Local and global pathwise solutions for a stochastically perturbed nonlinear dispersive PDE
Introduction
The integrable Novikov equation (NE) or in the equivalent form was 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 with . 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 under certain assumptions. Meanwhile, Ni and Zhou [44] proved that the NE is local well-posed in the critical Besov spaces . 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 with . 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) which involves the Novikov equation (1.1) with . It is shown that the mNE is locally well-posed in both the Besov spaces and Sobolev spaces 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 , for any and .
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 with . 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 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: where , and the one-dimensional torus denotes the spatial domain, namely, and periodic boundary conditions on the interval are assumed. The parameter , is a family of independent one-dimensional standard Brownian motions on some complete probability space , and is a sequence of locally Lipschitz and bounded functions. Recalling that the Bessel potential is a pseudo-differential operator which can be formulated by where the associated kernel function , , and stands for the biggest integer less than . Notice that , the direct calculation shows that the Cauchy problem (1.4) can be reformulated in the following nonlocal form: where Compared with the localized random perturbation considered in [47], the stochastic force in the paper is only localized in space but not in time, i.e., . Moreover, since , the perturbation in the equivalent form (1.5) can be regarded as a nonlocal stochastic force, which has not been investigated in [47].
Let be a complete filtration probability space, and be mutually independent real-valued standard Wiener processes relative to . Let be a complete orthonormal system in (i.e., the separable Hilbert space consists of all sequences of square summable real numbers), we define the cylindrical Wiener process with values in by We define the auxiliary space which is endowed with the norm , for any . Note that the embedding of is Hilbert–Schmidt (cf. [46]). Moreover, by using standard martingale arguments with the fact that each is almost surely continuous, we have that almost surely. In general, for any satisfying and a Banach space , we denote by the set of continuous functions with the norm .
Let be the square-integrable space on with the inner product and norm denoted by and respectively. For any , we define the Sobolev space where , . For each , is a Hilbert space with the inner product , where .
For and , the space consists of all measurable functions with Moreover, for , we define the fractional Sobolev space which is endowed with the norm
For , we consider the following assumptions:
(H1) For all , there exists a nondecreasing function , which is locally bounded and is independent of , such that
(H2) For all , , there exists a locally bounded and nondecreasing function such that
By virtue of the assumptions (H1)–(H2), for any with , the operator defined by belongs to (i.e., the collection of Hilbert–Schmidt operators from to ). Therefore, for any predictable process , we can define the following -valued Itô’s stochastic integral In what follows, denote generic constants only depending on , 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 , and be a -valued -measurable random variable such that .
(i) A local pathwise solution of SmNE is a pair , where a strictly positive stopping time (i.e., ) relative to , and is a -predictable process satisfying Moreover, for any and ,
(ii) The local pathwise solutions are said to be unique (or indistinguishable), if given any two pair of local pathwise solutions and , then where .
(iii) A maximal pathwise solution is a triple , if each pair is a local pathwise solution, increases with such that A maximal pathwise solution is said to be global if , -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 , is a -measurable -valued random element such that , and the conditions (H1)–(H2) hold. Then the Cauchy problem (1.5) has a unique local maximal pathwise solution 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 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 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 , 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 ), 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 . 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 , Notice that the corresponding results with nonlinear multiplicative is still an open problem.
Theorem 1.3 Global Pathwise Solution Let , . Assume that is a -measurable initial data, and for some
and Suppose that is the corresponding maximal pathwise solution to (1.5) with linear multiplicative noise (1.7). Then we have .
The above result covers the corresponding global existence results for the deterministic mNE (cf. [34], [42], [54]), which can be obtained by taking , ; or , . 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 , and be -measurable initial data. Assume that is the corresponding pathwise solution to (1.5) with linear multiplicative noise (1.7). (1) If , the solution exists globally, that is, . (2) If , the solution blows up in finite time if and only if , for a.e. . If , the corresponding solution blows up in finite time if and only if , for a.e. .
The peakon solutions to SmNE with linear multiplicative noise have also been studied.
Theorem 1.5 Decaying of Peakon Solutions (1) For any and , the mNE perturbed by linear multiplicative noise (1.7) has stochastic peakon solutions (in the sense of distribution): where (2) The above defined peakon solutions have the following decaying property: and the rate of propagating in -direction at time is given by , which tends to as almost surely.
Remark 1.6 Fig. 1 displays the dynamical evolution of peakon solutions (1.8) with at time . In Fig. 2, we display the time evolution of peakon solutions to SmNE with parameters .
Remark 1.7 Recalling that the function is called a weakly dissipative in the study of weakly dissipative NE (cf. [36], [53]): By a simple calculation, one can verify that is a solution to (1.3) if and only if , is a solution to the weakly dissipative mNE (1.9). If is a global solution of (1.3), then we observe that the global solution to (1.9) decaying to zero as , which is an obvious consequence of the exponential prefactor . So by Theorem 1.5, the linear multiplicative noise in (1.4) can be regarded as a stochastic weakly dissipative, but the parameter 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 , and then we remove the truncation 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 with . 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 , we choose a nonincreasing truncation function satisfying Let be a Schwartz function such that its Fourier transformation satisfies on and on . For any , we define the Friedrichs mollifier by with . Consider the mollified SmNE with truncation :
Local well-posedness in with
This section is devoted to the proof of Theorem 1.2. To this end, we consider a sequence of equations where is the Friedrichs mollifier defined in Section 2.1, and the nonlocal terms , are provided in (1.8). It follows from Proposition 2.14 that the functions in (3.1) are well-defined. By using an abstract Cauchy lemma, we shall prove that converges to an element up to a
Blow-up and global existence for linear multiplicative noise
The SmNE with linear multiplicative noise (1.7) can be written by where is a standard one-dimensional Brownian motion, and are the nonlocal terms defined as before. Now we transform (4.1) into a random partial differential equation by introducing a new process Applying the Itô’s product rule to we obtain
Peakon solutions
In this section, we study the dissipative effect of linear multiplicative noise on peakon solutions to deterministic mNE. For a.e. , we apply the Helmholtz operator to (4.2) and obtain with the initial data , .
Proof of Theorem 1.5 Let , where is an open set of . Assume that exists and is continuous except a single point and , where denotes the set of all functions 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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