Local and global pathwise solutions for a stochastically perturbed nonlinear dispersive PDE

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Abstract

In this paper, we consider the periodic Cauchy problem for a stochastically perturbed nonlinear dispersive partial differential equation with cubic nonlinearity, which involves the integrable Novikov equation arising from the shallow water wave theory as a special case. We first establish the existence and uniqueness of local pathwise solutions in Sobolev spaces Hs(T)(s>32) with nonlinear multiplicative noise, where the key ingredients are the stochastic compactness method, the Skorokhod representation theorem and the Gyöngy–Krylov characterization of convergence in probability. In the case of linear multiplicative noise, we investigate the conditions which lead to the blow-up phenomena and global existence of pathwise solution. Finally, we show that the linear multiplicative noise has a dissipative effect on the periodic peakon solutions to the associated deterministic Novikov equation.

Introduction

The integrable Novikov equation (NE) ututxx+4u2ux=3uuxuxx+u2uxxxor in the equivalent form mt+u2mx+3uuxm=0,m=uuxxwas discovered by Vladimir Novikov [45] in a symmetry classification of nonlocal partial differential equations with quadratic or cubic nonlinearity. After its derivation, the NE has been intensively investigated from the point of view of the Cauchy problem. In [28], Hone et al. obtained the non-smooth explicit soliton solutions with multiple peaks for NE by applying the scattering theory, which are common with the Camassa–Holm (CH) and Degasperis–Procesi (DP) equations (cf. [5], [11], [12], [13]). The quest for peaked traveling waves is motivated by the desire to find waves replicating a feature that is specific to the water waves of great height. There are the traveling waves of largest amplitude that are exact solutions of the governing equations for water waves, whether periodic or solitary (cf. [8], [9], [10], [50]). Similar to the CH and DP equations (cf. [14], [35], [38]), the peakons of NE are shown to be orbitally stable [39], which means that their shape is stable under small perturbations. Using the Galerkin-type approximation method, Himonas and Holliman [24] established the well-posedness of NE in Hs(R) with s>32. Sufficient conditions which lead to the formation of singularities in finite time for NE are provided by Jiang and Ni [30]. Applying the Littlewood–Paley theory, Yan et al. [52] established the local well-posedness in the Besov spaces Bp,rs(R) under certain assumptions. Meanwhile, Ni and Zhou [44] proved that the NE is local well-posed in the critical Besov spaces B2,132(R). Under some sign conditions on the initial data, it is shown by Lai et al. [34] that NE has a unique global weak solution in Sobolev space Hs(R) with 1s32. By vanishing viscosity method, Lai [33] established the global weak solutions for NE without any sign condition on the initial data.

Recently, Mi and Mu [42] investigated the following modified Novikov equation (mNE) ututxx+(b+1)u2ux=buuxuxx+u2uxxx,bR,which involves the Novikov equation (1.1) with b=3. It is shown that the mNE is locally well-posed in both the Besov spaces Bp,rs(R) and Sobolev spaces Hs(R) under certain assumptions. Given analytic initial data, Mi and Mu proved that the solutions are analytic in both variables, globally in space and locally in time. Moreover, in the non-periodic case, the mNE has peakon solutions in the form of u(t,x)=±ce|xctx0|, for any c>0 and x0=constant.

Stochastic partial differential equations (SPDEs) can be used to describe systems that are too complex to be described deterministically (cf. [3], [4], [21], [40], [41]), e.g., a flow of a chemical substance in a river subjected to wind and rain, an airflow around an airplane wing perturbed by the random state of the atmosphere and weather, etc. It is worth pointing out that the presence of stochastic perturbation in a system can lead to new phenomena. For example, in contrast to the deterministic case, it is shown in [3] that the two dimensional Navier–Stokes system driven by degenerate noise has a unique invariant measure and hence exhibits ergodic behavior in the sense that the time average of a solution is equal to the average over all possible initial data.

