REVIEW ON LONG TIME ASYMPTOTICS OF LARGE DATA IN SOME NONINTEGRABLE DISPERSIVE MODELS CLAUDIO MUÑOZ_

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REVIEW ON LONG TIME ASYMPTOTICS OF LARGE DATA IN

SOME NONINTEGRABLE DISPERSIVE MODELS

CLAUDIO MUÑOZ_

ABSTRACT. In this short note we review recent results concerning the long time

dynamics of large data solutions to several dispersive models. Starting with the

KdV case and ending with the KP models, we review the literature and present

new results where virial estimates allow one to prove local energy decay in dif-

ferent regions of space where solitons, lumps or solitary waves are not present.

1. INTRODUCTION

These notes are part of a talk given at the TYAN Virtual Thematic Workshop in

Mathematics, held on September 27th 2021. Here, I will describe some new results

concerning the time decay of solutions to Korteweg-de Vries (KdV) and related

equations with rough large data, mainly concentrated in regions where solitons are

not present. These are results obtained with my collaborators

‚Miguel A. Alejo (U. Córdoba, Spain),

‚Fernando Cortez (EPN, Ecuador),

‚Chulkwang Kwak (Ehwa Womans U., South Korea),

‚Argenis Mendez (PUCV Chile),

‚Felipe Poblete (UACh, Chile),

‚Gustavo Ponce (UCSB, USA),

‚Juan C. Pozo (U. Chile, Chile),

‚Jean-Claude Saut (U. Paris-Saclay, France),

to whom I deeply acknowledge.

Setting. First of all, we consider the generalized KdV equations posed on R:

(gKdV) "Btu`BxpB2

xu`upq “ 0,

u“upt,xq P R,t,xPR,p“2,3,4.

Some important remarks are in order:

(1) p“2,3,4: equation is globally well-posed in H1(Kato [52], Kenig, Ponce

and Vega [57]).

Date: October 21, 2022.

2020 Mathematics Subject Classiﬁcation. Primary: 35Q53. Secondary: 35Q05.

Key words and phrases. Dispersive equations, decay, long time dynamics, virial estimates, large

data.

_C.M.’s work was funded in part by Chilean research grants FONDECYT 1191412, project

France-Chile ECOS-Sud C18E06, MathAmSud EEQUADD II and Centro de Modelamiento

Matemático (CMM), ACE210010 and FB210005, BASAL funds for centers of excellence from

ANID-Chile.

arXiv:2210.10902v1 [math.AP] 19 Oct 2022

2 C. MUÑOZ

(2) p“2,3 are completely integrable models (KdV and mKdV, respectively),

see e.g. [1] and references therein.

(3) p“5 is the L2critical case, in the sense that the natural scaling in this case

upt,xq ÞÑ c1{2upc3t,cxqpreserves the L2norm.

(4) If pě5, blow up may occur (Martel-Merle [86]).

Essentially, one has global solutions for data in H1(or L2) and power pď4. For

more details, the reader can consult the monograph by Linares and Ponce [80].

Here, we are interested in the following

Key question: To describe the asymptotics of large, globally deﬁned solutions in

nonintegrable cases (a.k.a. the “soliton resolution conjecture”) and subcritical

nonlinearities.

Somehow this reduces the problem to the case p“4 in (gKdV), but we are

also interested in proofs that are independent of the integrable character of the

model. In particular, the results that I shall present here are also valid for some of

the integrable models, and are concentrated in the regions where solitons/solitary

waves/lumps are not present.

To try to explain the previous key question, we need some deﬁnitions.

Asoliton (or solitary wave) is a solution to (gKdV) of the form

upt,xq “ Qcpx´ctq,

with QcPH1pRqthe unique (up to translations) positive solution to

c´cQc`Qp

c“0,cą0.

The soliton resolution conjecture states that, except for some particular cases, glob-

ally deﬁned solutions to (gKdV) decompose, as time tends to inﬁnity, as the sum

of solitons and radiation. By radiation, we broadly mean a linear solution of

(gKdV). Mathematically,

upt,xq „L2ÿ

jě1

Qcjptqpx´ρjptqq``ptq,tÑ `8.

Here, cjptqand ρjptqare modulated scaling and shifts, respectively, and `ptqde-

notes a linear solution, namely Bt``B3

x`“0 (pě4, see explanations below).

Several questions appear immediately: are the cjptqalways different? What about

the behavior of ρjptq?

These are questions that still remain unsolved if p“4, for instance. Indeed,

except for integrable cases (to be described below), this conjecture is far from

being established in the nonintegrable subcritical setting.

2. A QUICK REVIEW ON THE LITERATURE

For the sake of completeness, we will consider a general nonlinearity fpuqin

(gKdV), which will be speciﬁed below. Therefore, we have

(gKdV) Btu`BxpB2

xu`fpuqq “ 0.

The following account is by far not 100% accurate and complete, but it describes

in a broad sense the recent literature on the subject.

LARGA DATA ASYMPTOTICS 3

Long time behavior of small solutions: this corresponds to the case where one as-

sumes small initial data in a well-chosen Banach space. The situation is quite good

for powers of the nonlinearity fpuq „ upwith pě3, but below this range things

are quite complicated and essentially only proved via inverse scattering techniques

and under weighted norms.

