by Geordon Haley Richards - University of Toronto T-Space...Geordon Haley Richards Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2012 This thesis contributes

Maximal-in-time behavior of deterministic and stochasticdispersive partial differential equations

by

Geordon Haley Richards

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

Copyright c© 2012 by Geordon Haley Richards

Abstract

Maximal-in-time behavior of deterministic and stochastic dispersive partial differential

equations

Geordon Haley Richards

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2012

This thesis contributes towards the maximal-in-time well-posedness theory of three

nonlinear dispersive partial differential equations (PDEs). We are interested in questions

that extend beyond the usual well-posedness theory: what is the ultimate fate of solu-

tions? How does Hamiltonian structure influence PDE dynamics? How does randomness,

within the PDE or the initial data, interact with well-posedness of the Cauchy problem?

The first topic of this thesis is the analysis of blow-up solutions to the elliptic-elliptic

Davey-Stewartson system, which appears in the description of surface water waves. We

prove a mass concentration property for H1(R2)-solutions, analogous to the one known for

the L2(R2)-critical nonlinear Schrödinger equation. We also prove a mass concentration

result for L2(R2)-solutions.

The second topic of this thesis is the invariance of the Gibbs measure for the (gauge

transformed) periodic quartic KdV equation. The Gibbs measure is a probability measure

supported on Hs(T) for s < 12, and local solutions to the quartic KdV cannot be obtained

below H12 (T) by using the standard fixed point method. We exhibit nonlinear smoothing

when the initial data are randomized, and establish almost sure local well-posedness

for the (gauge transformed) quartic KdV below H12 (T). Then, using the invariance of

the Gibbs measure for the finite-dimensional system of ODEs given by projection onto

the first N � 0 modes of the trigonometric basis, we extend the local solutions of the

(gauge transformed) quartic KdV to global solutions, and prove the invariance of the

ii

Gibbs measure under the flow. Inverting the gauge, we establish almost sure global

well-posedness of the (ungauged) periodic quartic KdV below H12 (T).

The third topic of this thesis is well-posedness of the stochastic KdV-Burgers equa-

tion. This equation is studied as a toy model for the stochastic Burgers equation, which

appears in the description of a randomly growing interface. We are interested in rig-

orously proving the invariance of white noise for the stochastic KdV-Burgers equation.

This thesis provides a result in this direction: after smoothing the additive noise (by a

fractional derivative), we establish (almost sure) local well-posedness of the stochastic

KdV-Burgers equation with white noise as initial data. We also prove a global well-

posedness result under an additional smoothing of the noise.

iii

Dedication

To Clara.

iv

Acknowledgements

I would like to thank my wonderful advisor, James Colliander, for the years of aca-

demic, intellectual and personal support. It has been an honor and a pleasure to learn

from James, and to participate in his virtuous and inspired approach to creating new

mathematics.

I am grateful to Tadahiro Oh, who, in his final years as a post-doc at the University

of Toronto, taught me a tremendous amount of new math, and helped narrow my focus

towards some very exciting and accessible directions for research. Tadahiro’s guidance

and generosity have been instrumental to my completion of this thesis.

There are numerous faculty members at the University of Toronto who have helped

in substantial ways over the years. I would particularly like to thank Catherine Sulem

and Jeremy Quastel, but I am also grateful to Almut Burchard, Michael Goldstein, Abe

Igelfeld, Robert Jerrard, Robert Mccann, Mary Pugh and Joe Repka, among others, for

their guidance and support.

The staff of the Department of Mathematics at the University of Toronto cannot

receive enough praise for their fantastic work. Particularly Ida Bulat, whose presence

makes the graduate program accessible and accommodating to all students. Ida’s pa-

tience, kindness, and excellence in her work are truly marvelous to behold.

I would also like to thank some of the people who sparked my interest in mathematics

at an early age. Particularly my brother Blake Richards, and my high school math

teacher, Irene Buckiewicz.

I am hugely indebted to my mother Rose Cullis and father Doug Richards for the

tremendous amount of emotional and physical support they have provided. I am ex-

tremely lucky to have such loving and brilliant parents.

Finally I am grateful for the presence of my friends and extended family. Particularly

my partner Clara, who I love very much, and whose encouragement has been so vital in

recent years.

This research was partially supported by grants from the governments of Ontario and

Canada.

v

Contents

1 Introduction 1

1.1 Maximal-in-time behavior of solutions . . . . . . . . . . . . . . . . . . . . 3

1.2 Invariant measures for Hamiltonian PDEs . . . . . . . . . . . . . . . . . 7

1.3 Well-posedness theory of stochastic PDEs . . . . . . . . . . . . . . . . . 12

2 Mass concentration for the Davey-Stewartson system 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Mass concentration for H1-solutions . . . . . . . . . . . . . . . . . . . . . 22

2.3 A compactness property . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Mass concentration for L2-solutions . . . . . . . . . . . . . . . . . . . . . 31

3 Invariance of the Gibbs measure for the quartic KdV 37

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Linear Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Nonlinear estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Local well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Global well-posedness and invariance of the Gibbs measure . . . . . . . . 74

4 Well-posedness of the stochastic KdV-Burgers equation 96

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 Local well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.3 Global well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Appendix 122

5.1 Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.1.1 Profile decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.1.2 Squares and tubes lemmata . . . . . . . . . . . . . . . . . . . . . 123

vi


5.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2.2 Probabilistic quadrilinear estimates . . . . . . . . . . . . . . . . . 133

5.2.3 Probabilistic heptilinear estimates . . . . . . . . . . . . . . . . . . 163

5.2.4 Deterministic nonlinear estimates . . . . . . . . . . . . . . . . . . 176

5.2.5 Proofs of lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.2.6 Nonlinear smoothing under randomization . . . . . . . . . . . . . 188


5.3.1 Bilinear estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5.3.2 Global well-posedness for the cut-off system . . . . . . . . . . . . 204

5.3.3 Invariance of white noise for the cut-off system . . . . . . . . . . . 211

vii

Chapter 1

Introduction

This thesis is concerned with maximal-in-time well-posedness properties of nonlinear

dispersive partial differential equations (PDEs). A basic example of a dispersive PDE is

the nonlinear Schrödinger equation (NLS)

i∂tu+ ∆u = ±|u|p−1u, x ∈M, t ∈ [0, T ], (NLS)

where u : [0, T ]×M → C, and M is a manifold (we will concentrate on the cases M = Rn

and M = Tn). This equation is said to be focusing in the (-) case, and defocusing in the(+) case. Another basic example is the generalized Korteweg-de Vries equation (gKdV)

∂tu+ ∂3xu =

1

p∂x(u

p), x ∈M, t ∈ [0, T ], (gKdV)

where u : [0, T ]×M → R, and M = R or M = T.Dispersive PDEs such as NLS and gKdV are canonical model equations that arise

in physics and engineering. For example, the cubic NLS (p = 3) is used to describe

the envelope of a monochromatic plane wave propagating in a weakly nonlinear medium

[69]. The (standard) KdV (gKdV with p = 2 and M = R) was introduced [42] to modelsolitary gravity waves in a shallow canal. These equations and their variants have been

the focus of intense mathematical study (see [71] and references therein). Interest in this

subject is reinforced by the underlying structure: NLS and gKdV can be interpreted as

infinite-dimensional Hamiltonian systems.

Stemming from breakthrough works of Bourgain [4], Klainerman-Machedon [41], and

Kenig-Ponce-Vega [40], the application of techniques from harmonic analysis has led

to spectacular progress in the well-posedness theory of dispersive PDEs via the fixed

1

Chapter 1. Introduction 2

point method1. This thesis is focused on three topics that extend beyond the usual

well-posedness theory.

1. What is the ultimate fate of solutions?

That is, when and how can local well-posedness of a given PDE be extended to global

well-posedness? For some dispersive PDEs, there exist (finite time) blow-up solutions,

which leave the initial data function space in finite time. In this context it is intriguing

to investigate further: what is the nature of the blow-up phenomenon?

• The main result of Chapter 2 is a mass concentration property for H1(R2) blow-upsolutions to the elliptic-elliptic Davey-Stewartson (DS) system [67], which appears

in the description of water waves. This property reveals that blow-up solutions

to the elliptic-elliptic DS concentrate (at least) a universal amount of mass into a

spatial singularity in finite time.

2. Can properties of finite-dimensional Hamiltonian dynamics be extended to infinite

dimensions?

We are particularly interested in proving the invariance of canonical probability mea-

sures, such as the Gibbs measure, under the flow of Hamiltonian PDEs in the periodic

setting. There are some technical difficulties: for a given PDE, the existence of solutions

in the support of the Gibbs measure (a necessary condition for invariance) may be far

from obvious. Indeed, the Gibbs measure is supported on functions of low regularity in

space, and this question of existence is non-trivial.

The Gibbs measure for periodic NLS (and gKdV) was first constructed in Lebowitz-

Rose-Speer [46]. A scheme for proving its invariance under the flow was developed by

Bourgain [6] (see also McKean-Vaninsky [47]). To establish the existence of global-in-

time dynamics, Bourgain used the invariant measure as a substitute for a conservation

law. More precisely, he showed that the invariance of the Gibbs measure under a finite-

dimensional system of approximating ODEs can be used to extend local solutions of the

PDE (evolving from data in the support of the Gibbs measure) to global solutions almost

surely. Conversely, the existence of these global solutions makes a proof of invariance

possible.

With this strategy in place, the crucial ingredient to proving invariance of the Gibbs

measure for a given Hamiltonian PDE is local well-posedness in the support of the Gibbs

1These authors used solution spaces of Xs,b-type, which first appeared in works of Rauch-Reed [66]and Beals [2].


measure (with “good” estimates on solutions). In fact, it suffices for this to be established

on a statistical ensemble of randomized initial data.

This randomization can (in some cases) provide crucial assistance. In studying the

well-posedness of a dispersive PDE, one often encounters a critical regularity, below

which the PDE is “ill-posed”. That is, below the critical regularity, there is no hope (at

least, in using standard techniques) to establish well-posedness for all initial data (see for

example Bourgain [5], Kenig-Ponce-Vega [40] and Christ-Colliander-Tao [17]). However,

by randomizing initial data, one can sometimes prove local well-posedness almost surely

below this critical threshold (see for example Burq-Tzvetkov [12]).

• The main result of Chapter 3 is the invariance of the Gibbs measure for the (gauge-transformed) periodic quartic KdV. We exhibit nonlinear smoothing when the ini-

tial data are randomized, and prove global well-posedness almost surely in H12−(T),

where this PDE is deterministically “ill-posed”.

3. Can invariant measures be used to prove global well-posedness for stochastic PDEs?

It is natural to inquire if randomness interacts with well-posedness in other ways. In

particular, we are interested in the well-posedness theory of stochastic PDEs, and in the

possibility of using invariant measures following [6] to extend local solutions to global

solutions.

• The main result of Chapter 4 concerns the stochastic KdV-Burgers equation. Weare interested in proving the invariance of white noise under the flow. In this

direction, we establish a local well-posedness result with white noise as initial data.

The remainder of this introduction is organized as follows: for each of these topics, we

will provide more background, then state the results of this thesis.

