On the local well-posedness of the Kadomtsev-Petviashvili II equation [Elektronische Ressource] / vorgelegt von Martin Hadac

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On the local well-posedness of theKadomtsev-Petviashvili II equationDissertationzur Erlangung des Grades einesDoktors der NaturwissenschaftenDem Fachbereich Mathematik der Universit˜ at Dortmundvorgelegt vonMartin Hadacim Mai 2007Gutachter der Dissertation: Professor Dr. Herbert Koch Jean-Claude SautProfessor Dr. Winfried KaballoTag der mund˜ lichen Prufung:˜ 22. August 2007Contents1 Introduction 12 Bourgain spaces and well-posedness 52.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Bourgain spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 The linear equation . . . . . . . . . . . . . . . . . . . . . . . . 112.4 The general well-posedness result . . . . . . . . . . . . . . . . 152.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . 233 Dispersive inequalities 253.1 Local smoothing estimates . . . . . . . . . . . . . . . . . . . . 253.2 Linear Strichartz . . . . . . . . . . . . . . . . . . . . 273.3 Bilinear Strichartz-type estimates: Generalities . . . . . . . . . 313.4 Strichartz type estimates in 2D . . . . . . . . . . . . . 343.5 Bilinear Strichartz type in 3D . . . . . . . . . . . . . 383.6 Notes and references . . . . . . . . . . . . . . . . . . . . . . . 444 The two dimensional case 454.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 The main bilinear estimate . . . . . . . . . . . . . . . . . . . . 474.3 Estimates for the Q . . . . . . . . . . . . .

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• Mathematics

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• Publié par: technischen_universitat_dortmund
• Publié le: 01 janvier 2007
• Nombre de lectures: 11
• Langue: English

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On the local well-posedness of the Kadomtsev-Petviashvili II equation

Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften

DemFachbereichMathematikderUniversita¨tDortmund vorgelegt von

Martin Hadac

im Mai 2007

Gutachter der Dissertation:

Tagdermu¨ndlichenPr¨fung: u

Professor Dr. Herbert Koch Professor Jean-Claude Saut Professor Dr. Winfried Kaballo

22. August 2007

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Contents

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4.4 4.5

three dimensional case Main results . . . . . . . . . . . . . The main bilinear estimate . . . . . Estimates for theQ ij . . . . . . . . Proof of the main bilinear estimates Notes and references . . . . . . . .

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Introduction

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Dispersive inequalities 3.1 Local smoothing estimates . . . . . . . . . . . . 3.2 Linear Strichartz estimates . . . . . . . . . . . . 3.3 Bilinear Strichartz-type estimates: Generalities . 3.4 Bilinear Strichartz type estimates in 2D . . . . . 3.5 Bilinear Strichartz type estimates in 3D . . . . . 3.6 Notes and references . . . . . . . . . . . . . . .

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Bourgain spaces and well-p 2.1 Preliminaries . . . . . . 2.2 Bourgain spaces . . . . . 2.3 The linear equation . . . 2.4 The general well-posednes 2.5 Notes and references . .

4.2 4.3

The 4.1

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two dimensional case

Main results . . . . . . . . . . . . . . . . . . . . The main bilinear estimate . . . . . . . . . . . . Estimates for theQ ij . . . . . . . . . . . . . . . Proof of the main bilinear estimate . . . . . . . Notes and references . . . . . . . . . . . . . . .

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osedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 7 11 15 23

Bibliography

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Chapter

1

Introduction

In this thesis we are concerned with the local well-posedness theory of the initial value problem for the Kadomtsev-Petviashvili II equation in two space dimensions

(u t +u xxx + (u 2 ) x ) x +u y = 0 y

as well as in three space dimensions

(u t +u xxx + (u 2 ) x ) x + Δ ~y u= 0

in (− TT )×R 2 

3 in (− TT )×R

u(0) =u 0

u(0) =u 0

and dispersive generalisations thereof. The Kadomtsev-Petviashvili II equations are universal models for the propagation of long weakly dispersive waves which are essentially one dimen- sional with weak transverse effects. 1 They can be seen as multidimensional generalisations of the Korteweg-de Vries equation2

u t +u xxx + (u 2 ) x = 0 in (−T  T)×R

u(0) =u 0

We consider initial valuesu 0 in non-isotropic Sobolev spacesHs 1 ,s 2 (Rd) and our goal is to show the local well-posedness for low regularity data, i. e. data inH s 1 ,s 2 (R d ) withs 1 ands 2  notion of Ouras small as possible. well-posedness comprises, for given regularitiess 1 ands 2 , theexistenceand uniquenessof solutions in a suitable space of space-time functions (or more generally distributions)X T , thepersistence of regularity, i. e. the solution uis a continuous function intwith values in the Banach spaceH s 1 ,s 2 (R d ),

1 See [16]. 2 For an explanation how the Kadomtsev-Petviashvili equations are (formally) obtained from the one dimensional models (also for more general dispersion terms), see also the introduction of [22].

