Representation of Manifolds for the Stochastic Swift-Hohenberg Equation with Multiplicative Noise

Abstract

Representation of approximation for manifolds of the stochastic Swift-Hohenberg equation with multiplicative noise has been investigated via non-Markovian reduced system. The approximate parameterizations of the small scales for the large scales are given in the process of seeking for stochastic parameterizing manifolds, which are obtained as pullback limits of some backward-forward systems depending on the time-history of the dynamics of the low modes in a mean square sense through the nonlinear terms. When the corresponding pullback limits of some backward-forward systems are efficiently determined, the corresponding non-Markovian reduced systems can be obtained for researching good modeling performances in practice.

1. Introduction

Recently, more and more authors have paid attention to considering the approximation problems of manifolds for the stochastic partial differential equations (SPDEs). For decades, various approximating methods have been given to solve these problems, such as amplitude equations approach [1–4] and the manifolds-based approaches [5–11].

In this paper, approximation of manifolds for the stochastic Swift-Hohenberg equation with multiplicative noise will be investigated in Stratonovich sense [12]. It is well known that there have been some authors to consider the approximation of manifolds in large probability sense [1, 2, 13] and they have obtained some results until now. In addition, approximation in parameterizing manifold for a stochastic Swift-Hohenberg equation with additive noise have been investigated by us in [14]. Furthermore, it is needed to consider the problems in Stratonovich sense. Until now, there have been few consideration from the point of view of approximation in parameterizing manifold under the pathwise sense for the stochastic Swift-Hohenberg equation with multiplicative noise. The ideas in [14] can be used to consider the approximation of manifold for some stochastic equations with multiplicative noise. Because the different difficulties come from different noise terms, there are some different methods and techniques in studying stochastic equations with multiplicative noise. Here, we investigate the corresponding problems for the stochastic Swift-Hohenberg equation with multiplicative noise with pathwise and obtain some new results for it. The results obtained in this paper are different from those in [14], although there are some similar sentences in some manuscripts. It is well known that various noises cause various stochastic processes for stochastic equations with different noises. The main differences from results in [14] are given by some formulas with various mathematics meanings, in which some different stochastic functions are used. Because the different difficulties mainly come from various noise terms, there are some new difficulties coming from the multiplicative noise solved in our manuscript. So, some different techniques are used in studying stochastic equations with multiplicative noise.

We will extend the strategy introduced in [5, 6] to the stochastic Swift-Hohenberg equation [12] with multiplicative noise and obtain the approximation of parameterizing manifolds and corresponding non-Markovian reduced system. The key idea is mainly based on the approximate parameterizations of the small scales for the large scales via the stochastic parameterizing manifolds. Random manifolds will improve the partial knowledge of the solutions of SPDEs in mean square error, when it is compared with its projection onto the resolved modes. Approximation of parameterizing manifolds can be obtained by representing the modes with high wave numbers as the pullback limit depend on the time-history of the modes with low wave numbers for the corresponding backward-forward systems. Some conditions with nonresonance conditions below are given and weaker than those in the classical stochastic invariant manifold theory (see [7, 15, 16] and references therein). On the base of these approximations of parameterizing manifolds, when the corresponding pullback limits of some backward-forward systems are efficiently determined, the corresponding non-Markovian stochastic reduced systems are given to reach good modeling performances in practice and take the form of stochastic differential equations with random coefficients, which convey memory effects via the history of the Wiener process and arise from the nonlinear interactions between the low modes embedded in the noise bath. These random coefficients show an exponential decay of correlations, whose rate depends explicitly on the gaps of the nonresonance conditions. In fact, it is possible to achieve very good parameterizing quality for the stochastic Swift-Hohenberg equation with multiplicative noise from our results. And the performances from the reduced system can be numerically assessed for a corresponding optimal or suboptimal control problem.

The paper is organized as follows. In Section 2, we give our functional framework, some definitions about parameterizing manifolds and some properties of some stochastic processes being used. We have devoted Section 3 to studying the representation of approximation of parameterizing manifolds as pullback limits of the corresponding backward-forward systems for a stochastic Swift-Hohenberg equation with multiplicative noise. In Section 4, on the basis of the approximation of parameterizing manifolds, the non-Markovian stochastic reduced systems involving random coefficients are obtained for the stochastic Swift-Hohenberg with multiplicative noise.

2. Preliminaries

The functional framework spaces are a pair of Hilbert spaces (, ) such that is compactly and densely embedded in . Let be a sectorial operator [16] such that is stable in the sense that its spectrum satisfies . And we consider interpolated spaces between and with along with the perturbations of the linear operator given by a one parameter family of bounded linear operators from to , depending continuously on . Define , which maps into .

A local stochastic Swift-Hohenberg equation with multiplicative noise in Stratonovich sense [1] is written as follows: with Dirichlet boundary conditions and initial condition , where is a parameterizing variable, is positive, and is some appropriate initial datum with and ; is a standard real valued one-dimensional Brownian motion [17] with paths in ; and being endowed with its corresponding Borel -algebra , its filtration , the Wiener measure .

Let , which is a continuous triple nonlinear mapping from into , where . Obviously, function is a mapping from into . Assume , where and is closed self-adjoint linear operator with dense domain in . The operator is self-adjoint with an orthonormal basis of eigenfunctions in with corresponding eigenvalues .

Then, problem (1) can be rewritten as with initial condition and Dirichlet boundary conditions . Now, we investigate the random dynamical systems of system (2) in the sense of parameterizing manifolds in [6, 17]. The stochastic parameterizing manifolds are mainly considered for local stochastic Swift-Hohenberg equation with multiplicative noise (2). Firstly, a stochastic parameterizing manifold is seen as the graph of a random function , which is a mapping from to and provides approximation parameterizations of the high part by using of the low part . The scalar Langevin equation, is given. A unique stationary solution of this equation is called the stationary Ornstein-Uhlenbeck (OU) process. By simply integrating on the both sides of (3), the identity holds, which is important for representation of approximation.