1.1 Maximal-in-time behavior of solutions

Consider NLS in Euclidean space M = Rd. This equation has a scaling symmetry:if u(t, x) solves NLS with data u0 for t ∈ [0, T ], then for any λ > 0, uλ(t, x) :=λ

2p−1u(λ2t, λx) solves NLS with data u0,λ(x) := λ

2p−1u0(λx) for t ∈ [0, T/λ2]. We compute

that

‖u0,λ‖Ḣs(Rd) = ‖λ2p−1u0(λ·)‖Ḣs(Rd) = λ

s−sc‖u0‖Ḣs(Rd), (1.1.1)


where sc :=d2− 2

p−1 . The number sc is referred to as the critical Sobolev index for NLS

with power p in (spatial) dimension d. This computation leads to a heuristic for local

well-posedness:

• If u0 ∈ Hs(Rd) with s > sc (sub-critical), the scaling (1.1.1) is consistent withlocal well-posedness, and the time of local existence is anticipated to be inversely

proportional to the Hs-norm of the initial data (the scaling produces small Hs-norm

data with corresponding solutions that exist for a long time, and large Hs-norm

data with solutions that exist for a short time).

• If u0 ∈ Hs(Rd) with s = sc (critical), the scaling symmetry is less informative, andlocal well-posedness is a delicate issue.

• If u0 ∈ Hs(Rd) with s < sc (super-critical), the scaling symmetry produces solutionsevolving from data of smallHs-norm which exist for a short time only. This suggests

a potential instability of the NLS data-to-solution map (from Hs to CtHs), and

(some form of) ill-posedness is anticipated.

1.1.1 The cubic NLS in R2

Consider the cubic NLS (p = 3) in R2. This equation is L2(R2)-critical with respectto scaling (sc = 0), and locally well-posed in H

s(R2) for s ≥ 0 [15]. The local theorycan be iterated, and the solution extended to C([0, T ∗];Hs(R2)) for a maximal time ofexistence T ∗ > 0. For s > 0 (sub-critical regime), the time of local existence is inversely

proportional to the Hs-norm of the initial data, and this leads to the classical blow-up

alternative: either T ∗ = ∞, or limt→T ∗ ‖u(t)‖Hs = ∞. Natural questions emerge: doblow-up solutions exist? What is the mechanism behind the blow-up phenomenon?

The NLS has a conserved quantity referred to as the energy2, given by

E(u) =1

2

∫R2|∇u|2dx± 1

4

∫R2|u|4dx.

The conservation of energy leads to different scenarios for the maximal-in-time behaviour

of H1(R2)-solutions in the focusing and defocusing cases. In the defocusing (+) case, theconserved energy bounds the Ḣ1(R2)-norm of the solution: blow-up is impossible, andthe solution exists globally-in-time.

2This is also the Hamiltonian for NLS. To avoid confusion with later chapters, we will refer to theNLS Hamiltonian as the energy E(u), and reserve H(u) for the gKdV Hamiltonian.


In the focusing (-) case, energy conservation has more subtle consequences. The

standing wave ansatz u(t, x) = W (x)eit, plugged into the focusing cubic NLS, produces

an elliptic PDE for W which has a unique positive solution W (x) = Q(x) > 0 known

as the ground state solution [19]. If ‖u0‖L2 < ‖Q‖L2 , then due to the sharp Gagliardo-Nirenberg inequality [77], ∫

R2|u|4dx ≤ 2

‖Q‖2L2‖∇u‖2L2‖u‖2L2 ,

the energy still bounds the Ḣ1(R2)-norm, and the solution exists globally-in-time. If theinitial data satisfies ‖u0‖L2 ≥ ‖Q‖L2 , blow-up for the focusing cubic NLS is not ruled outby energy conservation.

In fact, H1(R2) blow-up solutions to the focusing cubic NLS exist, and the blow-upphenomenon has a rich structure. The cubic NLS has a pseudo-conformal symmetry3,

which, applied to the ground state standing wave solution, produces a blow-up solution

with the minimal mass ‖u0‖L2 = ‖Q‖L2 . Up to other symmetries of NLS, this is the onlyminimal mass blow-up solution [48]. All blow-up solutions to cubic NLS in R2 concentratemass. More precisely, they concentrate at least the mass ‖Q‖L2 into a parabolicallyshrinking window as t approaches the blow-up time T ∗ [10, 35, 50]. For solutions with

mass ‖u0‖L2 slightly larger than ‖Q‖L2 , the existence of two distinct blow-up regimes,precise norm explosion rates, and asymptotic profile properties have been established

[52].

The first contribution of this thesis is a mass concentration result for blow-up solutions

to the elliptic-elliptic Davey-Stewartson system [67]. We proceed to introduce the Davey-

Stewartson system, and to state our first set of results.

1.1.2 The Davey-Stewartson system

The elliptic-elliptic Davey-Stewartson system is given by

iut + ∆u+ L(|u|2)u = 0, x ∈ R2, t ≥ 0. (1.1.2)

Here L = νId + γB, ν = ±1, γ > 0, and B is the pseudo-differential operator withsymbol B̂(f)(ξ) = ξ

21

ξ21+ξ22f̂(ξ). The equation (1.1.2) first appeared in the description of

the evolution of surface water waves [26]. This system resembles the 2-d cubic NLS; the

3In fact, this is a symmetry for the L2-critical NLS (p = 1 + 4d ).


same local theory in Hs(R2) for s ≥ 0 has been established, complete with the blow-upalternative [32].

Asymptotic and numerical analyses of Papanicolaou-Sulem-Sulem-Wang [64] revealed

similar characteristics for blow-up solutions to (1.1.2) and NLS. We prove a mass concen-

tration result for H1(R2)-solutions to (1.1.2), and separately prove a mass concentrationresult for L2(R2)-solutions [67]. Here is the first result we obtain:

Theorem 1.1 (Mass concentration for H1(R2) blow-up solutions). Let u be a solutionof (1.1.2) with u0 ∈ H1(R2) which blows up in finite time T ∗ > 0, and λ(t) > 0 anyfunction such that λ(t)‖∇u(t)‖L2 → ∞ as t → T ∗. Then, there exists y(t) ∈ R2 suchthat

lim inft→T ∗

∫|x−y(t)|≤λ(t)

|u(t, x)|2dx ≥ 2Copt

,

where 2Copt

= ‖R‖2L2 for a function R ∈ H1(R2) such that R(x)eit solves (1.1.2) andR(x) > 0 for all x ∈ R2.

Theorem 1.1 is analogous to the theorem of Merle-Tsutsumi [50] for H1(R2) blow-upsolutions to the cubic NLS. To establish Theorem 1.1, we adapt the method of Hmidi-

Keraani [35]. We use a profile decomposition and a Gagliardo-Nirenberg-type inequality

of Papanicolaou-Sulem-Sulem-Wang [64] (designed for (1.1.2)), to prove a compactness

property for bounded sequences in H1(R2). This property is then applied to spatiallyrescaled snapshots (in time) of a blow-up solution u to prove mass concentration.

Here is the second result we obtain:

Theorem 1.2 (Mass concentration for L2(R2) blow-up solutions). Let u be a solution of(1.1.2) with u0 ∈ L2(R2) which blows up in finite time T ∗ > 0. Take M := ‖u0‖L2. Thenthere exist constants η(M), C > 0 such that

lim supt→T ∗

supsquares Q⊂R2

of sidelength `(Q) η(M) > 0.

Theorem 1.2 is analogous to the theorem of Bourgain [8] on mass concentration for L2(R2)blow-up solutions to cubic NLS. The proof of Theorem 1.2 follows the proof from [8]; we

modify the crucial estimates for application to a nonlocal nonlinearity. In Chapter 2 we

will provide more background on this problem and present the proofs Theorem 1.1 and

Theorem 1.2.


1.2 Invariant measures for Hamiltonian PDEs

Canonical dispersive PDEs such as NLS and gKdV are among the simplest infinite-

dimensional (nonlinear) Hamiltonian systems that appear in physics. Some of these PDEs

are completely integrable (e.g. KdV and defocusing cubic NLS in one space dimension),

and in these contexts, a detailed understanding of the PDE dynamics has emerged (see,

for example, [31, 36, 78]). Beyond this handful of cases, however, there remain many open

problems motivated by our understanding of Hamiltonian systems in finite dimensions.

In this section we will focus on one facet of Hamiltonian structure: the existence of

invariant measures on phase space.

1.2.1 The periodic generalized KdV

For the remainder of this section, we restrict attention to gKdV on the torus M = T.This equation can be equipped with a Hamiltonian structure. In particular, this system

has a conserved (if it is finite) Hamiltonian H(u) given by

H(u) :=1

2

∫Tu2xdx+

1

p(p+ 1)

∫Tup+1dx.

Then gKdV can be reformulated as

∂tu = ∂x∂H

∂u, (1.2.1)

where ∂H∂u

is the Fréchet derivative with respect to the L2 inner product 〈u, v〉 =∫T uvdx.

4

This Hamiltonian structure leads to a natural question: is the Gibbs measure “dµ =

e−H(u)du” invariant under the flow of gKdV?

The Gibbs measure µ for gKdV, first constructed in Lebowitz-Rose-Speer[46]5, is a

finite Borel measure supported onH12−(T). To ask the question of its invariance under the

flow, it is required that the evolution of gKdV is well-defined (globally-in-time) for initial

data in the support of the Gibbs measure. Bourgain [6] rigorously proved the invariance

of the Gibbs measure for KdV (p = 2) and for mKdV (p = 3). The evolution of KdV is

well-defined for all u0 ∈ L2(T), globally-in-time by conservation of the L2(T)-norm [4],

4This is at least formally correct, for the rigorous definition of gKdV as a Hamiltonian system with

Poisson structure J = ∂∂x on the Sobolev space H− 120 (T) (mean-zero functions) equipped with a compat-

ible symplectic form, see [44].5In fact, they constructed the Gibbs measure for the periodic NLS, but the same argument applies

to gKdV, see [6].


so it is certainly well-defined, globally-in-time, in H12−(T). For mKdV, Bourgain proved

local well-posedness in H12−,1−(T) = H 12−(T) ∩ FL1−,∞, where FLs,p is the space of

functions u with Fourier transform û satisfying 〈n〉sû(n) ∈ lpn. Bourgain showed that thespace Hs1,s2(T) supports the Gibbs measure when 0 < s1 < 12 < s2 < 1. He could not,however, use L2(T)-conservation to extend his local solutions globally-in-time. The mainnew idea implemented in [6] was to use the invariance of the Gibbs measure under the flow

of the finite-dimensional system of ODEs given by the projection of mKdV6 to the first

N � 0 modes of the trigonometric basis (and an approximation argument) as a substitutefor a conservation law, extending the local solutions of mKdV to global solutions, and

subsequently proving the invariance of the Gibbs measure under the (infinite-dimensional)

flow. That is, Bourgain found a way to use the (structure behind the) invariant Gibbs

measure in order to extend solutions globally-in-time. This has impact from a purely

PDE point of view: the invariant Gibbs measure has become a tool for proving low

regularity global well-posedness.

We are interested in proving the invariance of the Gibbs measure for gKdV with higher

power nonlinearity (p ≥ 4). In this work, we restrict attention to the Gibbs measure µfor gKdV supported on functions of mean zero, but will often omit the prefix “mean

zero”. It should be noted that the equation (1.2.3) preserves the mean of the solution∫T u(t, x)dx =

∫T u0(x)dx.