In real world, the surface of fluid is non-constant pressure, or the bottom of fluid is not flat. In view of the wide usage of stochastic in fluid dynamics, there is an essential need to improve the mathematical foundations of the SPDEs of fluid flow, and in particular to study the modified Novikov equation (1.1) perturbed by some stochastic forcing. In fact, several recent works can be found concerning the stochastically perturbed integrable systems. For instance, Crisan and Holm [15] studied the wave breaking phenomena for the stochastic Camassa–Holm equation (SCHE) with an additional stochastic perturbation in the momentum transport velocity. For the numerical studies of the interactions of stochastic peakon solutions, we refer to the works by Holm and Tyranowski [26], [27]. In [7], by techniques developed in [2], [32], Chen et al. established the local well-posedness for SCHE with additive noise in the Sobolev spaces Hs(R) with s>32. The existence and uniqueness of local pathwise solution of SCHE is investigated by Tang [48]. Moreover, the blow-up criteria and global existence results for the SCHE are also investigated. By the weak convergence approach, Chen and Gao [6] studied the large deviation principle for the stochastic transport equations driven by Lévy noise. There are also related works related to SPDEs driven by the noise of the type u(t)dWt and blow-up of SPDEs, see for example [37], [51].

In order to examine how the stochastic noises affect the dynamic behavior of solutions to the mNE, we consider the following periodic Cauchy problem for the stochastically perturbed modified Novikov equation (SmNE, for short) with multiplicative noises: dm(t)+(u2mx+buuxm)dt=k1hk(t,m(t))dβtk,t>0,xT,m(0,x)=m0(x),xT,where m=(1x2)u, and the one-dimensional torus TR2πZ denotes the spatial domain, namely, xR and periodic boundary conditions on the interval [0,2π] are assumed. The parameter bR, {βk,k=1,2,} is a family of independent one-dimensional standard Brownian motions on some complete probability space (Ω,F,P), and {hk(,),k=1,2,} is a sequence of locally Lipschitz and bounded functions. Recalling that the Bessel potential (1x2)1 is a pseudo-differential operator which can be formulated by (1x2)1φ(x)=TG(xy)φ(y)dy,φL2(T),where the associated kernel function G(x)=cosh(x2π[x2π]π)2sinh(π), cosh(x)=ex+ex2, sinh(x)=exex2 and [x] stands for the biggest integer less than x. Notice that u=(1x2)1m, the direct calculation shows that the Cauchy problem (1.4) can be reformulated in the following nonlocal form: du(t)+(u2ux+L1(u)+L2(u))dt=k1(1x2)1hk(t,m(t))dβtk,t>0,xT,u(0,x)=u0(x),xT,where L1(u)=x(1x2)1(6b2uux2+b3u3),L2(u)=(1x2)1(b22ux3).Compared with the localized random perturbation η(t,x) considered in [47], the stochastic force in the paper is only localized in space but not in time, i.e., xT[0,2π]. Moreover, since (1x2)1hk(t,m(t))=TG(xy)hk(t,m(t,y))dy, the perturbation in the equivalent form (1.5) can be regarded as a nonlocal stochastic force, which has not been investigated in [47].

Let S=(Ω,F,(Ft)t0,P) be a complete filtration probability space, and (βk)k1 be mutually independent real-valued standard Wiener processes relative to (Ft)t0. Let (ek)k1 be a complete orthonormal system in l2 (i.e., the separable Hilbert space consists of all sequences of square summable real numbers), we define the cylindrical Wiener process W with values in l2 by W(t,,ω)=k=1ek()βk(t,ω).We define the auxiliary space U{u=k1akek;k1ak2k2<}l2,which is endowed with the norm uU2=k1ak2k2, for any u=k1akekU. Note that the embedding of l2U is Hilbert–Schmidt (cf. [46]). Moreover, by using standard martingale arguments with the fact that each βk is almost surely continuous, we have that WC([0,T];U) almost surely. In general, for any a,bR satisfying a<b and a Banach space X, we denote by C([a,b];X) the set of continuous functions f:[a,b]X with the norm fC([a,b];X)=maxt[a,b]f(t)X.

Let L2(T) be the square-integrable space on T with the inner product and norm denoted by (,)2 and L2 respectively. For any sR, we define the Sobolev space Hs(T)={fL2(T);fHs2=kZ(1+k2)s|f̂(k)|2<},where f̂(k)=Teikxf(x)dx, kZ. For each sR, Hs(T) is a Hilbert space with the inner product (f,g)Hs=(Λsf,Λsg)2, where Λs=(1x2)s2.

