

Therefore we get,

which implies that

This finishes the proof.

∬

Ω𝑇

𝑝𝑚
𝑎

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2 ≤ 𝐶,

∬

Ω𝑇

𝑝𝑚 (Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))2 ≤ 𝐶.

Remark 3.7.2. Integrals 𝐼1, 𝐼2 are new relative to [17, Theorem 3.2]. They do interfere with

achieving the 𝐿4 bound by introducing an 𝑚 on the right hand-side. This prevents us from bounding

the second derivative (Hessian) of the pressure by a constant independent of 𝑚. One may be able

to achieve the 𝐿4 bound in this setting by rearranging the diffusion portion as (3.13). Thus, we

settle for the 𝐿3 bound, which is sufficient. The most interesting terms are 𝐼3 and 𝐼4. Both appear in

[17, Theorem 3.2] where Φ only depends on the pressure and 𝑎 ≡ 1. Due to our generalization of

these terms, bounding each becomes a little more difficult due to having more sub-integrals (such as

𝐼4,1, 𝐼4,2, 𝐼4,3), however; the strategy is similar.

3.7.2 𝐿3 Aronson-Bénilan Estimate

Proposition 3.7.3 (𝐿3 AB-Estimate). Suppose Assumptions 3.2.1, 3.2.2, and 3.2.3 hold. For 𝑇 > 0

and 𝑚 > max{2, 5 − 4

𝑑 }, there exists 𝐶 > 0, independent of 𝑚, such that

∬

Ω𝑇

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚))3

− ≤ 𝐶 (𝑇, 𝑎, 𝑏, 𝑝 𝑀, Φ, |Ω𝑇 |)

90

and

∬

Ω𝑇

|Δ𝑝𝑚 | ≤ 𝐶 (𝑇, 𝑎, 𝑏, 𝑝 𝑀, Φ, |Ω𝑇 |).

Proof. For sake of simplicity, we drop the subscript 𝑚 and denote Φ(𝑥, 𝑡, 𝑝) as Φ. Define 𝑤 = Δ𝑝+Φ.

Taking the time derivative of 𝑤,

(3.12)

𝜕𝑡𝑤 = Δ𝜕𝑡 𝑝 + Φ𝑡 + Φ𝑝𝜕𝑡 𝑝.

Given that we know (3.4), we compute

Δ𝜕𝑡 𝑝 = Δ

(cid:19)

(cid:18) |∇𝑝|2
𝑎

+ (𝑚 − 1)Δ

(cid:18)

(cid:18) 𝑝
𝑎

∇𝑝 · ∇ log

(cid:19)(cid:19)(cid:19)

(cid:18) 𝑏
𝑎

+ (𝑚 − 1)Δ

(cid:17)

𝑤

(cid:16) 𝑝
𝑎

− (𝑚 − 1)Δ ( 𝑝𝜕𝑡 log(𝑏))

= 𝐷1 + (𝑚 − 1)(𝐷2 + 𝐷3 + 𝐷4).

For ease, we will denote 𝛼 = ∇ log(𝑏/𝑎)

𝑎

and 𝛽 = 𝜕𝑡 log(𝑏). Computing 𝐷1,

𝐷1 =

1
𝑎

Δ|∇𝑝|2 + 2∇|∇𝑝|2 · ∇

(cid:19)

(cid:18) 1
𝑎

+ |∇𝑝|2Δ

(cid:19)

(cid:18) 1
𝑎

= 𝐷1,1 + 𝐷1,2 + 𝐷1,3.

Using Young’s inequality (with 𝜖 = 1/4),

𝐷1,2 = −4

(𝜕2

𝑖, 𝑗 𝑝)

∇𝑝 · ∇𝑎
𝑎2

∑︁

𝑖, 𝑗

≥ −

1
𝑎

∑︁

𝑖, 𝑗

(𝜕2

𝑖, 𝑗 𝑝)2 − 4

|∇𝑎|2
𝑎3

|∇𝑝|2.

and so

𝐷1,1 + 𝐷1,2 ≥

2
𝑎

∑︁

𝑖, 𝑗

(𝜕2

𝑖, 𝑗 𝑝)2 +

2
𝑎

∇𝑝 · ∇Δ𝑝 −

1
𝑎

∑︁

𝑖, 𝑗

(𝜕2

𝑖, 𝑗 𝑝)2 − 4

|∇𝑎|2
𝑎3

|∇𝑝|2

≥

=

1
𝑎𝑑
1
𝑎𝑑

(Δ𝑝)2 +

2
𝑎

∇𝑝 · ∇Δ𝑝 − 4

|∇𝑎|2
𝑎3

|∇𝑝|2

(𝑤 − Φ)2 +

2
𝑎

∇𝑝 · ∇𝑤 − Φ𝑥𝑖 − Φ𝑝∇𝑝 − 4

|∇𝑎|2
𝑎3

|∇𝑝|2.

91

Going back to 𝐷4,

Now computing 𝐷2,

𝐷2 = Δ( 𝑝∇𝑝 · 𝛼)

𝐷4 = Δ𝑝𝛽 + 𝑝Δ𝛽 + 2∇𝑝 · ∇𝛽

= (𝑤 − Φ) 𝛽 + 𝑝Δ𝛽 + 2∇𝑝 · ∇𝛽.

= Δ𝑝∇𝑝 · 𝛼 + 𝑝∇Δ𝑝 · 𝛼 + 𝑝∇𝑝Δ (𝛼) + 2∇𝑝Δ𝑝𝛼

+ 2𝑝Δ𝑝∇ (𝛼) + 2|∇𝑝|2∇ (𝛼)

= 3Δ𝑝∇𝑝 · 𝛼 + 2𝑝Δ𝑝∇ (𝛼) + 𝑝∇Δ𝑝 · 𝛼 + 𝑝∇𝑝Δ (𝛼) + 2|∇𝑝|2∇ (𝛼)

= 3(𝑤 − Φ)∇𝑝 · 𝛼 + 2(𝑤 − Φ) 𝑝∇ (𝛼)

+ (∇𝑤 − Φ𝑥𝑖 − Φ𝑝∇𝑝) 𝑝𝛼 + 𝑝∇𝑝Δ (𝛼) + 2|∇𝑝|2∇ (𝛼)

= 𝐷2,1 + 𝐷2,2 + 𝐷2,3 + 𝐷2,4 + 𝐷2,5.

Multiplying (3.12) by −(𝑤)−,

1
2

𝜕𝑡 (𝑤)2

− = −𝜕𝑡𝑤(𝑤)− = −Δ𝜕𝑡 𝑝(𝑤)− − Φ𝑡 (𝑤)− − Φ𝑝𝜕𝑡 𝑝(𝑤)−.

Focusing on the last term,

−Φ𝑝𝜕𝑡 𝑝(𝑤)− = −Φ𝑝 (𝑤)−

|∇𝑝|2
𝑎
𝑝
𝑎

Φ𝑝

− (𝑚 − 1)

(cid:16)

(𝑤)−∇𝑝 · ∇ log(𝑏/𝑎) − Φ𝑝

𝑝
𝑎

(𝑤)2

− − Φ𝑝 𝑝(𝑤)−𝛽

(cid:17)

.

92

Updating −Δ𝜕𝑡 𝑝(𝑤)−,

−𝐷1,1(𝑤)− − 𝐷1,2(𝑤)− ≤

−1
𝑑𝑎
2
𝑎

+

(𝑤)3

− −

2
𝑑𝑎

(𝑤)2

−Φ −

1
𝑑𝑎

(𝑤)−Φ2 +

(∇𝑝(𝑤)−Φ𝑥𝑖 + Φ𝑝 |∇𝑝|2(𝑤)−) + 4

−𝐷1,3(𝑤)− = −(𝑤)−|∇𝑝|2Δ

(cid:19)

(cid:18) 1
𝑎

,

−𝐷2,1(𝑤)− = 3((𝑤)2

− + Φ(𝑤)−)∇𝑝 · 𝛼,

−𝐷2,2(𝑤)− = 2((𝑤)2

− + Φ(𝑤)−) 𝑝∇ (𝛼) ,

∇𝑝 · ∇(𝑤)2
−

1
𝑎
|∇𝑎|2
𝑎3

|∇𝑝|2(𝑤)−,

∇(𝑤)2

−𝐷2,3(𝑤)− =

1
2
−𝐷2,4(𝑤)− = −𝑝∇𝑝Δ (𝛼) (𝑤)−,

− 𝑝𝛼 + (Φ𝑥𝑖 (𝑤)− + Φ𝑝∇𝑝(𝑤)−) 𝑝𝛼,

−𝐷2,5(𝑤)− = −2|∇𝑝|2∇ (𝛼) (𝑤)−,
(cid:16) 𝑝
𝑎

−𝐷3(𝑤)− = Δ

(𝑤)−,

(𝑤)−

(cid:17)

−𝐷4(𝑤)− = ((𝑤)2

− + Φ(𝑤)−) 𝛽 − (𝑤)− 𝑝Δ𝛽 − 2(𝑤)−∇𝑝 · ∇𝛽.