By following the method in [6], to establish local-in-time dynamics in the support of

the (mean zero) Gibbs measure, it is enough to prove local well-posedness of gKdV in

Hs(T) for some s < 12. As we shall see in the next section (see references to [21] below),

this local well-posedness result is not readily available for p ≥ 4. Luckily, it suffices toprove something weaker: to establish that gKdV is locally well-posed almost surely with

randomized initial data of the form

u0,ω(x) =∑

n∈Z\{0}

gn(ω)

|n|einx, (1.2.2)

where {gn}∞n=1 is a sequence of independent complex-valued Gaussian random variablesof mean 0 and variance 1 on a probability space (Ω,F ,P), and g−n = gn (in order for u0,ωto be real-valued). The expression (1.2.2) represents a typical element in the support of

the (mean zero) Wiener measure. The hope behind pursuit of this weaker local result is

6Bourgain also used this argument to prove the invariance of the Gibbs measure for the periodic NLS,but we will focus on gKdV in this discussion.


that, with high probability, solutions of gKdV evolving from the randomized data (1.2.2)

may behave nicely.

The strategy of exploiting randomized data to prove the invariance of the Gibbs

measure has appeared in previous works. For example, Bourgain [7] established the

invariance of the Gibbs measure for the (Wick-ordered) cubic NLS on T2. In two spacedimensions the Gibbs measure is supported below L2(T2), where the cubic NLS is ill-posed due to scaling. The resolution in [7] shows that the combination of initial data

randomization and dispersion induces a nonlinear smoothing effect, and the Cauchy

problem for the (Wick-ordered) cubic NLS is locally well-posed almost surely below

L2(T2). It should be remarked that certain structural features of the nonlinearity (e.g.Wick-ordering) are crucial to the smoothing effect in [7]. See also Burq-Tzvetkov [11,

12, 13], Oh [61], and Colliander-Oh [22] for other works that invoke nonlinear smoothing

under randomization of initial data.

The methods developed by Bourgain in [5, 7] for proving the invariance of the Gibbs

measure have been adapted and expanded to other Hamiltonian PDEs. Among other

results, Tzvetkov [74, 75] proved the invariance of the Gibbs measure for radial NLS on the

disc, Burq-Tzvetkov [11, 12, 13] did this for super-critical 3-d nonlinear wave equations,

and Oh for KdV-type coupled systems and Schrödinger-Benjamin-Ono system [57, 58].

1.2.2 The periodic quartic KdV

Consider the Cauchy problem for the periodic quartic KdV{∂tu+ ∂

3xu =

14∂x(u

4), t ≥ 0, x ∈ T,u(0, x) = u0(x) ∈ Hs(T).

(1.2.3)

Higher power generalized KdV equations (p ≥ 4) have been studied mainly by math-ematicians; there is interest in exploring the balance of a stronger nonlinearity with

dispersion.

The study of well-posedness for (1.2.3) was initiated in [4]; a fixed point argument was

used to establish local well-posedness in Hs(T), for s > 32. This was improved to local

well-posedness in Hs(T) for s ≥ 1 by Staffilani [70], and then to s ≥ 12

be Colliander-

Keel-Staffilani-Takaoka-Tao [21]. Below H12 (T), however, (1.2.3) is analytically ill-posed

[21]. That is, the data-to-solution map for (1.2.3) is not analytic in Hs(T) for s < 12.

In fact, it is not C4 (see also [5]). This indicates that we cannot use the standard fixed


point machinery to establish local well-posedness in Hs(T) for s < 12: the contraction

principle necessitates analyticity of the data-to-solution map.

We are interested in proving the invariance of the Gibbs measure for (1.2.3). To estab-

lish local dynamics in the support of the Gibbs measure, we exhibit nonlinear smoothing

when the initial data are randomized (according to (1.2.2)), and use this to prove that

(1.2.3) is locally well-posed almost surely in H12−(T).

The analysis of well-posedness for (1.2.3) is simplified by a gauge transformation.

This transformation preserves the initial data, and it is invertible. A function u satisfies

(1.2.3) if and only if

v(x, t) = G(u(x, t)) := u(x−

(∫ t0

∫Tu3(x′, t′)dx′dt′

), t

).

satisfies {∂tv + ∂

3xv = P(v3)∂xv, t ≥ 0, x ∈ T,

v(0, x) = u0(x) ∈ Hs(T),(1.2.4)

where P(u) = u− 12π

∫T udx is the projection to functions with mean zero. The analysis

of well-posedness for (1.2.4) is simpler than for (1.2.3), but the data-to-solution map still

fails to be C4 below H12 (T) [21]. Before we state our main results, let us emphasize

that our notion of solution to (1.2.4) (and other PDEs) is that of strong solution to the

integral formulation

u(t) = S(t)u0 +

∫ t0

S(t− t′)P((u(t′))3)∂xu(t′)dt′, (1.2.5)

for the time of existence t ∈ [0, T ], where S(t) := e−∂3xt is the linear propagator for KdV.

Our first theorem in this section is almost sure local well-posedness (LWP) of (1.2.4)

with initial data given by (1.2.2).

Theorem 1.3 (Almost sure LWP of the gauge-transformed quartic KdV). The system

(1.2.4) is locally well-posed almost surely in H12−(T) (with initial data given by (1.2.2)).

In particular, (1.2.4) is locally well-posed with respect to the Gibbs measure µ.

Remark 1.1. We should pause to clarify our use of the phrase “locally well-posed almost

surely” in the statement of Theorem 1.3. More precisely, we have that, almost surely,

there exists a random time Tω > 0 and a solution u to (1.2.4) for t ∈ [0, Tω] withinitial data u0,ω given by (1.2.2). The solution u is unique in a weak sense only (for


the precise statement of uniqueness see Theorem 3.1 in Section 3.1), and the solution u

depends continuously on the random data in a neighbourhood of u0,ω with higher Sobolev

regularity. That is, the data-to-solution map is Lipschitz on a ball in H12

+(T) centeredat the randomized data u0,ω, almost surely.

Theorem 1.3 will be proven using the Fourier restriction norm method by Bourgain

and nonlinear smoothing due to initial data randomization. We will exploit the random-

ized data to bootstrap this nonlinear smoothing effect in the Xs,b-space of functions of

space-time adapted to KdV. To prove Theorem 1.3, we found that we could not take

b = 12, and perform a contraction argument on a ball in the space Zs,

12 centered at the

linear evolution of the randomized data, as is done for the 2-d cubic NLS in [7] (for the

definitions of these function spaces, see Section 3.2 below). Instead, we will establish a

priori bounds (which incorporate the randomized data) on the second iteration of the

integral formulation of quartic KdV in Xs,b with b < 12

(see [5, 59, 61]). Next, as in

[6, 7], we use the invariance of finite-dimensional Gibbs measures, and an approximation

argument, to extend the almost sure local well-posedness of Theorem 1.3 to almost sure

global well-posedness. Then we prove the invariance of the Gibbs measure µ under the

flow of (1.2.4).

Theorem 1.4 (Invariance of the Gibbs measure for the gauge-transformed quartic KdV).

The gauged system (1.2.4) is globally well-posed almost surely in H12−(T) (with initial data

given by (1.2.2)). Moreover, the Gibbs measure µ is invariant under the flow.

Finally, by inverting the gauge, we obtain almost sure global well-posedness of (1.2.3),

the ungauged quartic KdV, below H12 (T).

Corollary 1.1 (Almost sure GWP of quartic KdV). The system (1.2.3) is globally well-

posed almost surely in H12−(T) (with initial data given by (1.2.2)).

In terms of global theory, global well-posedness of (1.2.3) in Hs(T) for s > 56

was

established in [21] using the I-method. This is mentioned to emphasize that, to our

knowledge, Corollary 1.1 is the first result to provide global-in-time solutions to (1.2.3)

outside H56 (T). In Chapter 3, we will return to this topic, providing a more complete

background, as well as detailed statements and proofs of these results.


1.3 Well-posedness theory of stochastic PDEs

For canonical model equations such as NLS and gKdV, it is natural to study perturbations

of these systems, and in particular, to study stochastic perturbations. A stochastic PDE

with additive noise can be interpreted as a PDE with a random forcing term.

1.3.1 The stochastic KdV with additive noise

Consider, for example, the periodic stochastic KdV equation with additive noise

du = (−uxxx −1

2(u2)x)dt+ φdW, t ≥ 0, x ∈ T (1.3.1)

where W (t, x) represents space-time white noise, and φ is a bounded operator on L2(T).It has been used to model wave propagation in a noisy plasma [16].

By well-posedness of a stochastic PDE, we mean path-wise local well-posedness almost

surely. More precisely, in the context of (1.3.1), we mean path-wise local existence and

uniqueness of a strong solution to the integral formulation

u(t) = S(t)u0 −1

2

∫ t0

S(t− t′)∂x((u(t′))2

)dt′ +

∫ t0

S(t− t′)φdW (t′), (1.3.2)

almost surely. It is important to clarify that it is typically the additive noise, not the

initial data, which determines the spatial regularity of the solution to a stochastic PDE.

In this way, interest in the well-posedness of stochastic PDEs such as (1.3.1) provides

impetus to study well-posedness of dispersive PDEs at low regularities in space.

The low temporal regularity of space-time white noise provides an additional obstruc-

tion to adapting deterministic well-posedness theory to stochastic PDEs. For example in

the context of (1.3.1), the last term in (1.3.2), which will be referred to as the stochastic

convolution (and which corresponds to the solution of the linear stochastic KdV equation

with zero initial datum), has the regularity of Brownian motion as a function of time.

Recall that Brownian motion is almost surely Hölder continuous with power α < 12, but

not with α = 12. Any Banach space of functions of space-time where we may hope to con-

struct solutions to stochastic PDEs (such as (1.3.1)) must be designed to accommodate

this low regularity in time.

De Bouard-Debussche-Tsutsumi [28] proved a local well-posedness result for (1.3.1)

based on the result of Roynette [68] on regularity of Brownian motion in the Besov space

B12p,∞, with 1 ≤ p < ∞. They proved a bilinear estimate for KdV in (a Banach space


adapted to) their Besov space setting, establishing LWP of (1.3.1) using the fixed point

method. However, the bilinear estimate they proved required a slight regularization of

the noise in space via the bounded operator φ, so that the smoothed noise has spatial

regularity s > −12. In particular, they could not treat the case of space-time white noise

(φ = Id).

These developments are clarified in [59] with two observations. The first observation

is that the lpn-based function space b̂sp,∞ and its corresponding X

s,b-type space capture

the regularity of spatial and space-time white noise, respectively, for sp < −1 and b < 12.

The second is that a priori estimates on the second iteration of the integral formulation

(1.3.2) (in these spaces) are sufficient to establish local well-posedness of (1.3.1) with

φ = Id. Recall that (as discussed in the last section) we will establish a priori estimates

on the second iteration of the integral formulation of (1.2.4) in Xs,b with b < 12

during

the proof of Theorem 1.3. Let us pause to identify that the second iteration is a useful

trick, in general, for establishing local well-posedness in function spaces of low temporal

regularity: we can take b < 12

at the cost of conducting a higher order multilinear analysis.