For 1<q< and sR, the space Lq([0,T];Hs(T)) consists of all measurable functions f:[0,T]Hs(T) with fLq([0,T];Hs)=(0Tf(t)Hsqdt)1q. Moreover, for ϑ(0,1), we define the fractional Sobolev space Wϑ,q([0,T];Hs(T))={fLq([0,T];Hs(T));[f]ϑ,q,Tq=0T0Tf(t)f(t̄)Hsq|tt̄|1+ϑqdtdt̄<}, which is endowed with the norm fWϑ,q([0,T];Hs)q=fLq([0,T];Hs)q+[f]ϑ,q,Tq.

For s>12, we consider the following assumptions:

(H1) For all m=Λ2uHs(T), there exists a nondecreasing function ȷ:R+[1,), which is locally bounded and is independent of t, such that k1hk(t,m)Hs2ȷ(uW1,)(1+mHs2),t0.

(H2) For all m=Λ2u, n=Λ2vHs(T), there exists a locally bounded and nondecreasing function ϱ:R+R+ such that k1hk(t,m)hk(t,n)Hs2ϱ(uW1,+vW1,)mnHs2,t0.

By virtue of the assumptions (H1)–(H2), for any mHs(T) with s>12, the operator h(t,m)=(hk(t,m))k1:l2Hs(T) defined by h(t,m)a=k1hk(t,m)ak,a=(ak)kNl2belongs to 2(l2;Hs(T)) (i.e., the collection of Hilbert–Schmidt operators from l2 to Hs(T)). Therefore, for any predictable process mL2(Ω;Lloc2([0,);Hs(T))), we can define the following Hs(T)-valued Itô’s stochastic integral 0th(r,m(r))dWr=k10thk(r,m(r))dβrk,for anyt0.In what follows, Ca,b, denote generic constants only depending on a,b,, which may change from line to line.

With the mathematical preliminaries in hand, we make precise the notions of local, maximal and global pathwise solutions for SmNE.

Definition 1.1

Let s>32, and u0 be a Hs-valued F0-measurable random variable such that Eu0Hs2<.

(i) A local pathwise solution of SmNE is a pair (u,τ), where τ a strictly positive stopping time (i.e., P{τ>0}=1) relative to Ft, and u:Ω×[0,τ]Hs(T) is a Ft-predictable process satisfying u(τ)L2(Ω,C([0,),Hs(T))).Moreover, for any t0 and ϕL2(T), (u(tτ),ϕ)2(u0,ϕ)2+0tτ(u2ux+L1(u)+L2(u),ϕ)2dr=k10tτ((1x2)1hk(r,m),ϕ)2dβrk,P-a.s.

(ii) The local pathwise solutions are said to be unique (or indistinguishable), if given any two pair of local pathwise solutions (u1,τ1) and (u2,τ2), then P{1Ω0(u1(t,x)=u2(t,x)),(t,x)[0,τ1τ2]×T}=1,where Ω0P{u1(0)=u2(0)}Ω.

(iii) A maximal pathwise solution is a triple (u,τ,{τn}n1), if each pair (u,τn) is a local pathwise solution, τn increases with limnτn=τ such that limnsupt[0,τn]u(t)W1,=on the set{τ<},P-a.s.A maximal pathwise solution (u,τ,{τn}n1) is said to be global if τ=, P-a.s.

Our first result concerns the local well-posedness for the SmNE with nonlinear multiplicative noise can now be enunciated by the following theorem.

Theorem 1.2 Local Pathwise Solution

Assume that s>32, u0 is a F0-measurable Hs(T)-valued random element such that Eu0Hs2<, and the conditions (H1)–(H2) hold. Then the Cauchy problem (1.5) has a unique local maximal pathwise solution (u,τ,{τn}n1) in the sense of Definition 1.1.