Integrating in space and time,

∫ 𝑇

0

𝑑
𝑑𝑡

∫

Ω

(𝑤)2
−
2

≤ 𝐼1 + 𝐼2 + 𝐼3 + 𝐼4 + 𝐼5 + 𝐼6,

93

where

𝐼1 =

𝐼2 =

𝐼3 =

−1
𝑑
∬

Ω𝑇

∬

Ω𝑇

(cid:18)∬

Ω𝑇

(𝑤)3
−
𝑎

+ 2

(𝑤)2
−
𝑎

Φ +

∇𝑝 · ∇(𝑤)2

− + (𝑚 − 1)

(𝑤)−
𝑎
∬

Ω𝑇

(cid:19)

,

Φ2

(cid:16) 𝑝
𝑎

Δ

(cid:17)

(𝑤)−

(𝑤)− + (𝑚 − 1)

∬

Ω𝑇

∇(𝑤)2
−
2

𝑝𝛼,

− Δ

(cid:19)

(cid:18) 1
𝑎

+ 4

(cid:19)

|∇𝑎|2
𝑎3

|∇𝑝|2(𝑤)− − (𝑚 − 1)

∬

Ω𝑇

2∇ (𝛼) |∇𝑝|2(𝑤)−,

1
𝑎
(cid:18) Φ𝑝
𝑎

∬

𝐼4 = 3(𝑚 − 1)

∇𝑝(𝑤)2

−𝛼,

𝐼5 =

∬

Ω𝑇

Ω𝑇
2
𝑎

∇𝑝(𝑤)−

Φ𝑥𝑖

+ (𝑚 − 1)

∇𝑝(𝑤)− (3Φ𝛼 − 𝑝Δ (𝛼) − 2∇𝛽) ,

∬

Ω𝑇

𝐼6 = −

∬

Ω𝑇

Φ𝑡 (𝑤)− + (𝑚 − 1)

∬

Ω𝑇

2((𝑤)2

− + Φ(𝑤)−) 𝑝∇ (𝛼) + Φ𝑥𝑖 (𝑤)− 𝑝𝛼

+ (𝑚 − 1)

+ (𝑚 − 1)

∬

Ω𝑇

∬

Ω𝑇

((𝑤)2

− + Φ(𝑤)−) 𝛽 − (𝑤)− 𝑝Δ𝛽

Φ𝑝

𝑝
𝑎

(𝑤)2

− + Φ𝑝 𝑝(𝑤)−𝛽

We start with the most interesting integral 𝐼2 = 𝐼2,1 + 𝐼2,2 + 𝐼2,3. Using integration by parts for each

integral,

𝐼2,1 = −

∬

Δ𝑝
𝑎
Ω𝑇
(𝑤)3
−
𝑎

(𝑤)2

− + ∇𝑝 · ∇(1/𝑎) (𝑤)2
−

∬

=

Ω𝑇

∬

+

(𝑤)2
−
𝑎

Φ −

∬

Ω𝑇

−,
∇𝑝 · ∇(1/𝑎) (𝑤)2

(𝑤)−

(cid:17)

· ∇(𝑤)−

∬

Ω𝑇

∬

∬

Ω𝑇

Δ

Ω𝑇

∇

∇

(cid:16) 𝑝
𝑎
(cid:16) 𝑝
2𝑎
(cid:17)

(cid:16) 𝑝
𝑎

𝐼2,2 = −(𝑚 − 1)

= −(𝑚 − 1)

=

≤

=

𝐼3 =

(𝑚 − 1)
2
(𝑚 − 1)
2
(1 − 𝑚)
2
(1 − 𝑚)
2

Ω𝑇

∬

Ω𝑇

∬

Ω𝑇

∬

Ω𝑇

(cid:17)

· ∇(𝑤)2

− +

𝑝
𝑎

|∇(𝑤)−|2

(𝑤)2

− −

(cid:18) Δ𝑝
𝑎
(𝑤)2
−
𝑎

+

(𝑤)2
−

(𝑤)3
−
𝑎

(cid:19)

|∇(𝑤)−|2

2𝑝
𝑎
(cid:18) 1
𝑎
(𝑚 − 1)
2

Φ +

+ 𝑝Δ

+ 2∇𝑝 · ∇

(𝑤)2

−∇𝑝 · 𝛼 + (𝑤)2

− 𝑝∇ (𝛼) .

94

(cid:19)(cid:19)

(cid:18) 1
𝑎

∬

Ω𝑇

𝑝Δ

(cid:19)

(cid:18) 1
𝑎

(𝑤)2

− + 2∇𝑝 · ∇

(cid:19)

(cid:18) 1
𝑎

−,
(𝑤)2

Combining the sub-integrals above,

𝐼2 ≤

+

(3 − 𝑚)
2
(𝑚 − 1)
2

Define

Ω𝑇

∬

∬

Ω𝑇

∬

(𝑤)3
−
𝑎
(cid:18) 1
𝑎

𝑝Δ

+

(cid:19)

(𝑤)2
−
𝑎

Φ + (𝑚 − 2)

∬

Ω𝑇

∇𝑝 · ∇

(cid:19)

(cid:18) 1
𝑎

(𝑤)2
−

(𝑤)2

− − (𝑤)2

−∇𝑝 · 𝛼 − (𝑤)2

− 𝑝∇ (𝛼) .

𝐽1 =

(3 − 𝑚)
2

𝐽2 = (𝑚 − 2)

𝐽3 =

(𝑚 − 1)
2

Ω𝑇

∬

Ω𝑇

∬

Ω𝑇

(𝑤)3
−
𝑎

+

Φ,

(𝑤)2
−
𝑎
(cid:19)

∇𝑝 · ∇

(𝑤)2

− −

(cid:18) 1
𝑎

𝑝Δ

(cid:19)

(cid:18) 1
𝑎

(𝑤)2

− −

(𝑚 − 1)
2
(𝑚 − 1)
2

∬

Ω𝑇

∬

Ω𝑇

(𝑤)2

−∇𝑝 · 𝛼,

(𝑤)2

− 𝑝∇ (𝛼) .

Combing 𝐼1 and 𝐽1,

𝐼1 + 𝐽1 =

≤

∬

(cid:18) (3 − 𝑚)
2
1
𝑑
(cid:18) (3 − 𝑚)
2

−

−

(cid:19) ∬

1
𝑑
(𝑤)−
𝑎
(cid:19) ∬

Ω𝑇

Φ2

Ω𝑇

Ω𝑇

−

1
𝑑

(𝑤)3
−
𝑎

+

(cid:18) (3 − 𝑚)
2

−

2
𝑑

(cid:19) ∬

Ω𝑇

(𝑤)2
−
𝑎

Φ

(𝑤)3
−
𝑎

.

By Young’s inequality with 𝜖 and Proposition 3.7.1,

(cid:18)

𝐽2 ≤

(𝑚 − 2) +

(𝑚 − 1)
2

∬

(cid:19)

𝜖

≤

(3𝑚 − 5)𝜖
2

∬

Ω𝑇

(𝑤)3
−
𝑎

+

(𝑤)3
−
𝑎

Ω𝑇
(3𝑚 − 5)
2

(𝑚 − 2) +

(𝑚 − 1)
2

∬

(cid:19)

𝐶

Ω𝑇

|∇𝑝|3

(cid:18)

+

𝐶.

This type of bound is similar to what we will do for 𝐼3 and 𝐼4. By previous 𝐿∞ bounds,

𝐽3 ≤

(𝑚 − 1)
2

𝐶

∬

Ω𝑇

(𝑤)2
−
𝑎

.

This type of bound is similar to what we will do for 𝐼6. By Young’s inequality with 𝜖 and Proposition

3.7.1,

𝐼3 ≤ 𝑚𝜖

≤ 𝑚𝜖

∬

Ω𝑇

∬

Ω𝑇

(𝑤)3
−
𝑎
(𝑤)3
−
𝑎

∬

+ 𝑚𝐶

|∇𝑝|3

Ω𝑇

+ 𝑚𝐶.

95

Again using Young’s inequality with 𝜖 and Proposition 3.7.1,

𝐼4 ≤ 3(𝑚 − 1)𝜖

≤ 3(𝑚 − 1)𝜖

∬

Ω𝑇

∬

Ω𝑇

(𝑤)3
−
𝑎
(𝑤)3
−
𝑎

+ (𝑚 − 1)𝐶

∬

Ω𝑇

|∇𝑝|3

+ (𝑚 − 1)𝐶.

Using Cauchy’s inequality,

𝐼5 ≤ 𝑚

≤ 𝑚

∬

Ω𝑇

∬

Ω𝑇

(𝑤)2
−
𝑎
(𝑤)2
−
𝑎

∬

+ 𝑚𝐶

|∇𝑝|2

Ω𝑇

+ 𝑚𝐶.

𝐼6 ≤ 𝑚𝐶

∬

Ω𝑇

(𝑤)2
−
𝑎

∬

+ 𝑚𝐶

Ω𝑇

(𝑤)−
𝑎

+ 𝑚𝐶.