We will say that a solution uptq P H1of (gKdV) scatters to a linear one if there

exists `ptq P H1solution of Airy Bt``B3

x`“0 such that

}uptq´`ptq}H1Ñ0,tÑ `8.

This is formally the expected case if pě4. Sometimes one needs to perturb `ptqto

recover the asymptotic behavior at inﬁnity, this is the case of modiﬁed scattering

(pď3).

Concerning nonintegrable techniques, here one has:

(1) Kenig-Ponce-Vega [57]: scattering for small data solutions of the L2criti-

cal gKdV equation (p“5).

(2) Ponce-Vega [109]: for the case fpsq“|s|p,pąp9`?73q{4„4.39, small

data solutions in L1XH2lead to decay, with rate t´1{3(i.e. linear rate of

decay).

(3) Christ and Weinstein [14]: scattering of small data if fpsq“|s|p,pą

4p23 ´?57q „ 3.86.

(4) Hayashi and Naumkin [39,40] studied the case pą3, obtaining decay

estimates and asymptotic proﬁles for small data in the classical weighted

space H1,1.

Long time behavior of small solutions around solitons: In this case, we assume

initial data which is close in some sense to a soliton. Consequently, one seeks here

stability and asymptotic stability (AS) of the soliton in particular spaces.

(1) Bona-Souganidis-Strauss [10]: Stability and instability for the gKdV soli-

ton in the subcritical and supercritical cases.

(2) Pego-Weinstein [108]: First result of AS gKdV models: asymptotic stabil-

ity in exponentially weighted spaces.

(3) Martel-Merle [87,88,89,90]: Asymptotics in the energy space using virial

techniques, and studied the collision problem in quartic gKdV.

(4) Tao [117] considered the scattering of data for the quartic KdV in the space

H1X

H´1{6around the zero and the soliton solution.

(5) The ﬁnite energy condition above was then removed by Koch and Marzuola

[63] by using U´Vspaces.

(6) Côte [17] constructed solutions to the subcritical gKdV equations with a

given asymptotical behavior, for p“4 (quartic) and p“5 (L2critical).

(7) Germain, Pusateri and Rousset [33] dealt with the mKdV case (p“3)

around the zero background and the soliton, by using Fourier techniques

and estimates on space-time resonances. See also Harrop-Grifﬁths [37].

(8) Chen-Liu [13]: soliton resolution of mKdV using inverse scattering tech-

niques, in weighted Sobolev spaces. This results also includes the presence

of nonzero speed breathers (3.1), to be explained below.

4 C. MUÑOZ

No scattering results seems to hold for the quadratic power (p“2), which can

be considered as “supercritical” in terms of modiﬁed scattering.

3. A FIRST RESULT FOR THE KDVMODEL

Our ﬁrst result concerns the integrable quadratic case, which is nothing but KdV.

Using integrability techniques, it is possible to give a detailed description of dif-

ferent regions, see e.g. [24,26]. Assuming that the solution stays in the space

L8pR;L1pRqq, one can prove full decay around zero in the region |x|.t1{2´.

Theorem (G. Ponce-M. 2018 [104]). Assume u PCpR;H1pRqqXL8pR;L1pRqq

solution to (KdV) p “2, then

lim

tÑ8 żΩptq

u2px,tqdx“0,Ωptq:“!xPR:|x|.t1{2log´2t).

There are several comments that are important to address:

Remarks.

(1) Solitons do satisfy the last hypothesis (being in L8pR;L1pRqq).

(2) No size restriction on the data is needed.

(3) No use of integrable techniques, this result is valid for nonintegrable per-

turbations of KdV and small H1data, in the form

u2`ouÑ0pu2q.

One of the most important examples is the integrable Gardner model:

Btu`BxpB2

xu`u2`µu3q “ 0,µą0,

see [2] for more details.

(4) This decay result is not valid for p“3 (existence of breathers, which are

counterexamples, see below) and p“4 (scaling problems).

(5) Ifrim-Koch-Tataru [44] give a detailed description of asymptotics for weighted

small data Opεqand times of cubic order Opε´3q.

About breathers. It is well-known that both mKdV and Garner models have stable

breather solutions [1,2,4,5,7], that is to say, localized in space solutions which

are also periodic in time, up to the symmetries of the equation. An example of

these type of solutions is the mKdV breather: for any α,βą0,

Bpt,xq:“2?2Bxarctanˆβsinpαpx`δtqq

αcoshpβpx`γtqq˙,

δ“α2´3β2,γ“3α2´β2,

(3.1)

is a solution of mKdV with nontrivial time-periodic behavior, up to the translation

symmetries of the equation. Therefore, generalized KdV equations may have both

solitary waves and breathers as well, and both classes of solutions do not decay.

About the proof. Since we want to show that

lim

tÑ8 żΩptq

u2px,tqdx“0,

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REVIEWONLONGTIMEASYMPTOTICSOFLARGEDATAINSOMENONINTEGRABLEDISPERSIVEMODELSCLAUDIOMUÑOZ_ABSTRACT.Inthisshortnotewereviewrecentresultsconcerningthelongtimedynamicsoflargedatasolutionstoseveraldispersivemodels.StartingwiththeKdVcaseandendingwiththeKPmodels,wereviewtheliteratureandpresentnewresultswherev...

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REVIEW ON LONG TIME ASYMPTOTICS OF LARGE DATA IN SOME NONINTEGRABLE DISPERSIVE MODELS CLAUDIO MUÑOZ_

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