Lin LIN Wei YAN Jinqiao DUAN

Abstract This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition.Motivated by the works of Thomann and Tzvetkov (Nonlinearity,23 (2010),2771–2791),Tzvetkov (Probab.Theory Relat.Fields,146 (2010),4679–4714),Burq,Thomann and Tzvetkov (Ann.Fac.Sci.Toulouse Math.,27 (2018),527–597),the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity,and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem,Skorokhod convergence theorem and Gibbs measure.

Keywords Higher-order modified Camassa-Holm equation,The randomization of the initial value,Gibbs measure,Global solution

1 Introduction

In this paper,we consider the Gibbs measure and the Cauchy problem for the following periodic higher order modified Camassa-Holm equation under randomization on initial data:

Omitting the second term in (1.1) yields the Camassa-Holm equation in nonlocal form

which was introduced by Camassa and Holm as a nonlinear model for water wave motion in shallow channels and we recommend the reader to refer to [8]for details.Many researchers have investigated the Cauchy problem for the Camassa-Holm equation,for instance,see [4,8,11–12,26].

A Gibbs measure is an efficient tool to obtain a global solution of some dispersive equations by the continuation of local solution.It compensates the absence of conservation law in the Sobolev space with lower regularity to some extent.One can refer to [6–7,9,19,28]and the references therein,in which the Gibbs measure and its applications were demonstrated.The invariant measure in this paper is obtained by the nonlinear flow generated by the concerned modified Camassa-Holm equation.Actually,the construction of invariant measures under nonlinear flows for many partial differential equations with random initial data was introduced by Lebowitz-Rose-Speer [17],and since then the Gibbs measure has attracted much research attention due to its special feature and applications.There are rich existing results on this topic,for examples,by using some new spaces and a suitable randomization,Burq and Tzvetkov[6–7]constructed the local and global strong solutions for a large set of initial data to the supercritical wave equation; Colliander and Oh [10]studied almost surely well-posedness for the nonlinear cubic Schrdinger equation with the randomization on the initial data.Besides,we name but a few more references like [2–3,5,13–14,20–24]for interested readers.

For the Cauchy problem of (1.1) we are concerned with,many studies have carried out on the real line and in the periodic settings(see e.g.[1,15–16,18,25,27]).Different from existing results,we consider the Cauchy problem for the periodic modified Camassa-Holm equation with randomization on initial data in this paper.For this,inspired by [22–23],by using Prokhorov compactness theorem and Skorokhod convergence theorem,we construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity,and then establish the existence of global solution.

Next we clarify some notations used throughout this paper.Set We denote the spectral projector by ΠN.

Let (Ω,F,p) be a probability space and (gn(ω))n∈Zbe a sequence of independent complex normalized Gaussians,to be specific,gn∈NC(0,1) with a form below:

where (hn(ω))n∈Z,(ln(ω))n∈Zare independent standard real Gaussians with the distribution NR(0,1).

Also we use

to denote the complex vector space and

to denote the projection operator onEN.Settingcn=an+ibn,we consider the probability measure dθNon R2(2N+1)defined as

where

ThenθNcan be regarded as the distribution of theENvalued random variable

We denote byL2(Ω;Hσ(S1))the Banach space ofHσ(S1)-valued functions on Ω.Obviously,(ϕN) is a Cauchy sequence inL2(Ω;Hσ(S1)) which leads to the limit

In particular,ϕ(·,ω) ∈Hσ0(S1) almost surely and the mapω→ϕ(·,ω) is measurable from(Ω,A) to (Hσ0(S1),B),where B denotes the Borel sigma algebra ofHσ0(S1).Then,ϕ(·,ω)defines a measureθon (Hσ0(S1),B) and we define the measureθ=p ◦ϕ−1onLetχR:R →[0,1]be a continuous function with a compact support such thatχR(x) = 1 for |x|≤R.DenoteuN=ΠNu.Then we define the densityGN(u) as

and the measureµNonHσ(S1) as

where

To end this section,we show two main results in this paper.

Theorem 1.1(Constructions of Gibbs Measure) The sequenceGN(u) converges in measure asN→∞with respect to the measureθ.We denote byG(u) the limit of (1.3) asN→∞and define dµ(u)=G(u)dθ(u).Then,for everyp∈[1,∞),G(u)∈Lp(dθ(u)) and the sequenceGNconverges toGinLp(dθ(u)) asN→∞.

Theorem 1.2(Global Solution) There exists a set Σ of fullµmeasure such that for everyu0∈Σ,(1.1) has a solution

The rest of the paper is organized as follows.In the next section,we give two important conservation laws and several elementary inequalities.In Section 3,the hypercontractivity properties of the Ornstein-Uhlenbeck semi-group are recalled and a Gibbs measure with support inHσ(S1) for anyσ

2 Conservation Laws and Some Inequalities

To begin with,we give two important conservation laws and some useful estimates in the following lemmas in this section.