Using various 𝐿∞ bounds,

Collecting the bounds,

∫ 𝑇

0

𝑑
𝑑𝑡

∫

Ω

(𝑤)2
−
2

≤

(cid:18) (3 − 𝑚)
2

−

1
𝑑

+

(3𝑚 − 5)𝜖
2

(cid:19) ∬

+ 𝑚𝜖

Ω𝑇

(𝑤)3
−
𝑎

∬

+ 𝑚𝐶

Ω𝑇

(𝑤)2
−
𝑎

∬

+ 𝑚𝐶

Ω𝑇

(𝑤)−
𝑎

+ 𝑚𝐶.

Rearranging, choosing 𝜖 = 1/10, using Assumption 3.2.2 and Hölder’s inequality,

(cid:18) (𝑚 − 5)
4

+

1
𝑑

(cid:19) ∬

Ω𝑇

(𝑤)3
−
𝑎

≤ 𝑚𝐶

≤ 𝑚𝐶

∬

Ω𝑇
(cid:18)∬

(𝑤)2
−
𝑎

∬

+ 𝑚𝐶

Ω𝑇

+ 𝑚𝐶

(cid:19) 2/3

(𝑤)3
−
𝑎

(𝑤)−
𝑎

(cid:18)∬

Ω𝑇

+ 𝑚𝐶 +

∫

Ω

(𝑤0)2
−
2

(cid:19) 1/3

(𝑤)3
−
𝑎

+ 𝑚𝐶

Given the hypothesis on 𝑚,

Ω𝑇

Moreover,

∬

Ω𝑇

(𝑤)3
−
𝑎

≤ 𝐶.

∬

Ω𝑇

(𝑤)3

− ≤ 𝐶.

This gives us the first bound. For the second, using integration by parts,

∬

Ω𝑇

(Δ𝑝 + Φ) ≤ 𝐶.

96

So,

∬

Ω𝑇

|Δ𝑝 + Φ| =

∬

Ω𝑇

(Δ𝑝 + Φ) + 2

∬

(𝑤)−

Ω𝑇

(cid:19) 1/3

≤ 𝐶 + 𝐶

(cid:18)∬

Ω𝑇

(𝑤)3
−

≤ 𝐶.

As Φ is bounded,

This completes the proof.

∬

∬

|Δ𝑝| ≤

Ω𝑇

Ω𝑇

|Δ𝑝 + Φ| +

∬

Ω𝑇

|Φ|

≤ 𝐶.

Remark 3.7.4. Compared to [17, Theorem 3.1], we have approximately double the number of terms

because 𝑎, 𝑏 are non-constant functions of space and time. The integrals 𝐼3, 𝐼4 are completely new

and 𝐼5, 𝐼6 are mostly new. Integrals 𝐼5, 𝐼6 do not present an issue, however; integrals 𝐼3, 𝐼4 require us

to have an 𝐿3 bound on the gradient of the pressure. The term 𝐼2 is similar to one in [17, Theorem

3.1], but is more extensive. Though a similar strategy is used here, we require more. In particular,

as 𝑎 (cid:46) 1, we require 𝐿3 bound on the gradient of the pressure for 𝐼2 as well.

3.7.3 Complementarity Condition

Proposition 3.7.5 (Strong convergence of the pressure gradient). Suppose Assumptions 3.2.1, 3.2.2,

and 3.2.3 hold. For 𝑇 > 0, ∇𝑝𝑚 → ∇𝑝∞ strongly in 𝐿2(𝑄𝑇 ).

Proof. We first start by showing spatial compactness of the pressure gradient. For 𝜖 > 0, define the

continuous function,

𝜓(𝑠) =

−𝜖,

for 𝑠 < −𝜖,

𝑠,

𝜖,

for − 𝜖 ≤ 𝑠 ≤ 𝜖,

for 𝑠 > 𝜖 .






97

Using integration by parts for 𝑚, 𝑛 > 1,

∬

Ω𝑇

|∇𝑝𝑚 − ∇𝑝𝑛|2𝜓′( 𝑝𝑚 − 𝑝𝑛) 𝑑𝑥 𝑑𝑡 = −

∬

Ω𝑇

(Δ𝑝𝑚 − Δ𝑝𝑛)𝜓( 𝑝𝑚 − 𝑝𝑛) 𝑑𝑥 𝑑𝑡.

Defining the domain,

Ω𝑇,𝜖 = {(𝑥, 𝑡) ∈ Ω𝑇 : | 𝑝𝑚 (𝑥, 𝑡) − 𝑝𝑛 (𝑥, 𝑡)| ≤ 𝜖 },

and using Proposition 3.7.3,

∬

Ω𝑇 , 𝜖

|∇𝑝𝑚 − ∇𝑝𝑛|2 ≤ 𝐶𝜖 .

Thus we can use H older’s inequality so that

∬

Ω𝑇

|∇𝑝𝑚 − ∇𝑝𝑛| =

∬

Ω𝑇 , 𝜖

|∇𝑝𝑚 − ∇𝑝𝑛| +

∬

Ω𝑐

𝑇 , 𝜖

|∇𝑝𝑚 − ∇𝑝𝑛|

≤ 𝐶𝜖 1/2 + 𝐶 ∥ 𝑝𝑚 ∥ 𝐿2 (𝑄𝑇 ) |Ω𝑐

𝑇,𝜖 |1/2.

As 𝑝𝑚 has compact support, Lemma 3.4.2, 𝑝𝑚 is Cauchy and there exists 𝑁 (𝜖) large enough such

that for 𝑚, 𝑛 > 𝑁 (𝜖),

∬

Ω𝑇

|∇𝑝𝑚 − ∇𝑝𝑛| ≤ 𝐶𝜖 1/2 + 𝐶𝜖 .

This implies that ∇𝑝𝑚 is Cauchy in 𝐿1(𝑄𝑇 ) and thus, up to a subsequence, we have almost

everywhere convergence. By Proposition 3.7.1, we have, possibly up to a subsequence, ∇𝑝𝑚 weakly

converges to ∇𝑝∞ in 𝐿3(𝑄𝑇 ). So, the pressure gradient is compact in space for any 𝐿𝑞 (𝑄𝑇 ) for

1 ≤ 𝑞 < 3.

Now we move on to time compactness. Let 𝜙𝛼 = 𝛼−𝑑 𝜙(𝑥/𝛼), with 𝛼 > 0 be a nonnegative,

smooth mollifier where ∫

R𝑑 𝜙 = 1. We will comput the time shift of ∇𝑝𝑚 with ℎ > 0. In particular,

we add and subtract ∇𝑝𝑚 ∗ 𝜙𝛼 at time 𝑡 + ℎ and 𝑡 and use the triangle inequality,

∫ 𝑇−ℎ

0

∥∇𝑝𝑚 (𝑡 + ℎ) − ∇𝑝𝑚 (𝑡)∥ 𝐿1 (Ω) 𝑑𝑡 ≤

∥∇𝑝𝑚 (𝑡 + ℎ) − ∇𝑝𝑚 (𝑡 + ℎ) ∗ 𝜙𝛼 ∥ 𝐿1 (Ω) 𝑑𝑡

∥(∇𝑝𝑚 (𝑡 + ℎ) − ∇𝑝𝑚 (𝑡)) ∗ 𝜙𝛼 ∥ 𝐿1 (Ω) 𝑑𝑡

∥∇𝑝𝑚 (𝑡) ∗ 𝜙𝛼 − ∇𝑝𝑚 (𝑡) ∥ 𝐿1 (Ω) 𝑑𝑡.

∫ 𝑇−ℎ

+

+

0
∫ 𝑇−ℎ

0
∫ 𝑇−ℎ

0

98

Let us focus on controlling the middle term first. Using integration by parts in space and Young’s

convolution inequality,

∫ 𝑇−ℎ

∥(∇𝑝𝑚 (𝑡 + ℎ) − ∇𝑝𝑚 (𝑡)) ∗ 𝜙𝛼 ∥ 𝐿1 (Ω) 𝑑𝑡

𝜕𝑡 𝑝𝑚 (𝑡 + 𝑆) ∗ ∇𝜙𝛼 𝑑𝑠

(cid:13)
(cid:13)
(cid:13)
(cid:13)𝐿1 (Ω)

𝑑𝑡

0

=

∫ 𝑇−ℎ

∫ ℎ

(cid:13)
(cid:13)
(cid:13)
(cid:13)
∫ 𝑇−ℎ

0
∫ ℎ

∫

0

≤

|𝜕𝑡 𝑝𝑚 (𝑡 + 𝑠) ∗ ∇𝜙𝛼| 𝑑𝑥 𝑑𝑠 𝑑𝑡

0

0

≤ ∥∇𝜙𝛼 ∥ 𝐿1 (R𝑑)

Ω
∫ 𝑇−ℎ

∫ ℎ

∫

0

0

Ω

|𝜕𝑡 𝑝𝑚 (𝑥, 𝑡 + 𝑠)| 𝑑𝑥 𝑑𝑠 𝑑𝑡

By Lemma 3.4.7,

∥∇𝜙𝛼 ∥ 𝐿1 (R𝑑)

∫ 𝑇−ℎ

∫ ℎ

∫

|𝜕𝑡 𝑝𝑚 (𝑥, 𝑡 + 𝑠)| 𝑑𝑥 𝑑𝑠 𝑑𝑡

0

0
∫ 𝑇−ℎ

Ω
∫ 𝑡+ℎ

∫

|𝜕𝑡 𝑝𝑚 (𝑥, 𝑠)| 𝑑𝑥 𝑑𝑠 𝑑𝑡

0
∫ 𝑇−ℎ

𝑡

Ω

∫ min(𝑠,𝑇−ℎ)

∫

0

max(0,𝑠−ℎ)

Ω

|𝜕𝑡 𝑝𝑚 (𝑥, 𝑠)| 𝑑𝑥 𝑑𝑡 𝑑𝑠

= ∥∇𝜙𝛼 ∥ 𝐿1 (R𝑑)

= ∥∇𝜙𝛼 ∥ 𝐿1 (R𝑑)

≤ 𝐶

|ℎ|
𝛼

.