1

2

and thecontinuous dependenceof the solutions on the initial data, i. e. the flow map, which assigns the solutionuto the initial valueu 0 , is a continuous mapping fromHs 1 ,s 2 (Rd) toX T . In fact, all flow maps turn out to be analytic mappings. This stems from the fact that we use aPicard iteration method on the Duhamel formulation3of the Kadomtsev-Petviashvili II equation to construct the solution and from the fact that the nonlinearity is polynomial. The spacesX T where the solutions are constructed are modifications of the spaces first used byBourgain[5] in the context of the Kadomtsev- Petviashvili II equation (onT2rather than onR2).4Bourgain’sidea was to include the symbol of the linear part of the equation into the definition of the spaces, which makes it possible to easily exploit dispersive properties of the linear equation in the context of these spaces and which also allows to exploit certain algebraic properties of the symbol in order to overcome the loss of derivatives in the nonlinearity. The proof of local well-posedness then reduces to showing a suitable estimate for the nonlinearity in these spaces.5 By using the Picard iteration method in the modified Bourgain spaces, we show the local well-posedness of the Kadomtsev-Petviashvili II equation in two space dimensions fors 1 >− 1 2 ands 2 ≥ the scale of spaces0. On Hs 1 ,0(R2full subcritical range because the homogeneous) this includes the spaceH ˙ − 21 ,0 (R 2 ) is scale invariant for this problem. Since it is not possible to obtain the crucial bilinear estimate in the standard Bourgain spaces for − 2 1 < s 1 <− 3 1 which can be seen by the counterexamples in [31], we include a low frequency condition into the definition of the spaces. 6 The drawback of this low frequency condition is that the resulting spaces do not contain the (time localized) solutions of the linearized equation unless the initial value obeys the same low frequency condition. Therefore, we choose the spaceX T to be the sum of the low-frequency modified space and a standard space. This sum structure is the crucial ingredient to be able to lower thex-regularity without imposing a low frequency condition on the initial values.7 By the same method, we show the local well-posedness of the Kadomtsev-

3 More precisely, because the product in the nonlinearity does not make sense a priori for very rough initial values, we consider an operator equation which coincides with the Duhamel formulation for smooth functions. However, we will show in Theorem 3.3 that the solutions thus constructed are, in fact, distributional solutions of the original equation. 4 These spaces have already been used byouBairgn in the context of the[3, 4] Korteweg-deVriesandthenonlinearSchr¨odingerequation. 5 general scheme how to prove local well-posedness of theFor a good overview of the equation from the multilinear estimates see [6] or the first part of [7]. 6 A similar condition was already used byakTaaok[30] to get local well-posedness in the range− 1 2 < s 1 <− 3 1 initial value also satisfies a low frequencybut only if the condition, i. e. for initial da  ˙ 1 R 2 ) with suitably chosenε. ta inH s 1 ,0 (R 2 )∩H − 2 +ε,0 ( 7 Cf. also Remark 4.9.

Chapter 1.

Introduction

3

Petviashvili II equation in three space dimensions fors 1 > 2 1 ands 2 >0. In this case,H ˙ 21 ,0 (R 3 ) is scale invariant. More generally, we prove, by the method described above, the local well- posedness of thedispersion generalised Kadomtsev-Petviashvili II equation

(u t − |D x | α u x + (u 2 ) x ) x + Δ ~y u= 0 in (− TT )×R d  u(0) =u 0 (1.1)

for 4 3 < α≤6, ifd= 2, and 2≤α≤6, ifd equations are These= 3. multidimensional generalisations of the one dimensional models

u t − |D x | α u x + (u 2 ) x  ( in= 0− TT )×R u(0) =u 0

Ford (1.1) for2, we obtain local well-posedness of= s 1 >max µ 1− 3 4 α 1 4 − 3 8 α ¶

(1.2)

ands 2 ≥ the0. BecauseL2-norm of real valued solutions of (1.1) is con- served, this immediately implies global well-posedness for real-valued initial data inHs 1 ,0(R2) fors 1 ≥ note that if0. We 3 4 < α <2, we still get the full subcritical range on the scaleHs 1 ,0(R2). It is interesting that for theseαthe two dimensional models “behave better” than the one dimensional equation (1.2) in the sense that the flow map of the one dimensional model cannot be C 2 -differentiable at the origin in any Sobolev spaceH s (R also means). This that it is not possible to solve (1.2) inHs(R) with a Picard iteration scheme.8 The caseα= 4 is also known asfifth order Kadomtsev-Petviashvili II equation