3. Representation of Manifolds with Multiplicative Noise

Making use of the method in [6], we investigate the local stochastic Swift-Hohenberg (equation (2)) with multiplicative noise in Stratonovich sense. One considers the following backward-forward system associated with SPDE (2). where and . From system (5), (6), (7), and (8), we know that the initial value of is represented in fiber and the initial value of is prescribed in fiber .

It is possible to obtain the solution of system (5) and (6) by using a backward-forward integration procedure due to the partial coupling between the equations constituting this system, where forces the evolution equation of but not reciprocally. In addition, since is emanated backward from in and forces the equation ruling the evolution of , thus depends naturally on . One emphasizes this dependence as in the whole paper.

The nonresonance conditions should be given in following theorem, under which the pullback limit of exists. Now, representation of an analytical description of such parameterizing manifolds will be provided. In particularly, it emphasizes the dependence on the part of the noise path of the manifolds.

Theorem 1. Consider the SPDE (2) in the functional setting of Section 2, with assumed to be a trilinear function. Let with .
Suppose also for all . Furthermore, assume that the following nonresonance conditions for all , , hold. Then, the pullback limit of the solution of (7) and (8) exists and is given by where is the solution of (5) and (6) Moreover, has the following analytic expression: where , and

Proof. Firstly, from (5), (6), (7), and (8), one introduces two processes and for and as follows: Here, via the above transformation processes, the backward-forward system (5), (6), (7), and (8) is transformed into the following system of random differential equations: Using the variation of constants method, we can formally obtain the solution of (15) and (16), which is followed by making use of an integration by parts performed to the resulting stochastic convolution terms Similarly, the solution of (17) and (18) can be also obtained at , which is formed where is taken as a form of (19). When , since condition (9), the limit of (20) exists, which is formed Secondly, one investigates the analysis presentation of this limit. Propose that where Here, and , and According to the same assumptions and the inverse transformation, (11) can be immediately obtained from (19), and the analytic expression of has the form of (12).

Furthermore, the approximation can be provided by the above theorem, which constitutes a parameterizing manifold function of SPDE (2). Moreover, the random coefficients have decaying property of correlations when it is checked by similar calculations performing for the proof of Lemma 5.1 in [6], which are solutions of auxiliary SDEs

Remark 2. Here, the random coefficients satisfied the stochastic equation (25) and are different from in [14], since the stochastic processes’ transformations are various for stochastic equations with multiplicative noises. So, in this paper and in [14] have different representations by formulas. In this paper, the transformations of stochastic processes are more difficult than those in [14]. So, the random coefficients are much more complex than those in [14]. Then, these differences hold in the whole paper.

4. PM-Based Non-Markovian Reduced System with Multiplicative Noise

In this section, the PM-based non-Markovian reduced system of problem (2) is investigated in two cases that are in two subspaces, or , respectively.

When one projects (2) into the subspace , it yields that where with being the canonical projector on subspace . By replacing with (12), the pullback limit , one yields the following reduced system which provides an approximation of the SPDE dynamics projected onto the low modes.

From (12), the random coefficients of contained in the expansion of exhibit the decaying property of correlations. Therefore, extrinsic memory effects in the Stratonovich sense are conveyed by the drift part of (27), making such reduced systems be non-Markovian (see [18, 19]).

The analytic form of from (12) can be used. The nonlinear interactions have the following form. When ,

When ,

Firstly, we investigate the system in case . Since approximation of parameterizing manifolds have been obtained in Section 3, then one yields that where and

In this case, the approximation is simple. However, it is not enough to present the performances of the corresponding dynamics. Furthermore, parameterizing manifolds in two-dimensional case for low mode are considered, which perform more dynamics than in the above case.

Secondly, when , then one can obtain the more complex results than in the case of .

Here, where , , and

However, it is complex to use directly the analytic formula of to obtain the vector as is various in in spite of any case in fact. So, we can use to take the place of on the fly along a trajectory of interest, where is given by integrating on both sides of the backward-forward system (5), (6), (7), and (8), when is chosen sufficiently large [6]. Then, it is natural to study the reduced system as follows: where is appropriately chosen according to the SPDE initial datum and is given from the following system:

Now, we give the corresponding non-Markovian systems from the above system. Investigating them in two cases and .

Firstly, when , one denotes , with . Then, the system can be written as in coordinate form with , where and , are given from the following system: with

Secondly, when , one denotes , with . Then, for , the corresponding system can be written as in coordinate form with , , where and , are given from following system with .

From the above equations, the representations of approximation for manifold and the corresponding reduced non-Markovian systems for stochastic Swift-Hohenberg equation with multiplicative noise are obtained. And the performances given by the above non-Markovian reduced system should have approximate dynamics on the modes in modeling of the pathwise SPDE (2). It is more important that one should give partial dynamics in approximation sense on the modes in modeling of the pathwise SPDEs in practice. The performances from the reduced system may be numerically assessed for a corresponding optimal or suboptimal control problems in the deduced processes. The numerical results will be further shown in the future. The processes deduced in this manuscript offered an idea in order to further investigate the approximation of stochastic manifold for some quantum stochastic equations with multiplicative noise.

Data Availability

Some data and ideas in our previous work in reference [14] were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Nos. 11861013 and 11771444), Guangxi Natural Science Foundation (No. 2017GXNSFAA198221), Promotion of the Basic Capacity of Middle and Young Teachers in Guangxi Universities (No. 2017KY0340), the Fundamental Research Funds for the Central Universities, and the China University of Geosciences (Wuhan) (No. 2018061).