Now we deal with the first (and third) term. By Fréchet-Kolmogorov Theorem, the space shifts of

the pressure gradient converge as well. In particular, there exists a funtion 𝜔 : R → R≥0, such that

∬

Ω𝑇

|∇𝑝𝑚 (𝑥 + 𝑘, 𝑡) − ∇𝑝𝑛 (𝑥, 𝑡)| ≤ 𝜔(|𝑘 |),

with 𝜔(|𝑘 |) → 0 as |𝑘 | → 0. So,

∫ 𝑇−ℎ

0

≤

≤

≤

0
∫

R𝑑

∫

R𝑑

∥∇𝑝𝑚 (𝑡) ∗ 𝜙𝛼 − ∇𝑝𝑚 (𝑡) ∥ 𝐿1 (Ω) 𝑑𝑡

∫ 𝑇−ℎ

∫

∫

𝜙(𝑦) (∇𝑝𝑚 (𝑥 − 𝛼𝑦, 𝑡) − ∇𝑝𝑚 (𝑥, 𝑡))

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

𝑑𝑦 𝑑𝑥 𝑑𝑡

R𝑑

(cid:12)
(cid:12)
(cid:12)
(cid:12)
Ω
∫

𝜙(𝑦)

|∇𝑝𝑚 (𝑥 − 𝛼𝑦, 𝑡) − ∇𝑝𝑚 (𝑥, 𝑡)| 𝑑𝑥 𝑑𝑡 𝑑𝑦

Ω𝑇

𝜙(𝑦)𝜔(𝛼|𝑦|) 𝑑𝑦.

Combining the estimates,

∫ 𝑇−ℎ

0

∥∇𝑝𝑚 (𝑡 + ℎ) − ∇𝑝𝑚 (𝑡) ∥ 𝐿1 (Ω) 𝑑𝑡 ≤ 𝐶

|ℎ|
𝛼

+ 2

∫

R𝑑

𝜙(𝑦)𝜔(𝛼|𝑦|) 𝑑𝑦.

99

Choosing 𝛼 = |ℎ|1/2 and taking 𝛼 → 0,

∫ 𝑇−ℎ

0

∥∇𝑝𝑚 (𝑡 + ℎ) − ∇𝑝𝑚 (𝑡) ∥ 𝐿1 (Ω) 𝑑𝑡 → 0.

This gives us time compactness by Aubin-Lions lemma. Thus, ∇𝑝𝑚 → ∇𝑝∞ strongly in 𝐿𝑞 (𝑄𝑇 )

for 1 ≤ 𝑞 < 3 and in particular for 𝑞 = 2.

Proof of Theorem 3.3.3. By Proposition 3.7.5, ∇𝑝𝑚 → ∇𝑝∞ in 𝐿2(𝑄𝑇 ). Integrating the PDE that

the pressure satisfies (3.4) and rearranging gives us,

1
𝑚 − 1

∬

(cid:18)

𝑄𝑇

𝜕𝑡 𝑝𝑚 −

(cid:19)

|∇𝑝𝑚 |2
𝑎

𝜁 =

∬

𝑄𝑇

𝑝𝑚

(cid:18)

𝜁
𝑎

∇𝑝𝑚 · ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

(cid:19)

+ Δ𝑝𝑚

∬

+

𝑄𝑇

𝑝𝑚

𝜁
𝑎

(Φ(𝑥, 𝑡, 𝑝𝑚) − 𝑎𝜕𝑡 log(𝑏)) .

Integrating by parts,

∬

−1
𝑚 − 1
∬

=

𝑄𝑇

𝑄𝑇
𝑝𝑚
𝑎

𝜁

𝑝𝑚𝜕𝑡 𝜁 +

|∇𝑝𝑚 |2
𝑎

𝜁 −

∇𝑝𝑚 · ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

−

1
𝑚 − 1
∬

𝑄𝑇

∫

R𝑑

𝑝𝑚 (𝑥, 0)𝜁 (𝑥, 0)

|∇𝑝𝑚 |2

𝜁
𝑎

+ ∇𝑝𝑚 · ∇

(cid:19)

(cid:18) 𝜁
𝑎

𝑝𝑚

∬

+

𝑄𝑇

𝑝𝑚
𝑎

𝜁Φ(𝑥, 𝑡, 𝑝𝑚) − 𝑝𝑚 𝜁 𝜕𝑡 log(𝑏).

Taking the limit as 𝑚 → ∞ gives the result.

3.8 Velocity Law

Notice that

𝑎
𝑏

∇ ·

(cid:19)

∇𝑝

(cid:18) 𝑏
𝑎

= ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

· ∇𝑝 + Δ𝑝.

Thus, the complementarity condition in (3.6) can be rewritten as

(3.13)

−∇ ·

(cid:18) 𝑏
𝑎

∇𝑝∞

(cid:19)

=

𝑏
𝑎

Φ(𝑥, 𝑡, 𝑝∞) − 𝜕𝑡 𝑏.

We will use this representation in the upcoming proposition.

100

3.8.1 Comparison with barriers

Let D be a ball in R𝑑. For a time interval [𝑡1, 𝑡2] ⊂ [0, ∞), consider a function (that represent
the pressure) 𝜁 ∈ 𝐶𝑐 ( ¯𝐷 × [𝑡1, 𝑡2]) such that the initial density 𝑢1(𝑥) satisfies 𝑢1(𝑥) = 𝑏(𝑥, 𝑡1) in

{𝜁 (𝑡1) > 0}. For all 𝑥 ∉ {𝜁 (𝑡1) > 0}, we define 𝑡 (𝑥) as the last time that 𝜁 (𝑥, 𝑡) = 0 (with 𝑡 (𝑥) = 𝑡2

is 𝜁 (𝑥, 𝑡2 = 0)) and define the external density

𝑢𝐸
𝜁 (𝑥, 𝑡) = 𝑢1(𝑥) exp

(cid:18)∫ 𝑡

𝑡1

Φ(𝑥, 𝑠, 𝜁 (𝑥, 𝑠))
𝑎(𝑥, 𝑠)

(cid:19)

𝑑𝑠

for all 𝑡 < 𝑡 (𝑥). We assume that the external density satisfies,

𝑢𝐸
𝜁 (𝑥, 𝑡) < 𝑏(𝑥, 𝑡) in {𝜁 = 0}.

The external density solves 𝜕𝑡𝑢 = 𝑢

𝑎 Φ(𝑥, 𝑡, 𝜁). The density in 𝐷 × (𝑡1, 𝑡2) is defined by

𝑢𝜁 (𝑥, 𝑡) = 𝑏(𝑥, 𝑡) 𝜒{𝜁 >0} (𝑥) + 𝑢𝐸

𝜁 (𝑥, 𝑡) (1 − 𝜒{𝜁 >0} (𝑥))

=





𝑏(𝑥, 𝑡), in {𝜁 > 0}

𝑢𝐸
𝜁 (𝑥, 𝑡), in {𝜁 = 0}.

Proposition 3.8.1. Suppose that (𝑢𝜁 , 𝜁) are such that

1. 𝜁 ∈ 𝐶1({𝜁 > 0}) ∩ 𝐶2

𝑙𝑜𝑐 ({𝜁 > 0}) and Γ = 𝜕{𝜁 > 0} is 𝐶2 in space and 𝐶1 in time.

2. 𝜁 satisfies

(cid:16) 𝑏
𝑎 ∇𝜁
(cid:19)
𝑢𝐸
𝜁
𝑏

−∇ ·
(cid:18)

1 −





(cid:17)

≤ 𝑏

𝑎 Φ(𝑥, 𝑡, 𝜁) − 𝜕𝑡 𝑏, in {𝜁 > 0}

𝑉𝜁 ≤ |∇𝜁 |

𝑎 , on 𝜕{𝜁 > 0},

where 𝑉𝜁 denotes the normal velocity of 𝜕{𝜁 > 0}. Then (𝑢𝜁 , 𝜁) is a weak subsolution of the

limiting problem

𝜕𝑡𝑢𝜁 ≤ ∇ ·

(cid:19)

+

∇𝜁

(cid:18) 𝑏
𝑎

𝑢𝜁
𝑎

Φ(𝑥, 𝑡, 𝜁) in 𝐷 × (𝑡1, 𝑡2), 𝜁 ∈ 𝑃∞(𝑢𝜁 , 𝑏) a.e. in 𝐷 × (𝑡1, 𝑡2)

where the PDE holds in the sense that for every smooth, compactly supported test function

𝜓 : 𝐷 × (𝑡1, 𝑡2) → R with 𝜓(·, 𝑡2) = 0 and 𝜓(·, 𝑡) = 0 on 𝜕𝐷 × [𝑡1, 𝑡2], we have

∫

𝐷×[𝑡1,𝑡2]

𝑢𝜁 𝜕𝑡𝜓 −

𝑏
𝑎

∇𝜁 · ∇𝜓 +

𝑢𝜁
𝑎

Φ(𝑥, 𝑡, 𝜁)𝜓 ≥ −

∫

𝐷

𝑢1(𝑥)𝜓(𝑥, 𝑡1).