Turning to global theory, the methods for deterministic PDEs require a careful adap-

tation to the stochastic setting. We are particularly interested in adapting the method

discussed in the previous section of this introduction: the use of an invariant measure on

a phase space as a tool for extending local solutions to global solutions. A good candidate

for this strategy is the stochastic KdV-Burgers equation.

1.3.2 The stochastic KdV-Burgers equation

The stochastic Burgers equation, given by

du = (1

2uxx −

1

2(u2)x)dt+ φ∂xdW, t ≥ 0, x ∈ T (1.3.3)

is used to model a randomly growing interface [38] (it is often reformulated as the so-called

Kardar-Parisi-Zhang equation). We will study a stochastic KdV-Burgers equation

du = (uxx − uxxx −1

2(u2)x)dt+ φ∂xdW, t ≥ 0, x ∈ T (1.3.4)

where φ is a bounded operator on L2(T). We consider (1.3.4) as a toy-model (1.3.3).Indeed, with φ = Id, (1.3.3) and (1.3.4) share an important physical property: both of

these equations formally preserve spatial white noise. We are interested in rigorously

proving the invariance of white noise for (1.3.4) by adapting the method developed in [6]


(and implemented in Chapter 3) to the stochastic setting. By using this strategy we could

potentially exploit the white noise invariance (of a finite-dimensional approximation) in

order to prove global well-posedness for a stochastic PDE. This thesis provides a result

in this direction.

Spatial white noise is a probability measure supported on Hs(T) for s < −12. To

prove the invariance of white noise under the flow of (1.3.4), we must first establish local

well-posedness in the support of white noise. Until very recently, well-posedness of (1.3.3)

and (1.3.4) has been an open problem. New techniques involving the rough path method

have been developed by Hairer [34], and it appears that pathwise local well-posedness of

(1.3.3) has been established. This thesis is focused on more classical methods, since this

leads to a hope of proving global well-posedness for a stochastic PDE via the invariance

of white noise (following [6]).

Using classical methods, Da Prato-Debussche-Temam [25] considered (1.3.3) with a

smoothed noise (they placed the operator φ = ∂−1x in front of the additive noise), and

established local well-posedness in L2(T). Recall from our discussion in the last section,that local well-posedness of (1.3.1) was established in [28, 59], and that the additive

noise in (1.3.1) is also a full derivative smoother than the noise in (1.3.3). Using the

combination of dispersion and dissipation, we prove local well-posedness of (1.3.4) in

H−(12

+)(T) with a rougher additive noise than in [25, 28, 59]. More precisely, the followingtheorem holds:

Theorem 1.5 (LWP of (1.3.4)). Let φ = ∂−( 13

16+)

x . Then (1.3.4) is locally well-posed in

H−(12

+)(T). That is, if u0 ∈ H−(12

+)(T) with mean zero, then there exists a stopping timeTω > 0 and a (unique) process u ∈ C([0, Tω];H−(

12

+)(T)) satisfying (1.3.4) on [0, Tω]almost surely.

Theorem 1.5 represents progress towards proving global well-posedness of (1.3.4) and

the invariance of white noise under the flow, for two reasons: (i) the initial data u0 ∈H−(

12

+)(T) are in the support of spatial white noise, and (ii) the additive noise is smoothedvia φ by 13

16+ derivatives in space, which is less than the full derivative of smoothing

appearing in [25, 28, 59]. In the future, we hope to establish Theorem 1.5 with φ = Id.

The deterministic KdV-Burgers equation (KdV-B, (1.3.4) with φ = 0) is studied in

Molinet-Ribaud [52] and Molinet-Vento[53]; they established LWP of KdV-B in Hs(T)for s ≥ −1, with analytic dependence on the data. This is in contrast to the deterministicKdV and deterministic Burgers equations, which are both analytically ill-posed below


H−12 (T) (see [40, 29]). That is, the combination of dispersion and dissipation in KdV-B

leads to improved low regularity well-posedness results.

The proof of Theorem 1.5 is based on the following observations. With the com-

bination of dispersion and dissipation, we can obtain the bilinear estimates needed for

local well-posedness of KdV-B in the Xs,b-space (adapted to this equation) with s < −12

and b < 12. By taking s < −1

2, we can treat spatial white noise as initial data. More

importantly, with b < 12, we can estimate the stochastic convolution for (1.3.4) using the

Xs,b-norm (almost surely). That is, with b < 12, we have adapted to the low temporal

regularity of space-time white noise. Finally, the dissipative semigroup has a smoothing

effect in space, and we are able to treat a rougher additive noise than in [25, 28, 59].

Next we will establish global well-posedness of (1.3.4) in L2(T) with an additionalsmoothing applied to the noise. This will be proven by combining Theorem 1.5 (at a

higher regularity) with a priori bounds on the growth of the L2-norm.

Theorem 1.6 (GWP of (1.3.4)). Let φ = ∂−( 3

2+)

x . Then (1.3.4) is globally well-posed in

L2(T).

The final result of this thesis is a formal justification of the invariance of white noise

for (1.3.4). More precisely, we show that spatial white noise is invariant under the (finite-

dimensional) system of SDEs given by projection of (1.3.4) to the first N � 0 modesof the trigonometric basis. That is, consider the following finite-dimensional system of

SDEs {duN =

(uNxx − uNxxx − 12PN

[((uN)2)x

])dt+ PN∂xW, t ≥ 0, x ∈ T

uN(0, x) = uN0 (x) = PN(u0(x)),(1.3.5)

where PN is the Dirichlet projection to EN = span{sin(nx), cos(nx) : |n| ≤ N}, anduN = PNuN . We establish the following proposition.

Proposition 1.1. The flow of (1.3.5) preserves spatial white noise.

The proof of Proposition 1.1 is based on decomposing (1.3.5) into a frequency truncated

KdV equation, plus a rescaled Ornstein-Uhlenbeck process at each spatial frequency;

each of these evolutions individually preserve white noise, and this leads to invariance

for the superposition in the finite-dimensional setting.

Let us emphasize that, in collaboration with T. Oh and J. Quastel, we are developing

techniques for relaxing the smoothing hypothesis in the statement of Theorem 1.5. In


particular, if we can prove local well-posedness of (1.3.4) with φ = Id, then we may

be able to use Proposition 1.1 to extend this result to global well-posedness (following

Bourgain’s method [6]), and to prove the invariance of spatial white noise under the

flow. We will return to this topic in Chapter 4, providing more background and detailed

statements and proofs of our results.

This thesis is organized as follows. In Chapter 2 we discuss mass concentration for the

Davey-Stewartson system (Theorem 1.1 and Theorem 1.2). Chapter 3 is devoted to the

invariance of the Gibbs measure for the periodic quartic KdV (Theorem 1.3, Theorem

1.4 and Corollary 1.1), and Chapter 4 to the stochastic KdV-Burgers equation (Theorem

1.5, Theorem 1.6 and Proposition 1.1). Following these chapters is a combined appendix,

with proofs of some of the more technical estimates found in this thesis, as well as proofs

of various lemmata.

Chapter 2

Mass concentration for the

Davey-Stewartson system

2.1 Introduction

In this chapter we study the elliptic-elliptic Davey-Stewartson system in 2 space dimen-

sions {iut + ∆u+ L(|u|2)u = 0, t ≥ 0, x ∈ R2

u(0, x) = u0(x) ∈ H1(R2).(2.1.1)

Here L = νI + γB, ν = ±1, γ > 0, and B is the pseudo-differential operator with symbolB̂(f)(ξ) = ξ

21

ξ21+ξ22f̂(ξ).

The more general Davey-Stewartson system is given by{iut + σuxx + uyy + ν|u|2u− φxu = 0,αφxx + φyy + γ(|u|2)x = 0,

(2.1.2)

where σ, ν = ±1, γ > 0, and α can be positive or negative. These equations describe,to leading order, the (complex) amplitude u and (real) velocity potential φ of a weakly

nonlinear 3-dimensional water wave traveling predominantly in the x-direction ([26],[64]).

The constants σ, α, ν and γ depend on physical variables, such as the strength of gravity

and surface tension, the depth of the fluid domain and the wave number of the wave

packet. Depending on the signs of (σ, α), the cases (−,−), (−,+)(+,−) and (+,+)are classified as hyperbolic-hyperbolic, hyperbolic-elliptic, elliptic-hyperbolic and elliptic-

elliptic respectively. Following the picture from [30] (see also [1]) the elliptic-elliptic

17

Chapter 2. Mass concentration for the Davey-Stewartson system 18

equation describes a scenario with relatively large surface tension T , and a sufficiently

large depth h to achieve a “sub-sonic” flow: that is the group velocity of the wave packet

does not exceed the velocity√gh of long gravity waves.

We consider the elliptic-elliptic case of (2.1.2). By rescaling x and γ we can take

α = 1. Applying the Fourier transform to the second equation in (2.1.2), we find

(ξ21 + ξ22)φ̂ = −iγξ1 |̂u|2,

and, by differentiating in x,

(ξ21 + ξ22)φ̂x = −iξ1(ξ21 + ξ22)φ̂ = −γξ21 |̂u|2.

Thus, we can solve for φ̂x in terms of u,

φ̂x = −γξ21

ξ21 + ξ22

(|̂u|2) = −γB̂(|u|2).

By the Fourier inversion theorem φx = −γB(|u|2), and the system (2.1.2) reduces to(2.1.1).

We will also consider the relaxation of (2.1.1) to the case u0 ∈ L2(R2),{iut + ∆u+ L(|u|2)u = 0, t ≥ 0, x ∈ R2

u(0, x) = u0(x) ∈ L2(R2).(2.1.3)

Recall the existence theory for (2.1.1) and (2.1.3).

Theorem 2.1 (Ghidaglia-Saut [32]). For any u0 ∈ L2(R2), ∃ T ∗ > 0 and a uniquesolution u to (2.1.3) such that u ∈ C([0, T ∗);L2(R2)) ∩ L4((0, t)×R2) for all t ∈ (0, T ∗).Furthermore

(a) T ∗ is maximal in the sense that if T ∗


We review what is known (see also [18],[64]) about standing wave and blow-up solu-

tions to (2.1.1). There are standing wave solutions to (2.1.1) of the form u(t, x) = v(x)eit.

Theorem 2.3 (Papanicolaou et al [64]). Taking ν = 1 (focusing), we have the optimal

estimate ∫L(|u|2)|u|2dx ≤ Copt‖∇u‖22‖u‖22. (2.1.4)

Furthermore Copt =2‖R‖22

for some R ∈ H1(R2) such that R(x) > 0 ∀x ∈ R2, andu(t, x) = R(x)eit solves (2.1.1).

Remark 2.1. To the authors knowledge, the uniqueness of ground state standing wave

solutions to (2.1.1) (ie. solutions u(t, x) = R(x)eit with R(x) > 0 ∀x ∈ R2) is an openproblem. Thus the L2 norm (‖R‖2) for such a solution is not a priori well-defined. Rather,the inequality (2.1.4) is sharp [64], and the optimal constant Copt > 0 is therefore unique.