(u t −u xxxxx + (u 2 ) x ) x +u yy = 0

in (− TT )×R 2 

u(0) =u 0

Our result in this case shows local well-posedness fors 1 >− 5 4 ands 2 ≥0. 9 Ford= 3 and 2< α≤6, we obtain local well-posedness of (1.1) for s 1 >max µ 2 3 − α 2  4 1 − 42 5 α ¶

ands 2 ≥0. As in the two dimensional case, the global well-posedness for real-valued initial data inH s 1 ,0 (R 3 ) fors 1 ≥0 andα >3 follows.

8 This has been proven byMolinet, Saut and Tzvetkov[21]. Note, however, that a Picard iteration has been applied byHerr(cf. [8], Chapter 4) to prove well-posedness for initial values in Sobolev spaces which include a low frequency condition. 9 Note that well-posedness for the same class of initial data has recently been obtained byndzaj ´ Me,Lızaape ´ oasI[12].

4

In the caseα= 4 of thefifth order Kadomtsev-Petviashvili II equation in three space dimensions

(u t −u xxxxx + (u 2 ) x ) x + Δ y u= 0 ~

in (−T  T)×R 3 

u(0) =u 0

our result shows the local well-posedness fors 1 >− 1 2 ands 2 ≥0. We now give an overview of the organization of this thesis: In Chapter 2, after fixing some notation, we introduce the Bourgain spaces and show a general well-posedness result which reduces the question of lo- cal well-posedness for equation (1.1) to a bilinear estimate in the Bourgain spaces. In Chapter 3, we first give local smoothing as well as Strichartz estimates for solutions of the linear equation

(u t − |D x | α u x ) x + Δ ~y u= 0

in (−T  T)×R d 

u(0) =u 0

It is shown that the local smoothing estimate implies that solutions of the Duhamel formulation of (1.1) are actually solutions in the distributional sense. In Section 3.3 we give an overview of the techniques used to de- rive bilinear Strichartz type estimates and discuss some of their properties. Finally, in Section 3.4 and Section 3.5 we derive bilinear Strichartz type es- timates in the two dimensional, respectively three dimensional case. These estimates are the building blocks used to derive the bilinear estimate which is needed to apply the general well-posedness result of Section 2.4. In Chapter 4, the main results for the two dimensional case are proven. The main bilinear estimate for the two dimensional case is announced in Section 4.2 and proven in Section 4.3 and Section 4.4. This is done by first splitting the nonlinearity into various pieces and then using for each piece a pointwise estimate to reduce the case to an appropriate bilinear Strichartz type estimate of Section 3.4. In Chapter 5, the main results for the three dimensional case are proven. This is done analogously to the two dimensional case. I would like to thank my advisor Professor Dr. Herbert Koch for his constant support and encouragement as well as for many valuable suggestions and discussions on the subject. I would also like to thank Sebastian Herr for helpful discussions. Furthermore, I would like to thank Maren Martens and Christoph Hadac for proofreading parts of the manuscript.

Chapter

2

Bourgain spaces well-posedness

2.1

Preliminaries

and

Let us first recall some known facts about standard function spaces and fix some notation that will be used throughout this thesis:

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Forx∈Rnlethxi:= (1 +|x|2) 1 2 .

LetS(Rnthe Schwartz space, i. e. the space of all) denote u∈C such that for allj∈N

q j (u) := | m γ| a ≤ x j m ∈ a R x n hxi j |∂ γ u(x)|<∞ x

∞(Rn)

(2.1)

It is well known that, endowed with this family of seminorms,S(Rn) is a Frechet space, i. e. a completely metrizable topological vector space. ´ The dual spaceS0(Rn) is called the space of tempered distributions on Rn.

dalways denotes the number of space variables in the equation, i. e. d= 2 when we consider the two dimensional case andd= 3 when we consider the three dimensional case. The space variable will always be denoted by (~yx) wherex∈Randy~∈R d−1 . If we consider the case d= 2, we will often writeyinstead of~y. If we consider the cased= 3, we will write~y= (y y˜).

n:=d+ 1 always denotes the number of total variables (including the time variablet) in the equation.

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2.1.