101

Proof. Let us denote 𝑆(𝑡) = {𝜁 (·, 𝑡) > 0} = {𝑢(·, 𝑡) = 𝑏(·, 𝑡)} and Γ(𝑡) = 𝜕𝑆(𝑡) ∩ 𝐷. We also have

𝜈 as the outward normal of the boundary of either Γ(𝑡) or 𝜕𝐷 with respect to 𝑆(𝑡). Using integration

by parts,

∫

−

𝐷

𝑏
𝑎

∇𝜁 · ∇𝜓 = −

∫

𝑆(𝑡)

𝑏
𝑎

∇𝜁 · ∇𝜓

∫

=

≥ −

𝑆(𝑡)
∫

𝑏
𝑎

𝑆(𝑡)

∇ ·

(cid:19)

∇𝜁

(cid:18) 𝑏
𝑎

∫

𝜓 −

𝑏
𝑎

𝜓∇𝜁 · 𝜈 𝑑𝑆

Φ(𝑥, 𝑡, 𝜁)𝜓 +

𝜕𝑆(𝑡)
∫

𝜓𝜕𝑡 𝑏 +

∫

Γ(𝑡)

𝑏
𝑎

|∇𝜁 |𝜓,

𝑆(𝑡)

where it is used that 𝜁 = 0 and ∇𝜁 = |∇𝜁 |𝜈 on Γ(𝑡). Using the definition of 𝑢𝜁 ,

∫

𝐷

𝑢𝜁 𝜕𝑡𝜓 =

∫

𝑆(𝑡)

𝑏𝜕𝑡𝜓 +

∫

𝐷\𝑆(𝑡)

𝑢𝐸
𝜁 𝜕𝑡𝜓.

By product rule and differentiating moving regions,

Similarly,

Thus,

∫

∫

𝑆(𝑡)

𝑏𝜕𝑡𝜓 =

𝜕𝑡 (𝑏𝜓) −

∫

𝑆(𝑡)

𝜓𝜕𝑡 𝑏

𝑆(𝑡)
∫
𝑑
𝑑𝑡

∫

𝑏𝜓 −

Γ(𝑡)

𝑏𝜓𝑉𝜁 −

∫

𝑆(𝑡)

𝜓𝜕𝑡 𝑏.

𝑆(𝑡)

∫

𝐷\𝑆(𝑡)

𝑢𝐸
𝜁 𝜕𝑡𝜓 =

𝜕𝑡 (𝑢𝐸

𝜁 𝜓) −

∫

𝐷\𝑆(𝑡)

𝜓𝜕𝑡 (𝑢𝐸
𝜁 )

𝑢𝐸
𝜁 𝜓 −

∫

𝐷\𝑆(𝑡)

−𝑉𝜁 𝑢𝐸

𝜁 𝜓 −

∫

𝐷\𝑆(𝑡)

𝜓𝜕𝑡 (𝑢𝐸

𝜁 ).

=

∫

𝐷\𝑆(𝑡)
∫
𝑑
𝑑𝑡

𝐷\𝑆(𝑡)

=

∫

𝐷
∫

∫

𝐷

𝑢𝜁 𝜕𝑡𝜓 =

≥

𝑑
𝑑𝑡
𝑑
𝑑𝑡

∫

Γ(𝑡)
∫

𝑢𝜁 𝜓 −

𝑢𝜁 𝜓 −

(𝑏 − 𝑢𝐸

𝜁 )𝜓𝑉𝜁 −

∫

𝑆(𝑡)

𝜓𝜕𝑡 𝑏 −

∫

𝐷\𝑆(𝑡)

𝜓𝜕𝑡 (𝑢𝐸
𝜁 )

∫

|∇𝜁 |𝜓 −

𝑏
𝑎

∫

𝜓𝜕𝑡 𝑏 −

𝜓𝜕𝑡 (𝑢𝐸

𝜁 ).

Γ(𝑡)
Using the estimates above and recalling that the external density solves 𝜕𝑡𝑢 = 𝑢
∫

𝐷\𝑆(𝑡)

𝑆(𝑡)

∫

∫

∫

∫

𝐷

𝑢𝜁 𝜕𝑡𝜓 −

∇𝜁 · ∇𝜓 ≥

𝑢𝜁 𝜓 −

Φ(𝑥, 𝑡, 𝜁)𝜓 −

𝑎 Φ(𝑥, 𝑡, 𝜁),

𝜓𝜕𝑡 (𝑢𝐸
𝜁 )

𝑏
𝑎

𝐷

𝐷

𝑑
𝑑𝑡

=

=

𝑑
𝑑𝑡
𝑑
𝑑𝑡

𝐷

∫

𝐷

∫

𝐷

𝑏
𝑎

𝑆(𝑡)

∫

∫

𝑏
𝑎
𝑆(𝑡)
𝑢𝜁
𝑎

𝐷

102

𝑢𝜁 𝜓 −

𝑢𝜁 𝜓 −

Φ(𝑥, 𝑡, 𝜁)𝜓 −

Φ(𝑥, 𝑡, 𝜁)𝜓.

𝐷\𝑆(𝑡)

∫

𝐷\𝑆(𝑡)

𝑢𝐸
𝜁
𝑎

Φ(𝑥, 𝑡, 𝜁)𝜓

Integrating in time from 𝑡1 to 𝑡2 we obtain the result

∫

𝐷×[𝑡1,𝑡2]

𝑢𝜁 𝜕𝑡𝜓 −

𝑏
𝑎

∇𝜁 · ∇𝜓 +

𝑢𝜁
𝑎

Φ(𝑥, 𝑡, 𝜁)𝜓 ≥ −

∫

𝐷

𝑢1(𝑥)𝜓(𝑥, 𝑡1).

Thus the proof is complete.

Proposition 3.8.2. Suppose that (𝑢𝜁 , 𝜁) are such that

1. 𝜁 ∈ 𝐶1({𝜁 > 0}) ∩ 𝐶2

𝑙𝑜𝑐 ({𝜁 > 0}) and Γ = 𝜕{𝜁 > 0} is 𝐶2 in space and 𝐶1 in time.

2. 𝜁 satisfies

(cid:16) 𝑏
𝑎 ∇𝜁
(cid:19)
𝑢𝐸
𝜁
𝑏

−∇ ·
(cid:18)

1 −





(cid:17)

≥ 𝑏

𝑎 Φ(𝑥, 𝑡, 𝜁) − 𝜕𝑡 𝑏, in {𝜁 > 0}

𝑉𝜁 ≥ |∇𝜁 |

𝑎 , on 𝜕{𝜁 > 0},

where 𝑉𝜁 denotes the normal velocity of 𝜕{𝜁 > 0}. Then (𝑢𝜁 , 𝜁) is a weak supersolution of

the limiting problem

𝜕𝑡𝑢𝜁 ≥ ∇ ·

(cid:19)

+

∇𝜁

(cid:18) 𝑏
𝑎

𝑢𝜁
𝑎

Φ(𝑥, 𝑡, 𝜁) in 𝐷 × (𝑡1, 𝑡2), 𝜁 ∈ 𝑃∞(𝑢𝜁 , 𝑏) a.e. in 𝐷 × (𝑡1, 𝑡2)

where the PDE holds in the sense that for every smooth, compactly supported test function

𝜓 : 𝐷 × (𝑡1, 𝑡2) → R with 𝜓(·, 𝑡2) = 0 and 𝜓(·, 𝑡) = 0 on 𝜕𝐷 × [𝑡1, 𝑡2], we have

∫

𝐷×[𝑡1,𝑡2]

𝑢𝜁 𝜕𝑡𝜓 −

𝑏
𝑎

∇𝜁 · ∇𝜓 +

𝑢𝜁
𝑎

Φ(𝑥, 𝑡, 𝜁)𝜓 ≤ −

∫

𝐷

𝑢1(𝑥)𝜓(𝑥, 𝑡1).

By Proposition 3.8.1 and the comparison principle for weak solutions of the limiting problem

Corollary 3.6.2, we get the following corollary.

Corollary 3.8.3. Let (𝑢𝜁 , 𝜁) be a sub-solution as Proposition 3.8.1. If

1. 𝑢𝜁 (·, 𝑡1) ≤ 𝑢∞(·, 𝑡1) in 𝐷,

2. 𝜁 ≤ 𝑝∞ on 𝜕𝐷 × [𝑡1, 𝑡2],

then 𝑢𝜁 ≤ 𝑢∞ in 𝐷 × [𝑡1, 𝑡2].

In the viscosity sense or comparison with barriers, we have the motion law

(cid:18)

1 −

(cid:19)

𝑢𝐸
∞
𝑏

𝑉∞ ≤

|∇𝑝∞|
𝑎

.