Comparing with the deterministic mNE [42], the main difficulty in establishing the local well-posedness arises from the essential challenge of proving strong convergence for the random approximations, which lies in the fact that in the stochastic case we do not expect the solutions to be differentiable in time, and hence the usual compactness methods in [42], [52], [54] are invalid here. Motivated by the techniques in [21], [46], [48], we shall first consider the SmNE with a truncation θR in front of both the nonlinear drift and diffusion terms, which allows us to obtain uniform a priori estimates for approximations. However, due to the lack of cancellation property (cf. [21]; see also [17]) as that in stochastic Euler equation (SEE), one does not a priori know the existence of global approximations. Being inspired by [46], we shall further regard the truncated SmNE as Lipschitz stochastic differential equations (SDEs) in Hs(T) by mollifying the nonlinear terms. Then the classical theory of SDEs (cf. [20], [46]) ensures that the approximation equations admit global solutions, which is exactly the start point for us to construct the local pathwise solutions. The classical Yamada–Watanabe theorem tells us that, for finite dimensional systems at least, pathwise solutions exist whenever martingale solutions may be found, and pathwise uniqueness holds. Recently, a different proof of such results was developed in [30] which leans on an elementary characterization of convergence in probability (cf. Lemma 2.10). Thanks to a uniqueness result and this Gyöngy–Krylov characterization of convergence in probability, one can pass from martingale to pathwise solutions. Notice that if we attempt to prove pathwise uniqueness by estimating the difference of approximations in Hs(T)(s>32), we encounter the knotty terms involving Bessel operators and an excessive number of derivatives (cf. Proposition 2.9). Similar to [28], we shall first restrict ourselves to sufficiently regular spaces in order to overcome this problem (in particular we take Hs(T)(s>5)), and then use a density-stability argument depending on an abstract Cauchy lemma (cf. Lemma 3.1) to obtain the existence of local solutions evolving in Hs(T)(s>32). Notice that the structure of the SmNE introduces highly nonlinear and nonlocal terms, which makes the proof of several required nonlinear estimates for drift terms and diffusion terms somewhat delicate.

The next two theorems study the issue of global existence and blow-up phenomena of pathwise solutions to SmNE. To be more precise, we shall discuss the Cauchy problem for SmNE with linear multiplicative noise: for any αR, hk(t,m)=αm,ifk=1,0,otherwise,Notice that the corresponding results with nonlinear multiplicative is still an open problem.

Theorem 1.3 Global Pathwise Solution

Let s3, b=3. Assume that u0L2(Ω;Hs(T)) is a F0-measurable initial data, and for some p1,p2[0,1] P{Λ2u0(ω,x)0,for allxT}=p1and P{Λ2u0(ω,x)0,for allxT}=p2.Suppose that (u,τ,{τn}) is the corresponding maximal pathwise solution to (1.5) with linear multiplicative noise (1.7). Then we have P{τ=}p1+p2.

The above result covers the corresponding global existence results for the deterministic mNE (cf. [34], [42], [54]), which can be obtained by taking α=0, (p1,p2)=(1,0); or α=0, (p1,p2)=(0,1). By virtue of the special structure of equations, we have also the following blow-up and global existence result for SmNE without any sign conditions on initial profiles.

Theorem 1.4 Blow-up and Global Existence

Let s3, and u0L2(Ω;Hs(T)) be F0-measurable initial data. Assume that (u,τ,{τn}) is the corresponding pathwise solution to (1.5) with linear multiplicative noise (1.7).

(1) If b=1, the solution u exists globally, that is, P{τ=}=1.

(2) If b>1, the solution u blows up in finite time if and only if lim inftτ(uux)(ω,t)=, for a.e. ωΩ. If b<1, the corresponding solution u blows up in finite time if and only if lim suptτ(uux)(ω,t)=, for a.e. ωΩ.

The peakon solutions to SmNE with linear multiplicative noise have also been studied.

Theorem 1.5 Decaying of Peakon Solutions

(1) For any αR and c>0, the mNE perturbed by linear multiplicative noise (1.7) has stochastic peakon solutions (in the sense of distribution): u(t,x)=±ccosh(π)eαW(t)α22tcosh(xf(t)2π[xf(t)2π]π),a.s.,where f(t)=c0teαW(r)α22rdr,t>0.

(2) The above defined peakon solutions u(t,x) have the following decaying property: P{lim suptu(t,x)=0,xT}=1,for anyc>0,and the rate of u(t,x) propagating in x-direction at time t is given by ceαW(t)α22t, which tends to 0 as t almost surely.

Remark 1.6

Fig. 1 displays the dynamical evolution of peakon solutions (1.8) with α=0 at time t=0. In Fig. 2, we display the time evolution of peakon solutions to SmNE with parameters α=0.05,0.10.

Remark 1.7

Recalling that the function λ(uuxx) is called a weakly dissipative in the study of weakly dissipative NE (cf. [36], [53]): ututxx+(b+1)u2ux=buuxuxx+u2uxxxλ(uuxx),λ>0.By a simple calculation, one can verify that u(t,x) is a solution to (1.3) if and only if v(t,x)=eλtu(1e2λt2λ,x), λ>0 is a solution to the weakly dissipative mNE (1.9). If u is a global solution of (1.3), then we observe that the global solution v to (1.9) decaying to zero as t, which is an obvious consequence of the exponential prefactor eλt. So by Theorem 1.5, the linear multiplicative noise (αdWdt)(uuxx) in (1.4) can be regarded as a stochastic weakly dissipative, but the parameter α is allowed to be negative.