Preliminaries

Foru∈L1(Rn) theFourier transformFuofuis defined as ( Z R n u(x~yt)xd~ydtd(ξη~τ)∈R n Fu)(ξτ~η) :=e −i(tτ+xξ+y~∙η~)

(2.2)

It is well known thatF:S(Rn)→ S(Rn) is a topological and linear isomorphism with (F −1 v)(tx~y) = (2π) −n Z R e i(tτ+xξ+y~∙η~) v(τ~ηξ)~ηξdτdd(2.3) n

forv∈ S(Rn). Furthermore,Fcan be extended to a linear and con- tinuous isomorphism onS0(Rn we only consider a partial Fourier). If transform in some of the variables, we will denote this byF 1 for the Fourier transform in the first variable, etc.

Fors∈Rwe define the operatorsJs x ,Js y~ , and|D x |sasFourier multiplier operatorswith multiplierhξis,h~yis, and|ξ|s means,, respectively. This for example, that (F 2 Js x u)(tξ~y) =hξisF 2 u(tξ~y),ξ∈R.

The (non-isotropic) Sobolev spaceHs 1 ,s 2 (Rd) is the space ofu 0 ∈ S0(Rd) such that the norm

is finite.

ku 0 k H s 1 ,s 2 :=khξi s 1 hη~i s 2 Fu 0 k L ξ 2 η~

(2.4)

µ= (η~τξ)∈R3always denotes the Fourier variable dual to (xty~). In the cased= 2 we will again writeηinstead ofη~ the case. Ind= 3 we will writeη~= (η η˜).

Forµ= (~ητξ) let

~2 λ:=λ(µ) :=τ−ξ|ξ| α + η ξ

(2.5)

where~η2:=~η∙~η there are two frequency vari- Ifis the scalar product. ablesµandµ 1 , we will writeµ 2 :=µ−µ 1 ,λ 1 :=λ(µ 1 ),λ 2 :=λ(µ 2 ) for short. The elements ofµ 2 are also denoted by (τ 2  ξ 2 ~η 2 ). Further- more, let|λ max |:= max(|λ||λ 1 ||λ 2 |),|ξ max |:= max(|ξ||ξ 1 ||ξ 2 |), and |ξ min |:= min(|ξ||ξ 1 ||ξ 2 |).

A.Bmeans that there is a (harmless) constantCsuch thatA≤CB. A∼Bis equivalent toA.BandB.A.

Chapter 2.

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2.2

Bourgain spaces and well-posedness

7

For a Banach spaceXand a Hausdorff topological vector spaceA, the notationX ,→ Ameans that there is a continuous embedding fromX intoA. LetC b (R;X) denote the Banach space of all continuous and bounded functionsf:R→X Ifwith the sup-norm.X 1 andX 2 are Banach spaces withX i ,→ A, whereAis a Hausdorff topological vector space, we will often consider the two Banach spacesX 1 ∩X 2 , endowed with the norm

kxk X 1 ∩X 2 :=kxk X 1 +kxk X 2 

x∈X 1 ∩X 2

(2.6)

andX 1 +X 2 :={x∈ A |x=x 1 +x 2  x i ∈X i (i= 12)}, endowed with the norm

kxk X 1 +X 2 := inf{kx 1 k X 1 +kx 2 k X 2 |x=x 1 +x 2  x i ∈X i (i= 12)} (2.7)

Bourgain spaces

In this section we define the function spaces which are adapted to the linear part of equation (1.1). As the symbol of the linear operator has a singularity alongξ= 0 and as we want to be able to deal with a low frequency condition inξ, we will consider the following space of test functions.

Definition 2.1.

S −∞ :={φ∈ S(R n )|∂ k ξ Fφ(τ0η~) = 0∀k∈N 0 ∀(~τη)∈R n−1 }

(2.8)

Remark2.2.S −∞ capstehce´rFehtfesiaceoubspsedsacloS(Rn). The functions inS −∞ have the property that fork∈N 0 and for (τξ~η)∈Rn, we have|Fφ(~ηξτ)| ≤q k (φ)|ξ|k. Therefore, fors 1  s 2  b σ∈R, the following definition makes sense.

Definition 2.3.Lets 1  s 2  b σ∈R. Forφ∈ S −∞ let

kφk 1 ,s 2 :=k X σb,s |ξ| −σ hξi s 1 +σ hη~i s 2 hλi b Fφk L 2 µ

(2.9)

withλ define the space Weas defined in (2.5).X σ sb, 1 ,s 2 as the completion of S −∞ with respect to the norm (2.9).

Remark2.4.Ifs 2 = 0, we simply writeXbs, σ 1 instead ofXb,s σ 1 ,0. b,s 1 ,s 2 We can identifyX σ a subspace of tempered distributions onwith Rn, at least forσ >− 1 2 andb >− 2 1 −σ. In order to prove this, we shall need the following lemma.