In a similar manner, we have the analogous corollary.

103

Corollary 3.8.4. Let (𝑢𝜁 , 𝜁) be a super-solution as Proposition 3.8.2. If

1. 𝑢𝜁 (·, 𝑡1) ≥ 𝑢∞(·, 𝑡1) in 𝐷,

2. 𝜁 ≥ 𝑝∞ on 𝜕𝐷 × [𝑡1, 𝑡2],

then 𝑢𝜁 ≥ 𝑢∞ in 𝐷 × [𝑡1, 𝑡2].

In the viscosity sense or comparison with barriers, we have the motion law

(cid:18)

1 −

(cid:19)

𝑢𝐸
∞
𝑏

𝑉∞ ≥

|∇𝑝∞|
𝑎

.

Thus, we have achieved the velocity law in the viscosity sense. Moreover, we have proven the

following proposition.

Proposition 3.8.5 (Velocity Law). The external density, 𝑢𝐸

∞, is the limit of the density from

outside the saturated region {𝑥 : 𝑝∞(𝑥, 𝑡) > 0}. The normal velocity, 𝑉∞, of the free boundary

𝜕{𝑥 : 𝑝∞(𝑥, 𝑡) > 0} satisfies in a viscosity sense

(cid:18)

1 −

(cid:19)

𝑢𝐸
∞
𝑏

𝑉∞ =

|∇𝑝∞|
𝑎

.

104

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106

APPENDIX A

MOLLIFIER AND AGGREGATION KERNELS

Extra Details About the Mollifier

We first discuss some extra details about the mollifier.

Remark A.0.1. Notice the 𝐿1(R𝑑) norm of the positive mollifier 𝜁𝜖 is one by change of variables,

∫

R𝑑

𝜁𝜖 (𝑦) 𝑑𝑦 =

∫

R𝑑

1
𝜖 𝑑

𝜁

(cid:17)

(cid:16) 𝑦
𝜖

𝑑𝑦 =

∫

R𝑑

𝜁 (𝑧) 𝑑𝑧 = 1.

Remark A.0.2 (the first moment of 𝜁 is finite). First we split the integral where we focus inside the

unit ball and outside the unit ball,

𝑀1(𝜁) =

∫

(R𝑑)

|𝑦|𝜁 (𝑦) 𝑑𝑦 =

Inside the unit ball we have

∫

𝐵1

∫

|𝑦|𝜁 (𝑦) +

∫

R𝑑\𝐵1

|𝑦|𝜁 (𝑦) =: 𝐽1 + 𝐽2.

𝐵1
Outside the unit ball we can use polar coordinates along with |𝜁 |(𝑦) ≤ 𝐶𝜁 |𝑦|−𝑞 for 𝑞 > 𝑑 + 1,

𝐽1 ≤

𝜁 (𝑦) 𝑑𝑦 ≤ 1.

∫

𝐽2 ≤ 𝐶𝜁

= 𝐶𝜁,𝑑

= 𝐶𝜁,𝑑

< ∞

|𝑦|1−𝑞 𝑑𝑦

R𝑑\𝐵1
∫ ∞

𝑟 𝑑−𝑞 𝑑𝑟

1

−1
𝑑 − 𝑞 + 1

where using the power rule is justified as 𝑑 − 𝑞 < −1. Note that as 𝑟 → ∞, then 𝑟 𝑑−𝑞+1 → 0 as

𝑑 − 𝑞 + 1 < 0.

Lemma A.0.3. If |𝜁 | < 𝐶𝜁 |𝑥|−𝑞 for 𝑞 > 𝑑 + 𝑝, then 𝑀𝑝 (𝜁) < ∞. In particular,

𝑀𝑝 (𝜁𝜖 ∗ 𝜇) ≤ 2𝑝−1 (cid:0)𝑀𝑝 (𝜇) + 𝜖 𝑝 𝑀𝑝 (𝜁)(cid:1)

is finite.

107

Proof. The first statement follows from generalizing the 𝑀1(𝜁) calculation, namely,

𝐶𝜁

∫

R𝑑\𝐵1

|𝑦| 𝑝−𝑞 𝑑𝑦 = 𝐶𝜁,𝑑

∫ ∞

1

𝑟 𝑑−1+𝑝−𝑞 𝑑𝑟

= 𝐶𝜁,𝑑

−1
𝑑 + 𝑝 − 𝑞

< ∞

where the integration is justified as 0 > 𝑑 + 𝑝 − 𝑞. Let us focus on the second statement. We can

rewrite

𝑀𝑝 (𝜁𝜖 ∗ 𝜇) =

∬

𝜁𝜖 (𝑦)|𝑥 − 𝑦| 𝑝 𝑑𝑦 𝑑𝜇(𝑥).

Recall that |𝑥 − 𝑦| 𝑝 ≤ 2𝑝−1(|𝑥| 𝑝 + |𝑦| 𝑝) and 𝜇, 𝜁𝜖 are probability measures. As 𝑀𝑝 (𝜁𝜖 ) = 𝜖 𝑝 𝑀𝑝 (𝜁)

by change of variables, then we get

𝑀𝑝 (𝜁𝜖 ∗ 𝜇) ≤ 2𝑝−1 (cid:0)𝑀𝑝 (𝜇) + 𝜖 𝑝 𝑀𝑝 (𝜁)(cid:1) .

Thus as long 𝜖 is finite then so is 𝑀𝑝 (𝜁𝜖 ∗ 𝜇).

Bounds for Newtonian and Bessel Kernels

We now move on to some aggregation kernels. In particular, bounds for the Newtonian and

Bessel kernels.

Lemma A.0.4 (𝐿 𝑝 norm of the Newtonian and Bessel Kernels). Let 𝑅 > 0. For 𝑑 ≥ 3, N ∈
𝐿 𝑝 (𝑅𝑑\𝐵𝑅) for 𝑝 > 𝑑
𝑑−2. Similarly for 𝑑 ≥ 3, ∇N ∈ 𝐿 𝑝 (𝑅𝑑\𝐵𝑅) for
𝛼 . Furthermore, ∇B𝛼,𝑑 has

𝑑−2 and N ∈ 𝐿 𝑝 (𝐵𝑅) for 𝑝 < 𝑑

𝑑−1 and ∇N ∈ 𝐿 𝑝 (𝐵𝑅) for 𝑝 < 𝑑

𝑑−1. For 𝛼 > 0, ∥B𝛼,𝑑 ∥ 𝐿1 (𝐵𝑅) ≤ 1

𝑝 > 𝑑

the same 𝐿 𝑝-ness as ∇N for 𝑑 ≥ 3.

Proof. Let 0 < 𝜖 < 𝑅. Let 𝐴𝑅

𝜖 → 0, 𝐴𝑅

𝜖 → 𝐵𝑅 and as 𝑅 → ∞, 𝐴𝑅

𝜖 be an annulus with inner radius 𝜖 and outer radius 𝑅. Thus, as
𝜖 → R𝑑\𝐵𝜖 (this is written as R𝑑\𝐵𝑅 in the statement). The

first gives us information locally while the second gives us information away from the origin. For

𝑑 ≥ 3, computing via polar coordinates,

∥N ∥

𝑝
𝐿 𝑝 ( 𝐴𝑅
𝜖 )

= 𝐶 𝑝

𝑑 𝑑𝛼𝑑

∫ 𝑅

𝜖

𝑟 𝑑−1𝑟 (2−𝑑) 𝑝 𝑑𝑟 = 𝐶 𝑝

𝑑 𝑑𝛼𝑑

∫ 𝑅

𝜖

𝑟 𝑑−1+(2−𝑑) 𝑝 𝑑𝑟.

108

Away from the origin, we require 𝑑 − 1 + (2 − 𝑑) 𝑝 < −1. So, 𝑝 > 𝑑
𝑑 − 1 + (2 − 𝑑) 𝑝 > −1. So, 𝑝 < 𝑑

𝑑−2. For 𝑑 ≥ 3,

𝑑−2. Locally, we require

∥∇N ∥

𝑝
𝐿 𝑝 ( 𝐴𝑅
𝜖 )

= 𝐶 𝑝

𝑑 𝑑𝛼𝑑

∫ 𝑅

𝜖

𝑟 𝑑−1+(1−𝑑) 𝑝 𝑑𝑟.

Away from the origin, we require 𝑑 − 1 + (1 − 𝑑) 𝑝 < −1. So, 𝑝 > 𝑑
𝑑 − 1 + (1 − 𝑑) 𝑝 > −1. So, 𝑝 < 𝑑

𝑑−1.

𝑑−1. Locally, we require

Define 𝑔(𝑥, 𝑡) =

1
(4𝜋𝑡)𝑑/2

𝑒−|𝑥|2/(4𝑡), 𝑓 (𝑥, 𝑡) = 𝑒−𝛼𝑡𝑔(𝑥, 𝑡). Then,

B𝛼,𝑑 (𝑥) =

∫ ∞

0

𝑓 (𝑥, 𝑡) 𝑑𝑡.