Lemma 2.1Letube a 2πperiodic function which is the solution of (1.1).Then,F(u) =F(u0),where

ProofFirst note that

where

which implies

Inserting (1.1) into I2and integrating by parts,we conclude that

Combining (2.2)–(2.3) into (2.1),we have

This completes the proof of Lemma 2.1.

Lemma 2.2Letube a 2πperiodic function which is the solution of (1.1).Then,G(u) =G(u0),where

ProofMultiplying by 2uon both sides of (2.1) and integrating by parts with respect to[0,2π) yield

which completes the proof of Lemma 2.2.

Lemma 2.3Letube a 2πperiodic function which is the solution of (1.1).Then

Consequently,(1.1) possesses the Hamiltonian structure due to the fact thatis a Hamiltonian operator.

ProofBy an immediate computation,we have

Combining (2.5) with (1.1),we have (2.4) which leads to the desired result.

Lemma 2.4LetN∈Z andα>.Then there existsCβ>0 such that

for allβ <2α-1 when<α≤1 andβ=αwhenα>1.

For the proof of Lemma 2.4,we recommend the reader to refer to [5,Lemma 5.1].

Lemma 2.5Letσ <.Then there existsC >0 such that

for allp≥2.

For the proof of Lemma 2.5,we recommend the reader to refer to [5,Lemma 4.4].

3 Proof of Theorem 1.1

First we have a statement as below and one can refer to [22,Proposition 2.4]for the details of the proof.

Lemma 3.1(Wiener Chaos) Letd≥1,c(n1,···,nk) ∈C and (gn)1≤n≤d∈NC(0,1) be complexL2-normalized independent Gaussians.Fork≥1,denote byA(k,d)={(n1,···,nk)∈{1,···,d}k,n1≤···≤nk} and

Then for alld≥1 andp≥2,

Define the functionfN:Hσ(S1)→R by

Then we prove some lemmas in preparation to prove Theorem 1.1.

Lemma 3.2For the sequence(fN)N≥1defined above,it is a Cauchy sequence inL2(Hσ(S1),dθ).In particular,there existsC >0 such that for everyM >N≥1,

Moreover,for everyM >N≥1 andp≥2,

ProofNote that

which implies

From (3.3),we have

where

We now split (3.4) into three parts as below

where

Thus,we have

where

First,we show that there existsC >0 such that for allM >N≥1,

ForM >N≥1,set

Then we have

where (n1,n2,n3),(m1,m2,m3)∈Λ(N,M).Note that if (n1,n2,n3) and (m1,m2,m3) are two triples from Λ(N,M) such that {n1,n2,n3}{m1,m2,m3},then

We consider the following three cases:

(i) (n1,n2,n3)=(m1,m2,m3) or (n1,n2,n3)=(m1,m3,m2),

(ii) (n1,n2,n3)=(m2,m1,m3) or (n1,n2,n3)=(m2,m3,m1),

(iii) (n1,n2,n3)=(m3,m1,m2) or (n1,n2,n3)=(m3,m2,m1).

In fact,only the first case needs to be considered,and the other two cases can be proved in a similar way.For (n1,n2,n3)=(m1,m2,m3),it follows from (3.6) that

where

By the symmetry and Lemma 2.4,we further have

and

Combining (3.7) with (3.8),we have (3.5).

Next we prove that there existsC >0 such that for allM >N≥1,

For this,we write

ForM >N≥1,it turns out that

and

Combining (3.10) and (3.11),we deduce

Finally we prove that there existsC >0 such that for everyM >N≥1,

ForM >N≥1,it yields that

which implies (3.12).

Thus,combining(3.5),(3.9)with(3.12),we have(3.1).As for(3.2),it is a direct consequence of (3.1) and Lemma 3.1.

With Lemma 3.2,we immediately have the following corollary.

Corollary 3.1Denote byf(u)∈L2(Hσ(S1),B,dθ)the limit of(fN)N≥1.Then the sequence(fN)N≥1converges in measure tof.In particular,for every∊>0,

ProofThe lemma immediately follows from an application of the Chebyshev inequality,so we leave out the proof here to save space.

Then the following statement can be obtained.