In Section 2, we establish the local well-posedness for SmNE with smooth initial profiles. More precisely, in Section 2.1, we give the approximation scheme for SmNE. A priori estimates will be obtained in Section 2.2, and the tightness of measures for approximation solutions are also proved. In Section 2.3, applying the Gyöngy–Krylov lemma, we prove the existence of global pathwise solutions to Eq. (2.1). In Section 2.4, we first prove the existence of global pathwise solutions to SmNE with truncation θR, and then we remove the truncation θR to construct local pathwise solution for SmNE. In Section 3, by using the smooth solutions constructed in Section 2, we establish the local well-posedness for SmNE in Hs(T) with s>32. Section 4 is devoted to the global existence and blow-up phenomena for SmNE driven by linear multiplicative noises. In Section 5, we study the dissipative effect of the linear multiplicative noise on peakons.

Section snippets

Construction of approximate solutions

For any R>1, we choose a nonincreasing truncation function θR:[0,)[0,1] satisfying θR(x)=1,for|x|R,0,for|x|>2R.Let j(x) be a Schwartz function such that its Fourier transformation satisfies ĵ(ξ)=1 on [1,1] and 0ĵ(ξ)1 on R. For any ϵ(0,1], we define the Friedrichs mollifier Jϵ by Jϵu(x)jϵu(x)=Tjϵ(xy)u(y)dy,uHs(T)with jϵ(x)=1(ĵ(ϵξ)). Consider the mollified SmNE with truncation θR: du(t)+θR(uW1,)[Jϵ((Jϵu)2xJϵu)+L1(u)+L2(u)]dt=θR(uW1,)(1x2)1h(t,m)dW(t),t>0,xT,u(0,x)=u0(x

Local well-posedness in Hs(T) with s>32

This section is devoted to the proof of Theorem 1.2. To this end, we consider a sequence of equations duj(t)+((uj)2uxj+L1(uj)+L2(uj))dt=(1x2)1h(t,mj)dWt,t>0,xT,uj(0,x)=u0j(x)=(J1ju0)(x),xT,where J1j is the Friedrichs mollifier defined in Section 2.1, and the nonlocal terms L1(uj), L2(uj) are provided in (1.8). It follows from Proposition 2.14 that the functions uj(j1) in (3.1) are well-defined. By using an abstract Cauchy lemma, we shall prove that uj converges to an element u up to a

Blow-up and global existence for linear multiplicative noise

The SmNE with linear multiplicative noise (1.7) can be written by du(t)+(u2ux+L1(u)+L2(u))dt=αudWt,t>0,xT,u(0,x)=u0(x),xT,where W is a standard one-dimensional Brownian motion, L1(u) and L2(u) are the nonlocal terms defined as before. Now we transform (4.1) into a random partial differential equation by introducing a new process v(ω,t,x)=μ(ω,t)u(ω,t,x)=eα22tαW(t)u(ω,t,x),t0. Applying the Itô’s product rule to d(μu) we obtain d(μu)=μdu+udμ+dμdu=μ(u2ux+L1(u)+L2(u))dt+αμudWt+u(α2μdtαμdWt)α2

Peakon solutions

In this section, we study the dissipative effect of linear multiplicative noise on peakon solutions to deterministic mNE. For a.e. ωΩ, we apply the Helmholtz operator 1x2 to (4.2) and obtain t(1x2)v+13μ2x(1x2)(v3)+μ2x(b3v3+6b2vvx2)+b22μ2vx3=0,t>0,xT, with the initial data v(0,x)=u0(x), xT.

Proof of Theorem 1.5

Let fLloc1(O), where O is an open set of R. Assume that f exists and is continuous except a single point x0X and fLloc1(O), where Lloc1(O) denotes the set of all functions fL1(U) for

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The author was supported by the National Natural Science Foundation of China (Project # 11701198), and the Fundamental Research Funds for the Central Universities (FRFCU) (Project # 5003011025). The author would like to thank Xiuting Li for the meaningful discussion on MATLAB. The author also warmly thanks the anonymous referees for their valuable suggestions which improved the original version of present paper.

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