Given that 𝑔 is the heat kernel and its 𝐿1(R𝑑) norm is one, then

∥B𝛼,𝑑 ∥ 𝐿1 (R𝑑) ≤

∫ ∞

0

𝑒−𝛼𝑡 𝑑𝑡 =

1
𝛼

By lemma 2.4 of [8], |∇B𝛼,𝑑 (𝑥)| ≤ 𝐶𝑑 |𝑥|1−𝑑𝑔𝛼 (|𝑥|) where 𝑔𝛼 (|𝑥|) is a positive radial function

exponentially decreasing from 1 to 0 as |𝑥| → ∞. Moreover, |∇B𝛼,𝑑 (𝑥)| ≤ 𝐶𝑑 |𝑥|1−𝑑. Therefore,

|∇B𝛼,𝑑 (𝑥)| is proportional to |∇N (𝑥)|. Thus, ∇B𝛼,𝑑 has the same 𝐿 𝑝-ness as ∇N for 𝑑 ≥ 3.

109

APPENDIX B

CONSTRUCTION OF SUPERSOLUTION AND HEURISTICS

Construction of Supersolution

Here we will briefly talk about the construction of the barrier used for lemma 3.4.2. More details

can be seen in [12, Lemmas 8.1 - 8.3].

Lemma B.0.1 ([12] Lemma 8.1). Assume 𝑎, 𝑏 are radial in space. Then,

𝜑(|𝑥|, 𝑡) :=

1
𝑑

∫ |𝑥|

0

𝑎(𝑟, 𝑡)
𝑏(𝑟, 𝑡)

𝑟 𝑑𝑟

solves ∇ ·

(cid:17)

(cid:16) 𝑏
𝑎 ∇𝜑

= 1 in R𝑑 where there exists constants 𝜅𝑖 > 0 for 𝑖 = 1, 2, 3 such that 𝜅1|𝑥|2 ≤

𝜑(|𝑥|, 𝑡) ≤ 𝜅2|𝑥|2 and |∇𝜑(|𝑥|, 𝑡)| ≤ 𝜅3|𝑥|.

Lemma B.0.2 (Supersolutions to pressure equation for radial assumption, [12] Lemma 8.2). Let

𝑎, 𝑏 be radial in space and define 𝜑 as above. Define

𝑍 (𝑥, 𝑡) = 𝛼 |𝑅(𝑡) − 𝜑(|𝑥|, 𝑡)|+ , 𝑅(𝑡) = 𝛼 exp ((𝐶2 + ∥(1/𝑎) ∥ 𝐿∞𝐶1𝛼)𝛼𝑡) ,

for constants 𝐶1, 𝐶2 > 0 and 𝛼 > ∥Φ𝑏/𝑎∥ 𝐿∞ + ∥𝜕𝑡 𝑏∥ 𝐿∞. Then, 𝑍 is a supersolution for (3.4).

Proof. Note that we are only interested in the region {𝑅(𝑡) ≥ 𝜑(|𝑥|, 𝑡)} by the definition of 𝑍. We

can rewrite (3.4) as

𝜕𝑡 𝑝𝑚 =

|∇𝑝𝑚 |2
𝑎

+ (𝑚 − 1)

𝑝𝑚
𝑏

(cid:18)

∇ ·

(cid:18) 𝑏
𝑎

∇𝑝𝑚

(cid:19)

+

𝑏
𝑎

Φ(𝑥, 𝑡, 𝑝𝑚) − 𝜕𝑡 𝑏

(cid:19)

.

By the previous lemma and the hypothesis of this lemma,

∇ ·

(cid:19)

+

∇𝑍

(cid:18) 𝑏
𝑎

𝑏
𝑎

Φ(𝑥, 𝑡, 0) − 𝜕𝑡 𝑏 = −𝛼 +

𝑏
𝑎

Φ(𝑥, 𝑡, 0) − 𝜕𝑡 𝑏

Thus, it is left to show that

< 0.

𝜕𝑡 𝑍 ≥

𝛼2|∇𝜑|2
𝑎

.

110

By the bounds in Lemma B.0.1,

and

|∇𝜑|2 ≤ 𝜅2

3|𝑥|2 ≤

𝜅2
3
𝜅1

|𝜑| ≤

𝜅2
3
𝜅1

𝑅(𝑡),

|𝜕𝑡 𝜑| ≤ 𝜅4|𝜑| ≤ 𝜅4𝑅(𝑡),

for some constant 𝜅4. So for some constants 𝐶1, 𝐶2 > 0,

𝜕𝑡 𝑍 −

𝛼2|∇𝜑|2
𝑎

≥ 𝛼𝜕𝑡 𝑅 − (𝛼𝐶2 + ∥(1/𝑎) ∥ 𝐿∞𝛼2𝐶1)𝑅 ≥ 0.

This ends the proof.

Lemma B.0.3 (Supersolutions to pressure equation for fast enough decay assumption, [12] Lemma

8.3). Suppose Assumption 3.2.3 holds. Define

𝑍 (𝑥, 𝑡) = 𝛼 |𝑅(𝑡) − 𝜑(|𝑥|, 𝑡)|+ , 𝑅(𝑡) = 𝛼 exp ((𝐶2 + ∥(1/𝑎) ∥ 𝐿∞𝐶1𝛼)𝛼𝑡) ,

for constants 𝐶1, 𝐶2 > 0 and 𝛼 > ∥Φ𝑏/𝑎∥ 𝐿∞ + ∥𝜕𝑡 𝑏∥ 𝐿∞. Then, 𝑍 is a supersolution for (3.4).

Moreover, 𝑍 is bounded in 𝐿∞(𝑄𝑇 ) and is compactly supported for an fixed time.

Proof. We construct a positive subsolution of ∇ ·

(cid:17)

(cid:16) 𝑏
𝑎 ∇𝑢

= 1 in R𝑑. We start with the construction

inside a ball centered at the origin 𝐵𝑅. Let us examine the unique solution of

∇ ·

(cid:17)

(cid:16) 𝑏
𝑎 ∇𝜙

= 1

in 𝐵𝑅 × [0, 𝑇],

𝜙(𝑥, 𝑡) = 1 on 𝜕𝐵𝑅.





By standard estimate for uniformly elliptic PDEs ([20]), there is a constant 𝐶 = 𝐶 (𝑎, 𝑏, 𝑅, 𝑑) such

that

sup
(𝑥,𝑡)∈𝐵𝑅×[0,𝑇]

|𝜙(𝑥, 𝑡)| ≤ 𝐶.

As 𝜙(𝑥, 𝑡) + 2𝐶 is positive and solves the same PDE with constant boundary data, we may assume

that 𝜙(𝑥, 𝑡) > 0 on 𝐵𝑅 × [0, 𝑇]. Now we focus on the regularity in time. By estimates in [20], there

is a constant 𝐶 = 𝐶 (𝑎, 𝑏, 𝑅, 𝑑) such that

sup
(𝑥,𝑡)∈𝐵𝑅×[0,𝑇]

𝑑
∑︁

𝑖=1

|𝜕𝑖𝜙(𝑥, 𝑡)| +

sup
(𝑥,𝑡)∈𝐵𝑅×[0,𝑇]

𝑑
∑︁

𝑖, 𝑗=1

|𝜕2

𝑖, 𝑗 𝜙(𝑥, 𝑡)| ≤ 𝐶.

111

Differentiating the elliptic equation in time,

∇ ·

(cid:16)

𝜕𝑡

(cid:17)

(cid:16) 𝑏
𝑎

(cid:17)

∇𝜙

+ ∇ ·

(cid:16) 𝑏
𝑎 ∇𝜕𝑡 𝜙

(cid:17)

= 0

in 𝐵𝑅 × [0, 𝑇],

𝜕𝑡 𝜙(𝑥, 𝑡) = 0 on 𝜕𝐵𝑅.





As ∇ ·

(cid:16)

𝜕𝑡

(cid:17)

(cid:16) 𝑏
𝑎

(cid:17)

∇𝜙

is smooth in space and bounded 𝐵𝑅 × [0, 𝑇], again by uniformly elliptic estimates

in [20], there is a constant 𝐶 = 𝐶 (𝑎, 𝑏, 𝑅, 𝑑) such that

|𝜕𝑡 𝜙(𝑥, 𝑡)| ≤ 𝐶.

For |𝑥| ≥ 𝑅 we have by Assumption 3.2.3,

∇ ·

(cid:19)

∇|𝑥|2

(cid:18) 𝑏
𝑎

= 2𝑑

𝑏
𝑎

+ 2∇

(cid:19)

(cid:18) 𝑏
𝑎

· 𝑥 ≥

2𝑑
Λ2

− 2𝜖 ≥

.

1
Λ2

Define 𝑤(𝑥) = 1 + 𝐶 (|𝑥|2 − 𝑅2(𝑡)). From the regularity of 𝜙, we can choose 𝐶 ≥ Λ2 large enough

so that for if 𝑥 ∈ 𝜕𝐵𝑅, then

Thus if we define

𝜕𝑤
𝜕|𝑥|

(𝑥) >

𝜕𝜙
𝜕|𝑥|

(𝑥).