All sufficiently well-localized negative energy initial data blow-up in finite time:

Theorem 2.4 (Ghidaglia-Saut [32]). Let Σ := {v ∈ H1(R2) : |x|v ∈ L2(R2)}. Thefollowing holds true

(a) If −ν ≥ γ, all solutions of (2.1.1) are global in time.

(b) An element v ∈ Σ satisfying E(v) < 0 exists if and only if −ν < γ.

(c) If u0 ∈ Σ satisfies E(u0) < 0, the corresponding solution u to (2.1.1) blows up infinite time.

Remark 2.2. Part (c) of Theorem 2.4 follows from the following virial identity, first

obtained in [1], ∫|x|2|u(t)|2dx = 4E(u0)t2 + ct+

∫|x|2|u0|2dx. (2.1.5)

For a non-zero solution u to (2.1.1), the left-hand side of (2.1.5) is manifestly positive.

But, if E(u0) < 0, the right-hand side of (2.1.5) becomes negative in finite time. The

corresponding solution u cannot, therefore, be global in time. By part (a) of Theorem

2.2, u is a blow-up solution.


Global existence in H1(R2) for solutions to (2.1.1) with ‖u0‖2 <√

2Copt

is a corollary

of Theorem 2.3 [64]. This result is optimal (with respect to mass) due to an explicit blow-

up solution induced by a pseudo-conformal invariance: If u solves (2.1.1) for t ∈ [1,∞),then

pc[u](t, x) := e−i|x|24t

1

|t|u(−1

t,x

t) (2.1.6)

solves (2.1.1) for t ∈ [−1, 0). We can apply this symmetry to the ground state standingwave solution uR(t, x) := R(x)e

it, with ‖R‖2 =√

2Copt

, to find a blow-up solution given

by

pc[uR](t, x) = e−i |x|

2

4t+ it

1

|t|R(x

t). (2.1.7)

This solution to (2.1.1) satisfies ‖pc[uR]‖2 = ‖R‖2 =√

2Copt

and ‖∇pc[uR](t)‖2 ∼ 1|t| as

t ↑ 0; pc[uR] is a blow-up solution with the minimal mass√

2Copt

.

2.1.1 Results

Taking γ = 0, ν = 1, (2.1.1) reduces to the focusing cubic NLS,{iut + ∆u = −|u|2u, t ≥ 0, x ∈ R2

u(0, x) = u0(x) ∈ H1(R2).(2.1.8)

The results of this chapter [67] are motivated by asymptotic and numerical analyses

[64] which reveal a similar blow-up phenomenon for (2.1.1) and (2.1.8). We prove a mass

concentration property for blow-up solutions to (2.1.1), analogous to the one known for

(2.1.8), by adapting methods from [35]. We use a profile decomposition and Theorem

2.3 to prove a compactness property for bounded sequences in H1(R2). This property isthen applied to spatially rescaled snapshots (in time) of a blow-up solution u to prove

mass concentration. Here are the results we obtain:

Theorem 2.5. Let {vn}∞n=1 be a bounded sequence in H1(R2) such that

lim supn→∞

‖∇vn‖22 ≤M2, lim supn→∞

∫L(|vn|2)|vn|2dx ≥ m4, (2.1.9)

for some 0 < m,M


Theorem 2.6. Let u be a solution of (2.1.1) which blows up in finite time T ∗ > 0, and

λ(t) > 0 any function such that λ(t)‖∇u(t)‖2 → +∞ as t ↑ T ∗. Then, ∃ y(t) ∈ R2 suchthat

lim inft↑T ∗

∫|x−y(t)|≤λ(t)


.

Remark 2.3. Recall from Theorem 2.3 that 2Copt

= ‖R‖22 for a function R ∈ H1(R2) suchthat u(t, x) = R(x)eit solves (2.1.1) and R(x) > 0 ∀x ∈ R2.

Remark 2.4. If λ(t) & (T ∗ − t)1/2−� for a small � > 0, then by the scaling lower boundfor NLS, λ(t)‖∇u(t)‖2 ≥ C(T ∗ − t)−� →∞ as t ↑ T ∗. Thus Theorem 2.6 holds for sucha function λ(t), and implies a (nearly) parabolic mass concentration effect for blow-up

solutions to (2.1.1). Observe that the blow-up solution (2.1.7) concentrates mass within

a smaller conic window at time T ∗ = 0: λ(t) & |t|1−.

Bourgain has proven [8] a mass concentration property for solutions to cubic NLS

posed in L2(R2) with a finite lifespan (T ∗ < ∞). We adapt Bourgain’s proof to (2.1.3),and obtain the following result:

Theorem 2.7. Let u be a solution of (2.1.3) which satisfies T ∗ 0 such that

lim supt↑T ∗

supsquares Q⊂R2

of sidelength `(Q) η(M) > 0. (2.1.10)

Remark 2.5. Refinements of Theorem 2.7 relating the window size of mass concentration

and rate of explosion of the L4[0,t]×R2 norm also hold [23]. That is, the window of mass

concentration will have sidelength (T ∗ − t) 1+β2 if and only if the L4[0,t]×R2 norm grows intime no slower than (T ∗ − t)−β.

The remainder of this chapter is organized as follows. In Section 2.2 we use Theorem

2.5 to prove the mass concentration result for H1(R2) solutions (Theorem 2.6). Theproof of Theorem 2.5 is presented in Section 2.3. Mass concentration for L2(R2) solutions(Theorem 2.7) is proven in Section 2.4. The appendix to this thesis is found in Section

5.1; here we prove several lemmata and a proposition used in our analysis.


2.2 Mass concentration for H1-solutions

Proof of Theorem 2.6. Assume that H1(R2) 3 u0 7→ u(t) is any finite time blow-upsolution to (2.1.1). We introduce a shrinking parameter which encodes the core size of

the blow-up region:

ρ(t) =1

‖∇u(t)‖2,

and rescale

v(t, x) = ρu(t, ρx). (2.2.1)

Let {tn}∞n=1 be an arbitrary sequence such that tn ↑ T ∗, ρn = ρ(tn) and vn = v(tn, ·).The mass of u is invariant under both the rescaling (2.2.1), and the flow of (2.1.1), and

therefore

‖vn‖2 = ‖u(tn)‖2 = ‖u0‖2.

We then compute, by choice of ρ, that

‖∇vn‖2 = 1.

From linearity of L, conservation of energy and blow-up, we have

E(vn) =1

2− 1

4

∫L(|vn|2)|vn|2dx

=1

2

∫|∇vn|2dx−

1

4

∫L(|vn|2)|vn|2dx

=1

2

∫|∇(ρnu(tn, ρnx))|2dx−

1

4

∫L(|ρnu(tn, ρnx)|2)|ρnu(tn, ρnx)|2dx

= ρ4n

(12

∫|(∇u)(tn, ρnx)|2dx−

1

4

∫L(|u(tn)|2)(ρnx)|u(tn, ρnx)|2dx

)= ρ2nE(u(tn))

=1

‖∇u(tn)‖22E(u0)→ 0, as n→∞.

In particular,

limn→∞

∫L(|vn|2)|vn|2dx = 2.


The sequence {vn} satisfies the hypotheses (2.1.9) of Theorem 2.5 with

m4 = 2, M = 1.

Let us assume Theorem 2.5 holds true (we prove Theorem 2.5 in the next section).

By Theorem 2.5 there is a sequence {xn}∞n=1 ⊂ R2 and a profile V ∈ H1(R2), with‖V ‖2 ≥ m

2

M√Copt

=√

2Copt

, such that, up to a subsequence

vn = ρnu(tn, ρn ·+xn) ⇀ V weakly in H1(R2).

Then for any A > 0, χ|x|≤Aρnu(tn, ρn · +xn) ⇀ χ|x|≤AV weakly in L2(R2). By lowersemi-continuity of the norm in the weak limit,

lim infn→∞

∫|x|≤A

ρ2n|u(tn, ρnx+ xn)|2dx ≥∫|x|≤A

|V |2dx.

With a change of variables we have

lim infn→∞

∫|x−xn|≤ρnA

|u(tn, x)|2dx ≥∫|x|≤A

|V |2dx.

From the assumption of Theorem 2.6, that λ(t)‖∇u(t)‖2 →∞ as t ↑ T ∗,

λ(tn)

ρn= λ(tn)‖∇u(tn)‖2 →∞.

Thus λ(tn) ≥ ρnA for n sufficiently large, and

lim infn→∞

supy∈R2

∫|x−y|≤λ(tn)

|u(tn, x)|2dx ≥ lim infn→∞

supy∈R2

∫|x−y|≤ρnA

|u(tn, x)|2dx

≥ lim infn→∞

∫|x−xn|≤ρnA

|u(tn, x)|2dx

≥∫|x|≤A

|V |2dx for every A > 0.

Taking A to infinity,

lim infn→∞

supy∈R2

∫|x−y|≤λ(tn)

|u(tn, x)|2dx ≥ ‖V ‖22 ≥2

Copt.

Since the sequence {tn}∞n=1 is arbitrary,

lim inft↑T ∗

supy∈R2

∫|x−y|≤λ(t)


. (2.2.2)


For every t ∈ [0, T ∗), the function y 7→∫|x−y|≤λ(t) |u(t, x)|

2dx is continuous and vanishes

at infinity. The supremum in (2.2.2) is therefore a maximum and there exists y(t) ∈ R2

such that

supy∈R2

∫|x−y|≤λ(t)

|u(t, x)|2dx =∫|x−y(t)|≤λ(t)

|u(t, x)|2dx. (2.2.3)

Combining (2.2.2) and (2.2.3),

lim inft↑T ∗

∫|x−y(t)|≤λ(t)


.

This completes the proof of Theorem 2.6 under the assumption that Theorem 2.5 holds

true.

2.3 A compactness property

The proof of Theorem 2.5 relies on a profile decomposition for bounded sequences in

H1(R2).

Proposition 2.1. [35] Let {vn}∞n=1 be a bounded sequence in H1(R2). Then there exists asubsequence of {vn}∞n=1(still denoted {vn}∞n=1), a family {xjn}∞j=1 for each n, and a boundedsequence {V j}∞j=1 ⊂ H1(R2), such that

(a) ∀ i 6= j,

|xin − xjn| → ∞ as n→∞. (2.3.1)

(b) ∀ k ≥ 1 and ∀x ∈ R2

vn(x) =k∑j=1

V j(x− xjn) + vkn(x) (2.3.2)

with

lim supn→∞

‖vkn‖p → 0 as k →∞ for every p ∈ (2,∞). (2.3.3)

Moreover, as n→∞,

‖vn‖22 =k∑j=1

‖V j‖22 + ‖vkn‖22 + o(1), (2.3.4)


‖∇vn‖22 =k∑j=1

‖∇V j‖22 + ‖∇vkn‖22 + o(1). (2.3.5)

Here o(1) represents terms that go to zero as n→∞.

See also [53] for related asymptotic compactness modulo symmetries results. We

require an additional lemma describing a property of the profiles obtained in Proposition

2.1.