Lemma 3.3For the sequence (rN)N≥1defined in (3.13),it is a Cauchy sequence inL2(Hσ0(S1),B,dθ).In particular,there existsC >0 such that for everyM >N≥1,

Moreover,denote byr(u) the limit ofrN(u) inL2(Hσ0(S1),B,dθ),the sequence (rN)N≥1converges tor(u) in measure,i.e.,

ProofWe write

Combining (3.15)with (3.16),we have

SetRn(ω) = |gn(ω)|2- 1.Fornm,sincegnandgmare mutually independent andE[|gn(ω)|2]=1,it follows that

By an immediate computation,we have

Obviously,

From (3.17)–(3.20),we conclude

This yields (3.14).Finally,the convergence of (rN)N≥1in measure follows from the Chebyshev inequality,which puts an end of the proof of Lemma 3.3.

Lemma 3.4There existsC >0 such that for everyM >N≥1,p>2,

ProofNotice

By Lemma 3.1,we have

which implies

This puts an end of the proof for Lemma 3.4.

We are now able to define the function

by

Then it is no risk to say thatG(u) is the limit in measure ofN→∞.Actually,based on [23,Corollary 4.4],Corollary 3.1 and Lemma 3.3,we have the convergence of ‖uN‖L3(S1),fN(u) andχR(rN(u)) to ‖u‖L3(S1),f(u) andχR(r(u)) in theθmeasure,respectively.Then by composition and multiplication of continuous functions,the conclusion thatGN(u) converges toG(u) in measure follows immediately.

Lemma 3.5For 1 ≤p<∞,there existsC >0 such that for everyN≥1,

ProofFirst recall that

Thus it follows that

which ends the proof of Lemma 3.5.

Now we are well prepared to prove Theorem 1.1.

Consider the sequence(GN(u))N≥1.Noticing the convergence ofGN(u)toG(u)in measure,we know that there exists a subsequenceNksuch thatGNk(u)→G(u),θa.s.Note that Lemma 3.5 implies that there exists a constantC >0 such that‖GNK(u)‖Lp(dθ(u))≤C,∀k∈N.Thus,by Fatou lemma,we have

4 Proof of Theorem 1.2

The proof of Theorem 1.2 is divided into Lemmas 4.1–4.3 in this section.

We first state some useful notation.Setting(∂xuN)2),we consider the approximation of (1.1):

Set Π0=1-Π0,the orthogonal projection on 0-mean functions.We then prove the following lemma inspired by [23,Lemma 5.1].Lemma 4.1Forσ≤-2∊,hN(uN,∂xuN)is a Cauchy sequence inMore precisely,for allp≥2,there existη >0 andC >0 such that for all 1 ≤M

If we denote its limit by Π0h(u,∂xu),we further have

ProofAccording to the definition of the measureθ,we know

Also,by definition ofϕN,we have

where

Since (gn)n∈Z∗is a sequence of mutually independent complex normalised Gaussian,we arrive at

Consequently,due toσ≤-2∊,we obtain

This completes the proof of Lemma 4.1.

Next we consider the probability measureµNdefined by (1.4).Define the measureνNonC([-T,T];X1

2) as the image measure ofµNby the map

where ΦNis the flow generated by(4.1).Moreover,for any measurableF:C([-T,T];)→R,

Then we are able to prove the following estimates.

Lemma 4.2Forσ≤-∊,there existsC >0 such that

and

for allN≥1 andp≥2.

ProofThe bound (4.3) is obtained thanks to Lemmas 2.5 and 3.5.We now turn to (4.4).From (4.1),it follows that

Thus

Obviously,by using Lemma 4.1 and (4.2),we have

which ends the proof of Lemma 4.2.

By Lemma 4.2,similar to [5,Proposition 4.11],we can prove the following lemma.

Lemma 4.3ForT >0 andσ≤-∊,the family of measures

is tight inC([-T,T];Hσ(S1)).

Finally,we are well prepared to prove Theorem 1.2.Based on Lemma 4.3,Prokhorov theorem can be used,i.e.,for eachT >0,there exists a sub-sequenceνNkand a limit measureνon the spacesuch thatνNk→νweakly onC([-T,T];Hσ(S1)) for allσ≤-∊.Thanks to the Skorokhod theorem,there exists a probability spacesequence of random variablesand a random variableuwith values inso that

We can see that all terms of (4.5) converges ask→∞,among which the convergence of the nonlinear terms is derived from [5,Lemma 5.6].Therefore,it follows that,forσ≤-∊,there existsC >0 such that

inL2([-T,T];Hσ(S1)) ask→∞.This finishes the proof of Theorem 1.2.

AcknowledgementPart of this work was completed while Lin Lin and Wei Yan were visiting Center for Mathematical Sciences (mathcenter.hust.edu.cn),Huazhong University of Science and Technology,Wuhan,China,as the members of a Research Team on stochastic partial differential equations,whose support is greatly appreciated.