𝜑(𝑥, 𝑡) =

𝜙(𝑥, 𝑡)

for |𝑥| ≤ 𝑅,

𝑤(𝑥)

for |𝑥| ≥ 𝑅,





then we have that 𝜑 is a viscosity solution of ∇ ·

(cid:17)

(cid:16) 𝑏
𝑎 ∇𝑢

≥ 1. It follows that 𝜕𝑡 𝜑 is bounded as

|𝜕𝑡 𝜙| ≤ 𝐶 when |𝑥| ≤ 𝑅 and 𝜕𝑡𝑤 = 0 for |𝑥| ≥ 𝑅. The results on 𝑍 are achieved in the same way as

in the case when 𝑎, 𝑏 are radial in space.

Complementarity Condition Heuristics

We first examine the heuristics of the complementarity condition. We will find the equation for

the pressure using density equation (3.1). From there, we will see the complementarity condition.

Given the definition of 𝑝𝑚,

∇𝑝𝑚 = 𝑚

(cid:16) 𝑢𝑚
𝑏

(cid:17) 𝑚−2 (cid:18) ∇𝑢𝑚
𝑏

(cid:19)

𝑢𝑚∇𝑏
𝑏2

−

⇒ 𝑚

(cid:16) 𝑢𝑚
𝑏

(cid:17) 𝑚−2 ∇𝑢𝑚

𝑏

= ∇𝑝𝑚 + (𝑚 − 1) 𝑝𝑚

∇𝑏
𝑏

.

112

Similarly,

𝜕𝑡 𝑝𝑚 = 𝑚

(cid:16) 𝑢𝑚
𝑏

(cid:17) 𝑚−2 (cid:18) 𝜕𝑡𝑢𝑚
𝑏

(cid:19)

𝑢𝑚𝜕𝑡 𝑏
𝑏2

−

⇒ 𝑚

(cid:16) 𝑢𝑚
𝑏

(cid:17) 𝑚−2 𝜕𝑡𝑢𝑚

𝑏

= 𝜕𝑡 𝑝𝑚 + (𝑚 − 1) 𝑝𝑚

𝜕𝑡 𝑏
𝑏

.

We multiply (3.1) by 𝑚
𝑏

(cid:0) 𝑢𝑚
𝑏

(cid:1) 𝑚−2 to achieve,

𝜕𝑡 𝑝𝑚 + (𝑚 − 1) 𝑝𝑚

𝜕𝑡 𝑏
𝑏

𝑚
𝑏

(cid:16) 𝑢𝑚
𝑏

=

(cid:17) 𝑚−2 (cid:16)

∇

(cid:17)

(cid:16) 𝑢𝑚
𝑎

(cid:17)

· ∇𝑝𝑚

+ (𝑚 − 1)

𝑝𝑚
𝑎

(Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚)).

Focusing on the first term on the right-hand side,

𝑚
𝑏

(cid:16) 𝑢𝑚
𝑏

(cid:17) 𝑚−2 (cid:16)

∇

(cid:17)

(cid:16) 𝑢𝑚
𝑎

(cid:17)

· ∇𝑝𝑚

=

=

=

(cid:18)

𝑚∇𝑝𝑚
𝑏
∇𝑝𝑚
𝑎
|∇𝑝𝑚 |2
𝑎

(cid:16) 𝑢𝑚
𝑏

(cid:17) 𝑚−2 (cid:18) ∇𝑢𝑚
𝑎

−

(cid:19)

𝑢𝑚∇𝑎
𝑎2
(cid:19)

∇𝑏
𝑏
(cid:18) ∇𝑏
𝑏

∇𝑝𝑚 + (𝑚 − 1) 𝑝𝑚

− (𝑚 − 1)

+ (𝑚 − 1)

𝑝𝑚∇𝑝𝑚
𝑎

(cid:19)

.

∇𝑎
𝑎

−

𝑝𝑚
𝑎

∇𝑝𝑚 · ∇𝑎
𝑎

As an aside, we could write ∇𝑏

𝑏 − ∇𝑎

𝑎 = ∇ log

(cid:17)

(cid:16) 𝑏
𝑎

if desired. Therefore, the equation for the pressure

is

𝜕𝑡 𝑝𝑚 + (𝑚 − 1) 𝑝𝑚𝜕𝑡 log(𝑏) =

|∇𝑝𝑚 |2
𝑎

+ (𝑚 − 1)

(cid:18)

𝑝𝑚
𝑎

∇𝑝𝑚 · ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

+ Δ𝑝𝑚 + Φ(𝑥, 𝑡, 𝑝𝑚)

(cid:19)

.

Formally, letting 𝑚 → ∞ we find the complementarity condition

(cid:18)

𝑝∞
𝑎

∇𝑝∞ · ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

+ Δ𝑝∞ + Φ(𝑥, 𝑡, 𝑝∞) − 𝑎𝜕𝑡 log(𝑏)

(cid:19)

= 0

in

{𝑝∞(𝑥, 𝑡) > 0}.

Furthermore, we get the Hele-Shaw free boundary problem,

(FBP)





−Δ𝑝∞ = ∇𝑝∞ · ∇ log

(cid:17)

(cid:16) 𝑏
𝑎

+ Φ(𝑥, 𝑡, 𝑝∞) − 𝑎𝜕𝑡 log(𝑏)

in {𝑥 : 𝑝∞(𝑥, 𝑡) > 0},

𝑉 =

𝜕𝑡 𝑝∞
|∇𝑝∞| = 1

𝑎 |∇𝑝∞|

on 𝜕{𝑥 : 𝑝∞(𝑥, 𝑡) > 0}.

For some calculations it is easier to evaluate the normalized density 𝑣𝑚 = 𝑢𝑚

𝑏 . Similar to the

calculation above, we have the equation for the normalized density

𝜕𝑡𝑣𝑚 + 𝑣𝑚𝜕𝑡 log(𝑏) =

(cid:18)

1
𝑎

∇𝑣𝑚

𝑚 · ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

+ Δ𝑣𝑚

𝑚 + 𝑣𝑚Φ(𝑥, 𝑡, 𝑝𝑚)

(cid:19)

.

113

In particular, see that

𝜕𝑡𝑣𝑚 =

=

𝜕𝑡𝑢𝑚
𝑏
𝜕𝑡𝑢𝑚
𝑏

−

𝑢𝑚
𝑏

𝜕𝑡 𝑏
𝑏

− 𝑣𝑚𝜕𝑡 log(𝑏).

Dividing (3.2) by 𝑏,

𝜕𝑡𝑣𝑚 + 𝑣𝑚𝜕𝑡 log(𝑏) =

=

=

∇ ·

1
𝑏
Δ𝑣𝑚
𝑚
𝑎
Δ𝑣𝑚
𝑚
𝑎

+

+

1
𝑏

1
𝑎

(cid:19)

∇𝑣𝑚
𝑚

(cid:18) 𝑏
𝑎

(cid:18) 𝑏
𝑎

∇

+

(cid:19)

𝑣𝑚
𝑎

Φ(𝑥, 𝑡, 𝑝𝑚)

· ∇𝑣𝑚

𝑚 +

𝑣𝑚
𝑎

Φ(𝑥, 𝑡, 𝑝𝑚)

∇𝑣𝑚

𝑚 · ∇ log

(cid:19)

(cid:18) 𝑏
𝑎

+

𝑣𝑚
𝑎

Φ(𝑥, 𝑡, 𝑝𝑚).

Velocity Law Heuristics

We now move on to the heuristics of the velocity law. The heuristics here is similar to the

heuristics in [21]. Denote 𝑢𝐼

∞, 𝑢𝐸
respectively. Starting with (3.2) (after formally taking 𝑚 → ∞),

∞ as the internal and external density of Ω(𝑡) = {𝑥 : 𝑝∞(𝑥, 𝑡) > 0},

∫

R𝑑

𝑢∞
𝑎

Φ =

=

=

=

=

𝑑
𝑑𝑡
𝑑
𝑑𝑡
∫

𝑢∞

∫

R𝑑
(cid:18)∫

𝑢∞ +

Ω(𝑡)
𝜕𝑡𝑢𝐼

∞ +

∫

∫

R𝑑\Ω(𝑡)

(cid:19)

𝑢∞

∫

𝜕𝑡𝑢𝐸

∞ +

𝑉 (𝑢𝐼

∞ − 𝑢𝐸
∞)

Ω(𝑡)

∫

Ω(𝑡)

∫

𝜕Ω(𝑡)

R𝑑\Ω(𝑡)
(cid:19)

∫

∇𝑝∞

+

∇ ·

(cid:18) 𝑏
𝑎

𝑢∞
𝑎

R𝑑

𝜕Ω(𝑡)
∫

Φ +

𝑏
𝑎

∇𝑝∞ · 𝜈 + 𝑉 (𝑏 − 𝑢𝐸

∞) +

∫

𝜕Ω(𝑡)
𝑢∞
𝑎

R𝑑

Φ.

𝑉 (𝑢𝐼

∞ − 𝑢𝐸
∞)

This suggests that in the presence of the mushy region (where 0 < 𝑢∞ < 𝑏) the normal boundary

velocity of 𝜕Ω(𝑡) satisfies (1 − 𝑢𝐸

∞/𝑏)𝑉 = −(1/𝑎)∇𝑝∞ · 𝜈 since 𝑏 > 0. When the external density

vanishes on the boundary, then we get the velocity law as expected.

114