Lemma 2.1. Given bounded sequences {vn}∞n=1, {Vj}∞j=1 ⊂ H1(R2) and a family of se-quences {xjn}∞j=1 ⊂ R2 such that properties (2.3.1)-(2.3.5) are satisfied, then

lim supn→∞

∫L(|vn|2)|vn|2dx ≤

∞∑j=1

∫L(|V j|2)|V j|2dx. (2.3.6)

Remark 2.6. (2.3.6) is a modification of an inequality from [35, p. 2825-2826], (3.29)-

(3.31), for the NLS case (2.1.8). It is designed to be used with Theorem 2.3 to adapt

the arguments of [35] to the system (2.1.1). When we prove Lemma 2.1, and when we

apply Theorem 2.3, we invoke properties of (2.1.1) distinct from (2.1.8). Otherwise, our

arguments are identical to those from [35].

The proof of Lemma 2.1 is postponed to the end of this section.

Proof of Theorem 2.5. Given a bounded sequence {vn}∞n=1 ⊂ H1(R2) satisfying (2.1.9),we apply Proposition 2.1 to get a sequence of functions {Vj}∞j=1 ⊂ H1(R2), and a sequenceof points {xjn}∞j=1 ⊂ R2 for each n, which satisfy properties (2.3.1)-(2.3.5). By (2.1.9),Lemma 2.1 and Theorem 2.3, we find

m4 ≤ lim supn→∞

∫L(|vn|2)|vn|2dx

≤∞∑j=1

∫L(|V j|2)|V j|2dx

≤ Copt∞∑j=1

‖V j‖22‖∇V j‖22

≤ Copt(

supj≥1‖V j‖22

) ∞∑j=1

‖∇V j‖22. (2.3.7)


Then by (2.3.5) and (2.1.9),

∞∑j=1

‖∇V j‖22 ≤ lim supn→∞

‖∇vn‖22 ≤M2. (2.3.8)

Combining (2.3.7) and (2.3.8), we have

supj≥1‖V j‖22 ≥

m4

M2Copt. (2.3.9)

By (2.3.4) the series∑∞

j=1 ‖V j‖22 converges, limj→∞ ‖V j‖2 = 0, and the supremum in(2.3.9) is attained. That is, there exists some j0 such that

‖V j0‖2 ≥m2

M√Copt

. (2.3.10)

With a change of variables, for any k ≥ j0

vn(x+ xj0n ) = V

j0(x) +∑

1≤j≤k,j 6=j0

V j(x+ xj0n − xjn) + ṽkn(x), (2.3.11)

where ṽkn = vkn(x+ x

j0n ). Then by (2.3.1), for j 6= j0

V j(·+ xj0n − xjn) ⇀ 0 weakly in H1(R2) as n→∞. (2.3.12)

Up to a subsequence, we can assume that

ṽkn ⇀ ṽk weakly in H1(R2) as n→∞ (2.3.13)

for some ṽk ∈ H1(R2), and all k ≥ 0. We justify this assumption. Fixing k = 1, {ṽ1n}∞n=1is bounded in H1(R2). By extracting a subsequence {ṽ1

n1i}∞i=1, we can ensure ṽ1n1i ⇀ ṽ

1 as

i→∞. For k = 2, {ṽ2n1i}∞i=1 is bounded in H1(R2), so once again extracting a subsequence

{n2i } ⊂ {n1i }, we can ensure ṽ2n2i ⇀ ṽ2 as i→∞. Proceeding this way, such that ṽk

nki⇀ ṽk

as i→∞ for each k ≥ 1, we can extract the diagonal subsequence {ṽknii}∞i=1 which satisfies

ṽknii⇀ ṽk as i→∞ for all k ≥ 1, since {nii} ⊂ {nki } for i sufficiently large. Rename this

subsequence {ṽkn}∞n=1, and (2.3.13) is justified.By (2.3.11), (2.3.12) and (2.3.13),

vn(x+ xj0n ) ⇀ V

j0 + ṽk weakly in H1(R2) as n→∞.

The sequence vn(x+ xj0n ) is independent of k, and hence the weak limit V

j0 + ṽk is also

independent of k. Therefore, for all k ≥ j0, ṽk = ṽj0 for some ṽj0 ∈ H1(R2). By lowersemi-continuity of the L4(R2) norm in the weak limit,

‖ṽj0‖4 ≤ lim supn→∞

‖ṽkn‖4 = lim supn→∞

‖vkn‖4 → 0 as k →∞, by (2.3.3).


Thus ṽj0 = 0, and,

vn(x+ xj0n ) ⇀ V

j0 weakly in H1(R2) as n→∞. (2.3.14)

By (2.3.10) and (2.3.14), the function V j0 and the sequence {xj0n }∞n=1 satisfy the claimsof Theorem 2.5. This completes the proof of Theorem 2.5 under the assumption that

Lemma 2.1 holds true.

The proof of Lemma 2.1 requires two elementary results.

Lemma 2.2. Suppose φ, ψ ∈ H1(R2), if |x1n − x2n| → ∞ as n→∞, then

‖φ(· − x1n)ψ(· − x2n)‖2 → 0 as n→∞.

Proof. By Sobolev embedding φ, ψ ∈ L4(R2). Given � > 0, we use the density of C∞0 (R2)in L4(R2) to find φ0, ψ0 ∈ C∞0 (R2) such that ‖φ− φ0‖4 < �2‖ψ‖4 , and ‖ψ − ψ0‖4 <

�2‖φ0‖4 .

Applying the triangle and Hölder inequalities

‖φ(· − x1n)ψ(· − x2n)‖2 = ‖φ0(· − x1n)(ψ(· − x2n)− ψ0(· − x2n))

+ (φ(· − x1n)− φ0(· − x1n))ψ(· − x2n) + φ0(· − x1n)ψ0(· − x2n)‖2≤ ‖φ0‖4‖ψ − ψ0‖4 + ‖ψ‖4‖φ− φ0‖4 + ‖φ0(· − x1n)ψ0(· − x2n)‖2<�

2+�

2+ 0 (2.3.15)

= �.

(2.3.16) follows from ‖φ0(· − x1n)ψ0(· − x2n)‖2 = 0 for n sufficiently large, which is due to|x1n − x2n| → ∞ as n→∞, and φ0, ψ0 ∈ C∞0 (R2).

Lemma 2.3. Suppose φ ∈ L2(R2), ψ ∈ H1(R2), if |x1n − x2n| → ∞ as n→∞, then∫|φ(x− x1n)||ψ(x− x2n)|2dx→ 0 as n→∞.

Proof. We choose φ0, ψ0 ∈ C∞0 (R2) such that ‖φ − φ0‖2 < �2‖ψ‖24 , and ‖ψ − ψ0‖4 <�B

,


where B will be determined afterward. Then,∫|φ(x− x1n)||ψ(x− x2n)|2dx =

∫|(φ(x− x1n)− φ0(x− x1n))(ψ(x− x2n))2

+ φ0(x− x1n)((ψ(x− x2n))2 − (ψ0(x− x2n))2)

+ φ0(x− x1n)(ψ0(x− x2n))2|dx

≤ ‖ψ2‖2‖φ− φ0‖2 + ‖φ0‖2‖ψ2 − ψ20‖2

+

∫|φ0(x− x1n)||ψ0(x− x2n)|2dx

≤ ‖ψ‖24‖φ− φ0‖2 + ‖φ0‖2‖ψ + ψ0‖4‖ψ − ψ0‖4

+

∫|φ0(x− x1n)||ψ0(x− x2n)|2dx

≤ ‖ψ‖24( �

2‖ψ‖24

)+ ‖φ0‖2(2‖ψ‖4 + ‖ψ − ψ0‖4)‖ψ − ψ0‖4

+

∫|φ0(x− x1n)||ψ0(x− x2n)|2dx

<�

2+ ‖φ0‖2

(2‖ψ‖4 +

�

B

) �B

+ 0

≤ �

for n large, as long as

1

2B2 − 2‖φ0‖2‖ψ‖4B − �‖φ0‖2 ≥ 0

which holds for B sufficiently large.

Proof of Lemma 2.1. Observe that the infinite sum on the right-hand side of (2.3.6)

converges by (2.3.7), (2.3.8) and boundedness of {vn} ⊂ H1(R2). It therefore suffices toshow that ∀ � > 0, ∃K(�) > 0 such that for k ≥ K(�),

lim supn→∞

∫L(|vn|2)|vn|2dx ≤

k∑j=1

∫L(|V j|2)|V j|2dx+ �.

Letting W kn :=∑k

j=1 Vj(x− xjn), we compute∫L(|vn|2)|vn|2dx =

∫L(|W kn + vkn|2)|W kn + vkn|2dx

=

∫L(|W kn |2)|W kn |2dx+Rkn (2.3.16)

where the remainder term is

Rkn =

∫L(2Re(W knvkn) + |vkn|2)|vn|2dx+

∫L(|W kn |2)(2Re(W knvkn) + |vkn|2)dx.


We also take ∫L(|W kn |2)|W kn |2dx =

k∑j=1

∫L(|V j|2)|V j|2dx+ Ckn (2.3.17)

where the mixed cross term is

Ckn =∑

1≤i1,i2,i3,i4≤kim 6=ij

for some m 6=j

∫L(Vi1(x− xi1n )V i2(x− xi2n ))Vi3(x− xi3n )V i4(x− xi4n )dx.

We claim that given � > 0,∃K(�) > 0 such that for k ≥ K(�)

lim supn→∞

|Rkn| ≤ � (2.3.18)

lim supn→∞

|Ckn| = 0. (2.3.19)

By Plancherel’s theorem and the definition of the operator B,

‖L(f)‖2 = ‖νf + γB(f)‖2 ≤ ‖f‖2 + γ‖B(f)‖2 = ‖f‖2 + γ‖B̂(f)‖2

= ‖f‖2 + γ( ∫| ξ

21

|ξ|2f̂(ξ)|2dξ

)1/2≤ ‖f‖2 + γ‖f̂‖2= (1 + γ)‖f‖2. (2.3.20)

We then compute, with repeated use of Hölder and triangle inequalities,

|Rkn| ≤∫|L(2Re(W knvkn) + |vkn|2)|vn|2

+ L(|W kn |2)(2Re(W knvkn) + |vkn|2)|dx

≤(‖L(2Re(W knvkn) + |vkn|2)‖2‖|vn|2‖2

+ ‖L(|W kn |2)‖2‖2Re(W knvkn) + |vkn|2‖2)

≤ (1 + γ)‖2Re(W knvkn) + |vkn|2‖2(‖|vn|2‖2 + ‖|W kn |2‖2

)by (2.3.20),

≤ (1 + γ)(

2‖W knvkn‖2 + ‖vkn‖24)(‖vn‖24 + ‖W kn‖24

)≤ (1 + γ)‖vkn‖4

(2‖W kn‖4 + ‖vkn‖4

)(‖vn‖24 + ‖W kn‖24

)≤ (1 + γ)‖vkn‖4

(2‖vn‖4 + 3‖vkn‖4

)(2‖vn‖24 + 2‖vn‖4‖vkn‖4 + ‖vkn‖24

)≤ (1 + γ)‖vkn‖4

(C1 + 3‖vkn‖4

)(C2 + C3‖vkn‖4 + ‖vkn‖24

),


where C1, C2, C3 > 0 are constants. Here we have obtained the second last line by

writing W kn = vn − vkn, and the last line follows from boundedness of {vn} ⊂ H1(R2),since ‖vn‖24 ≤ C‖vn‖2‖∇vn‖2 < C̃, independent of n. By (2.3.3)

lim supn→∞

|Rkn| ≤ lim supn→∞

(1 + γ)‖vkn‖4(C1 + 3 lim sup

n→∞‖vkn‖4

)·(C2 + C3 lim sup

n→∞‖vkn‖4 +

(lim supn→∞

‖vkn‖4)2)

≤ �,

for k ≥ K(�) sufficiently large. Having justified (2.3.18), it remains to justify (2.3.19).For k ≥ K(�)

|Ckn| = |∑


for some m6=j

∫L(Vi1(x− xi1n )V i2(x− xi2n ))Vi3(x− xi3n )V i4(x− xi4n )dx|

≤∑


for some m 6=j

∫|L(Vi1(x− xi1n )V i2(x− xi2n ))Vi3(x− xi3n )V i4(x− xi4n )|dx.

We organize the terms of this sum as follows

(a) i1 6= i2

(b) i1 = i2, i3 6= i4

(c) i1 = i2 6= i3 = i4

In case (a),

lim supn→∞

∫|L(Vi1(x− xi1n )V i2(x− xi2n ))Vi3(x− xi3n )V i4(x− xi4n )|dx

≤ lim supn→∞

‖L(Vi1(· − xi1n )V i2(· − xi2n ))‖2‖Vi3(· − xi3n )V i4(· − xi4n )‖2

≤ lim supn→∞

(1 + γ)‖Vi1(· − xi1n )V i2(· − xi2n )‖2‖Vi3‖4‖Vi4‖4 by (2.3.20)

= 0

by Lemma 2.2. Similarly, in case (b), by Lemma 2.2,

lim supn→∞


≤ lim supn→∞

(1 + γ)‖Vi1‖24‖Vi3(· − xi3n )V i4(· − xi4n )‖2

= 0.


In case (c), Vi1 ∈ H1(R2) ⊂ L4(R2), and thus |Vi1|2 ∈ L2(R2). By (2.3.20), L(|Vi1|2) ∈L2(R2). Applying Lemma 2.3,

lim supn→∞

∫|L(|Vi1(x− xi1n )|2)||Vi3(x− xi3n )|2dx = lim sup

n→∞

∫|L(|Vi1|2)(x− xi1n )||Vi3(x− xi3n )|2dx

= 0.

Therefore

lim supn→∞

|Ckn| ≤∑


for some m 6=j

lim supn→∞


= 0

for k ≥ K(�), and (2.3.19) holds. Combining (2.30),(2.3.17),(2.3.18) and (2.3.19)

lim supn→∞

∫L(|vn|2)|vn|2dx ≤

k∑j=1

∫L(|V j|2)|V j|2dx+ lim sup

n→∞|Rkn|+ lim sup

n→∞|Ckn|

≤k∑j=1

∫L(|V j|2)|V j|2dx+ �

for k ≥ K(�). This completes the proof of Lemma 2.1.

2.4 Mass concentration for L2-solutions

For the proof of Theorem 2.7 we require two lemmas from [8]. The proofs of these

lemmata can be found in the first section of the appendix to this thesis.

Lemma 2.4 (Squares Lemma). Given f ∈ L2(R2), � > 0, ∃ (fr)1≤r 0.


(d) ‖eit∆f − Σ1≤r 0, ∃ tubes (Qs)1≤s


Here the third line from the bottom follows from the operator bound ‖eis∆‖L4/3→L4 .1|s|1/2 , and the second last line follows from ‖

∫1

|t−s|1/2ψ(s)ds‖4 . ‖ψ‖4/3. By Hölder’sinequality, and (2.3.20),∫

R2|u|4/3|L(|u|2)|4/3dx ≤

( ∫R2|u|4dx

)1/3( ∫R2|L(|u|2)|2dx

)2/3≤( ∫

R2|u|4dx

)1/3((1 + γ)2

∫R2|u|4dx

)2/3= (1 + γ)4/3‖u‖4L4

R2.

This gives

‖u(t)− ei(t−tj)∆u(tj)‖L4Ij×R2

. (1 + γ)‖u‖3L4Ij×R2

. λ3, (2.4.2)

and then

‖ei(t−tj)∆u(tj)‖L4Ij×R2

≤ ‖u(t)‖L4Ij×R2

+ ‖ei(t−tj)∆u(tj)− u(t)‖L4Ij×R2

. λ+ λ3

. λ. (2.4.3)

Justifying (2.4.2) and (2.4.3) as above is the only place where structure specific to (2.1.1)

will be invoked. The remainder of our proof of Theorem 2.7 will mimic the proof from

[8] exactly. By (2.4.1), (2.4.2), (2.4.3), and Hölder’s inequality, we find

λ4 =

∫Ij

∫R2|u(t)|4dxdt

=

∫Ij

∫R2u(t)(ei(t−tj)∆u(tj) + (u(t)− ei(t−tj)∆u(tj)))

· (ei(t−tj)∆u(tj) + (u(t)− ei(t−tj)∆u(tj))) (ei(t−tj)∆u(tj) + (u(t)− ei(t−tj)∆u(tj)))dxdt

=

∫Ij

∫R2u(t)(ei(t−tj)∆u(tj))(ei(t−tj)∆u(tj))

2dxdt+O(λ6).

Applying Lemma 2.4 to f := u(tj), with � = λ2, ∃ functions (fr)1≤r


Therefore there is a choice of r1, r2, r3 < R(λ2) such that∫

Ij

∫R2u(t)(ei(t−tj)∆fr1)(e

i(t−tj)∆fr2)(ei(t−tj)∆fr3)dxdt >

λ4

(R(λ2))3=: η > 0.

Here supp (F(ei(t−tj)∆fri)) = supp (f̂ri) ⊂ τri , a square of side `ri > 0. Assume `r1 ≥`r2 ≥ `r3 . Letting ψi := ei(t−tj)∆fri , by Plancherel’s theorem we can write∫

R2u(x)ψ1(x)ψ2(x)ψ3(x)dx =

∫∫∫(R2)3

û(ξ)ψ̂1(ξ1 − ξ)ψ̂2(ξ2)ψ̂3(ξ1 − ξ2)dξ2dξ1dξ.

The product û(ξ)ψ̂1(ξ1 − ξ)ψ̂2(ξ2)ψ̂3(ξ1 − ξ2) is non-zero only when ξ1 − ξ ∈ τr1 , ξ2 ∈ τr2 ,and ξ1− ξ2 ∈ τr3 . This implies that ξ ∈ τ, a square of sidelength ` := 3`r1 . Let Pτ be theFourier restriction operator defined by P̂τf = χτ f̂ , where χτ is the characteristic function

of the square τ . We find∫R2u(x)ψ1(x)ψ2(x)ψ3(x)dx =

∫∫∫(R2)3

û(ξ)ψ̂1(ξ1 − ξ)ψ̂2(ξ2)ψ̂3(ξ1 − ξ2)dξ2dξ1dξ

=

∫∫∫(R2)3

P̂τu(ξ)ψ̂1(ξ1 − ξ)ψ̂2(ξ2)ψ̂3(ξ1 − ξ2)dξ2dξ1dξ

=

∫R2Pτu(x)ψ1(x)ψ2(x)ψ3(x)dx.

This leads to

η <

∫∫Ij×R2

u(t)(ei(t−tj)∆fr1)(ei(t−tj)∆fr2)(e

i(t−tj)∆fr3)dxdt

=

∫∫Ij×R2

Pτu(t)(ei(t−tj)∆fr1)(e

i(t−tj)∆fr2)(ei(t−tj)∆fr3)dxdt

≤( ∫∫

Ij×R2|Pτu|2|ei(t−tj)∆fr1|2dxdt

)1/2‖ei(t−tj)∆fr2‖L4Ij×R2‖ei(t−tj)∆fr3‖L4

Ij×R2

.( ∫∫


)1/2‖fr2‖2‖fr3‖2.( ∫∫


)1/2.

That is, we have

cη2 <

∫∫Ij×R2

|Pτu|2|ei(t−tj)∆fr1|2dxdt. (2.4.4)

Applying Lemma 2.5 to g = e−itj∆fr1 with � = η10, there are tubes (Qs)1≤s


Combining (2.4.4) and (2.4.5), there is a choice of Q = {(t, x) : x+2tξ0 ∈ K, t ∈ J∩Ij} ∈(Qs)1≤sη2

S(η10)=: η1 > 0.

⇒∫∫{(t,x):x+2tξ0∈K,t∈J∩Ij}

|Pτu|4dxdt > cη21, (2.4.6)

by Hölder’s inequality, a Strichartz estimate, and fr1 ∈ L2(R2). Now observe that

‖Pτu(t)‖L∞x ≤ ‖P̂τu(t)‖L1ξ ≤ |τ |1/2‖u0‖2 ≤ C`. (2.4.7)

For a small constant δ > 0, we apply (2.4.6), split up the integral, and peel out two

factors of ‖Pτu(t)‖L∞x to find

cη21 <

∫J∩[tj ,tj+1−

δη21`2

]

∫x∈K−2tξ0

|Pτu(t)|4dx+∫J∩[tj+1−

δη21`2,tj+1]

∫x∈K−2tξ0

|Pτu(t)|4dx

<1

`2sup

t∈J∩[tj ,tj+1−δη21`2

]

∫x∈K−2tξ0

|Pτu(t)|4dx+∫ tj+1tj+1−

δη21`2

∫x∈K−2tξ0

|Pτu(t)|4dx

≤ C supt∈J∩[tj ,tj+1−

δη21`2

]

∫x∈K−2tξ0

|Pτu(t)|2dx+ C`2δη21`2‖u0‖2

= C sup

t∈J∩[tj ,tj+1−δη21`2

]

∫x∈K−2tξ0

|Pτu(t)|2dx+ δCη21.

Choosing δ sufficiently small, there is some t ∈ [tj, tj+1− δη1`2 ] and a square E1 = K−2tξ0of sidelength 1

`such that ∫

E1

|Pτu(t)|2dx > cη21. (2.4.8)

From t ∈ [tj, tj+1 − δη1`2 ], we have t < tj+1 −δη21`2< T ∗ − δη

21

`2, and therefore

side(E1) =1

`< C(T ∗ − t)1/2.

Furthermore

|Pτu(t)| . |u(t)| ∗ φ`,

where φ`(x) = `2φ(`x) and φ is a smooth bump function supported on [−1, 1]2. This

gives

|Pτu(t)|2 . |u(t)|2 ∗ φ`. (2.4.9)


Combining (2.4.8) and (2.4.9)

cη21 .∫E1

|u(t)|2 ∗ φ`dx = 〈|u(t)|2, χE1 ∗ φ`〉 .∫E2

|u(t)|2dx,

for some square E2 of sidelength side(E2) = C(side(E1)) < C′(T ∗ − t)1/2. Thus∫

E2

|u(t)|2dx > cη21 =: c′,

where c′ > 0 is a constant independent of j. As this argument applies for each j, the

expression (2.1.10) follows. This concludes the proof of Theorem 2.7.

Chapter 3

Invariance of the Gibbs measure for

the periodic quartic KdV

3.1 Introduction

Consider the Cauchy problem for the periodic quartic Korteweg-de Vries (KdV) equation{∂tu+ ∂

3xu =

14∂x(u

4), t ∈ R, x ∈ T,u(0, x) = u0(x) ∈ Hs(T).

(3.1.1)

In the introduction to this thesis, we discussed the method developed by Bourgain [6]

for proving the invariance of the Gibbs measure for the periodic gKdV equation. In

this chapter we prove the invariance of the Gibbs measure for the (gauge-transformed)

periodic quartic KdV, given by{∂tu+ ∂

3xu = P(u3)∂xu, t ≥ 0, x ∈ T,

u(0, x) = u0(x) ∈ Hs(T).(3.1.2)

As a corollary we will obtain almost sure global well-posedness of the (ungauged) periodic

quartic KdV (3.1.1) below H12 (T). Before stating and proving these results precisely, let

us begin by answering the most basic questions. What is the Gibbs measure? Why is

invariance of the Gibbs measure anticipated for Hamiltonian PDEs?

Lebowitz, Rose, and Speer [46] initiated the study of invariant Gibbs measures for

Hamiltonian PDEs. They constructed the Gibbs measure as a weighted Wiener measure.

Recall that the Wiener measure1, ρ, is the probability measure supported on⋂s< 1

2Hs(T)

1This is the mean zero Wiener measure, but we restrict attention to measures, data, and solutionswith spatial mean zero throughout this thesis, and will often omit the pre-fix “mean zero”.

37

Chapter 3. Invariance of the Gibbs measure for the quartic KdV 38

with density

dρ = Z−10 e− 1

2

∫u2xdx

∏x∈T

du(x), u mean zero. (3.1.3)

This is a purely formal expression, but it provides intuition. We can in fact define ρ as

the weak limit of a sequence of finite-dimensional Wiener measures ρN . Each ρN is the

probability measure on CN (the space of Fourier coefficients) with density

dρN = Z−1N e

− 12

∑0


where PN denotes Dirichlet projection to EN . The flow of (3.1.7) leaves the followingquantities invariant:

(i) The Hamiltonian H(uN).

(ii) The L2-norm ‖uN‖L2 = (∑

0


To properly state our results, we need one more definition. Let ΦN(t) denote the flow

map of the finite-dimensional system of ODEs obtained by projecting (3.1.2) to the first

N > 0 modes of the trigonometric basis:{∂tu

N + ∂3xuN = PN(P((uN)3)∂xuN), t ∈ R, x ∈ T,

uN(0, x) = PN(u0(x)), u0 mean zero.(3.1.10)

Here PN denotes Dirichlet projection to EN = span{sin(nx), cos(nx) : 1 ≤ n ≤ N}. Thefirst Theorem of this chapter is almost sure local well-posedness of (3.1.2) with initial

data given by (3.1.9).

Theorem 3.1 (Almost sure local well-posedness). The gauge-transformed periodic quar-

tic KdV (3.1.2) is locally well-posed almost surely in H1/2−(T). More precisely, for all0 < δ1 < δ, with δ sufficiently small, there exists 0 < β < δ − δ1, and c > 0 such that foreach T � 1, there is a set ΩT ∈ F with the following properties:

(i) The complemental measure of ΩT is small. More precisely, we have

P (ΩcT ) = ρ ◦ u0(ΩcT ) < e− cTβ

, where ρ is the Wiener measure defined in (3.1.3), and the initial data (given by

(3.1.9)) is viewed as a map u0 : Ω→ H1/2−(T).

(ii) For each ω ∈ ΩT there exists a solution u to (3.1.2) with data u0,ω satisfying

u ∈ S(t)u0,ω + C([−T, T ];H1/2+δ(T)) ⊂ C([−T, T ];H1/2−(T)).

(iii) This solution is unique in {S(t)u0,ω + BK}, for some K > 0, where BK denotes aball of radius K in the space X

12

+δ, 12−δ

T .

(iv) The solution u depends continuously on the initial data, in the sense that, for each

ω ∈ ΩT , the solution map

Φ :{u0,ω + {‖ · ‖H 12 +δ ≤ R}

}→{S(t)u0,ω + {‖ · ‖C([−T,T ];H 12 +δ) ≤ R̃}

}is well-defined and Lipschitz, for some fixed R, R̃ ∼ 1.

(v) The solution u is well-approximated by the solution of (3.1.10). More precisely,

‖u− S(t)u0,ω − (ΦN(t)− S(t))PNu0,ω‖C([0,T ];H 12 +δ1 ) . N−β. (3.1.11)


Following the method developed in [6], we use the invariance of the finite-dimensional

Gibbs measures µN under the flow of (3.1.10), and an approximation argument, to extend

the local solutions of (3.1.2) (obtained from Theorem 3.1) to global solutions, and to prove

the invariance of the Gibbs measure under the flow.

Theorem 3.2 (Invariance of the Gibbs measure). The gauge-transformed periodic quartic

KdV (3.1.2) is globally well-posed almost surely in H1/2−(T). More precisely, for δ2 > 0sufficiently small, it holds that for almost every ω ∈ Ω, given any T > 0, there is a(unique) solution u to (3.1.2) with data u0,ω given by (3.1.9) satisfying

u ∈ S(t)u0,ω + C([0, T ];H1/2+δ2(T)) ⊂ C([0, T ];H1/2−(T)).

Furthermore, the Gibbs measure µ (given by (3.1.6)) is invariant under the flow.

By inverting the Gauge transformation, we obtain the following corollary.

Corollary 3.1 (Almost sure global well-posedness). The periodic quartic KdV (3.1.1)

is globally well-posed almost surely in H1/2−(T). More precisely, for almost every ω ∈ Ω,for every T > 0, there exists a solution u to (3.1.1) for t ∈ [0, T ] with data u0,ω given by(3.1.9).

Remark 3.1. The solution of (3.1.2) produced by Theorem (3.2) is unique in a mild sense

only. For the technical statement, see Remark 3.7 in Section 4.3. For the solution of

(3.1.1) produced by Corollary 3.1, we have an existence result only.

Remark 3.2. By composing with the (time-dependent) gauge transformation Gt, we canobtain a time-dependent measure νt := µ ◦Gt, supported on Hs(T) for s < 12 , which (dueto Theorem 3.2) satisfies Φ(t)∗νt = µ for each t ≥ 0, where Φ is the evolution operator forthe ungauged quartic KdV (well-defined in the support of the Gibbs measure by Corollary

3.1). We cannot show (at this time) that νt = µ for all t ≥ 0, and this leads to a naturalquestion for future investigation: is the time-dependent measure νt = µ ◦ Gt absolutelycontinuous with respect to the Gibbs measure µ? Do we in fact have invariance of the

Gibbs measure for the ungauged quartic KdV (3.1.1)? This type of issue was recently

resolved in the affirmative for the periodic derivative NLS [56].

For the remainder of this section we provide more background on this problem, then

outline the methods involved and the challenges confronted in the proofs of Theorem 3.1

and Theorem 3.2.


3.1.2 Background and method

Nonlinear smoothing

As discussed above, we can construct the Gibbs measure for the quartic KdV (3.1.1)

as in [46, 6]. However, the Gibbs measure is supported below H12 (T), and local well-

posedness of the quartic KdV (both gauged and ungauged) cannot be established in

Hs(T) for s < 12

by applying the contraction principle to an equivalent integral equation;

the data-to-solution map is not C4 [21].

The same issue was confronted by Bourgain in [7]. He considered the cubic NLS on

T2 with randomized data given by

ũ0,ω(x) =∑n∈Z2

gn(ω)√1 + |n|2

ein·x, (3.1.12)

where {gn}n∈Z2 is a collection of complex-valued independent and identically distributedGaussian random variables of mean 0 and variance 1 on a probability space (Ω,F ,P).The data (3.1.12) represents a typical element in the support of the 2-dimensional Wiener

measure (which is, as above, the Gaussian part of the Gibbs measure for 2-dimensional

cubic NLS). Bourgain quantified the nonlinear smoothing effect by proving that, with

high probability, the nonlinear part of the solution to the cubic NLS lies in a smoother

space - C([0, T ];Hs(T2)) for some s > 0 - than the linear evolution of the randomizeddata - which almost surely stays below L2(T2) for all time. More precisely, for all T > 0sufficiently small, he constructs a set ΩT ⊂ Ω corresponding to “good” randomized dataũ0,ω, such that ΩT is exponentially likely as a function of T ↓ 0, and such that foreach ω ∈ ΩT , one can prove local existence and uniqueness of the solution to the cubicNLS with data ũ0,ω for t ∈ [0, T ] by performing a contraction argument in the space{eit∆ũ0,ω + B}, where B is a ball in Z

s, 12

T ⊂ C([0, T ];Hs(T2)) for some s > 0 (for thedefinition of the function space Z

s, 12

T , consult section 3.2). By taking an appropriate

union over sets of this type, with T ↓ 0, one obtains local well-posedness almost surelyfor the Wick-ordered cubic NLS below L2(T2). For other works that have used nonlinearsmoothing to establish local dynamics in the support of measures on phase space, see

Burq-Tzvetkov [11, 12, 13], Oh [61], and Colliander-Oh [22].

For (3.1.2), we consider randomized initial data of the form (3.1.9). We could not

follow the method of [7] directly, and perform a contraction argument for (3.1.2) (with

exponential likelihood) in {S(t)u0,ω +BR}, where S(t) := eit∂3x and BR is a ball of radius

R in the Banach space Zs, 1

2T . In particular, we found that the square of the Z

s, 12

T -norm of


the first Picard iterate for (3.1.1) has infinite expectation. This is due to the temporal

regularity b = 12

of the Zs, 1

2T space.

To avoid this obstruction, we will establish estimates on the nonlinear part of the

Duhamel formulation for the gauge-transformed quartic KdV (3.1.2) in Xs,bT , with s >12

and b < 12. Unfortunately, with b < 1

2, we cannot perform (with exponential likelihood) a

contraction argument for (3.1.2) in {S(t)u0,ω +BR}, where BR is a ball of radius R in theBanach space Xs,bT . Indeed, by taking b <

12, there are regions of frequency space which

(produce contributions that) make the nonlinear estimates required for a contraction

argument impossible.

This is resolved by establishing estimates on the second iteration of the Duhamel

formulation of (3.1.2). More precisely, the local-in-time solution u to (3.1.2) will be

constructed as the limit in Xs,bT of a sequence of smooth solutions uN evolving from

frequency truncated data uN0,ω = PN(u0,ω). Each uN will satisfy the Duhamel formulation

uN(t) = S(t)uN0,ω +

∫ t0

S(t− s)N (uN(s))ds, (3.1.13)

where N (u) = ux(u3 − 1

2π

∫T u

3dx)

is the gauge-transformed nonlinearity. We will esti-

mate ‖uN‖Xs,bT , ‖uN−uM‖Xs,bT in order to establish convergence to a solution u as N →∞.

During the nonlinear estimates, in the troublesome regions of frequency space (created

by taking b < 12) we will substitute (3.1.13) into an appropriately chosen factor of the

nonlinearity. This will resolve the technical obstruction (due to b < 12), but by consid-

ering

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by Geordon Haley Richards - University of Toronto T-Space...Geordon Haley Richards Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2012 This thesis contributes