Wei Wang

Wei Wang
Nanjing University | NJU · Department of Mathematics

About

39
Publications
2,868
Reads
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666
Citations
Citations since 2016
1 Research Item
409 Citations
2016201720182019202020212022020406080
2016201720182019202020212022020406080
2016201720182019202020212022020406080
2016201720182019202020212022020406080
Additional affiliations
October 2008 - January 2012
University of Adelaide
Position
  • PostDoc Position
June 2005 - June 2007
Chinese Academy of Sciences
Position
  • PostDoc Position

Publications

Publications (39)
Article
Full-text available
A dynamical approximation of a stochastic wave equation with large interaction is derived. A random invariant manifold is discussed. By a key linear transformation , the random invariant manifold is shown to be close to the random invariant manifold of a second-order stochastic ordinary dierential equation.
Article
The three-dimensional (3D) viscous primitive equations describing the large-scale oceanic motions under fast oscillating random perturbation are studied. Under some assumptions on the random force, the solution to the initial boundary value problem (IBVP) of the 3D random primitive equations converges in distribution to that of IBVP of the limiting...
Chapter
Randomness may enter a heterogeneous system at the physical boundary of small scale obstacles as well as the interior of the physical medium. Such a system may be modeled by stochastic partial differential equations defined on a perforated domain. Homogenization for this stochastic system is derived and the approximation errors are estimated.
Chapter
Motivation for extracting effective dynamics of stochastic partial differential equations is discussed, and some examples of stochastic partial differential equations and outlines of this book are presented.
Chapter
A few examples of deterministic partial differential equations (PDEs) together with their solutions by Fourier series or Fourier transforms are briefly presented. Then some equalities and inequalities useful for estimating solutions of both deterministic and stochastic, partial differential equations are recalled.
Chapter
Averaging principles for a class of slow-fast stochastic partial differential equations are presented. The errors of the averaging approximation are quantified via normal and large deviations.
Chapter
Basic probability concepts and Brownian motion in Euclidean space and in Hilbert space are recalled. Then Fréchet derivatives and Ĝateaux derivatives as needed for Itô’s formula are reviewed. Finally stochastic calculus in Hilbert space, including a version of Itô’s formula useful for analyzing stochastic partial differential equations are discusse...
Chapter
Some basic facts about stochastic partial differential equations, including various solution concepts such as weak, strong, mild and martingale solutions are reviewed. Furthermore, sufficient conditions under which these solutions exist are discussed. Moreover, infinite dimensional stochastic dynamical systems are also discussed through a few examp...
Chapter
A random center manifold reduction method for class of stochastic evolutionary equations is given. Then, a random slow manifold reduction method for slow-fast stochastic partial differential equations is presented.
Article
This paper derives a Schrödinger approximation for weakly dissipative stochastic Klein-Gordon-Schrödinger equations with a singular perturbation and scaled small noises on a bounded domain. Detail uniform estimates are given to pass the limit as perturbation and noise disappear. Approximations in two different spaces are considered. Furthermore, a...
Book
Full-text available
Effective Dynamics of Stochastic Partial Differential Equations focuses on stochastic partial differential equations with slow and fast time scales, or large and small spatial scales. The authors have developed basic techniques, such as averaging, slow manifolds, and homogenization, to extract effective dynamics from these stochastic partial differ...
Article
A large deviation principle is derived for a class of stochastic reaction–diffusion partial differential equations with slow–fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This resul...
Article
Full-text available
By applying Rohlin's result on the classification of homomorphisms of Lebesgue space, the random inertial manifold of a stochastic damped nonlinear wave equations with singular perturbation is proved to be approximated almost surely by that of a stochastic nonlinear heat equation which is driven by a new Wiener process depending on the singular per...
Article
Full-text available
This paper is devoted to provide a theoretical underpinning for ensemble forecasting with rapid fluctuations in body forcing and in boundary conditions. Ensemble averaging principles are proved under suitable `mixing' conditions on random boundary conditions and on random body forcing. The ensemble averaged model is a nonlinear stochastic partial d...
Article
Full-text available
Self-similarity of Burgers' equation with some stochastic advection is studied. In self-similar variables a stationary solution is constructed which establishes the existence of a stochastically self-similar solution for the stochastic Burgers' equation. The analysis assumes that the stochastic coefficient of advection is transformed to a white noi...
Article
Full-text available
Similarity solutions play an important role in many fields of science: we consider here similarity in stochastic dynamics. Important issues are not only the existence of stochastic similarity, but also whether a similarity solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of s...
Article
Macroscopic reduction methods, such as averaging, homogenization and slow manifold approximation, were proposed for stochastic partial differential equations (SPDEs) with separated time and/or spatial scales in recent years. Here, we survey some very recent results of applying these methods to derive effective reduced models for stochastic partial...
Article
Full-text available
We explore the relation between fast waves, damping and imposed noise for different scalings by considering the singularly perturbed stochastic nonlinear wave equations \nu u_{tt}+u_t=\D u+f(u)+\nu^\alpha\dot{W} on a bounded spatial domain. An asymptotic approximation to the stochastic wave equation is constructed by a special transformation and sp...
Article
The asymptotic behavior of the stochastic FitzHugh–Nagumo system with small excitability is concerned. It is proved that solutions of the stochastic FitzHugh–Nagumo system converge in probability to the unique solution of the limit system as the excitability tends to zero. In our approach the proof of tightness of the distributions of solutions in...
Article
Full-text available
The qualitative properties of local random invariant manifolds for stochastic partial differential equations with quadratic nonlinearities and multiplicative noise is studied by a cut off technique. By a detail estimates on the Perron fixed point equation describing the local random invariant manifold, the structure near a bifurcation is given. Ke...
Article
Full-text available
Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class of stochastic reaction-diffusion partial differential equations with cubic nonlinearity. Dividing space into o...
Article
Full-text available
A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This also confirms the effectiveness of...
Article
Dynamical behavior of the following nonlinear stochastic damped wave equations νu tt +u t =Δu+f(u)+εW ˙(1) on an open bounded domain D⊂ℝ n , 1≤n≤3, is studied in the sense of distribution for small ν,ε. Here ν is the parameter that describes the singular perturbation. First, by a decomposition of Markov semigroup defined by (1), a stationary soluti...
Article
Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any Lipschitz assumption on the slow modes. The rate of convergence in probability is obtained as a byproduct. Impo...
Article
We consider multiscale stochastic dynamical systems. In this article an \emph{intermediate} reduced model is obtained for a slow-fast system with fast mode driven by white noise. First, the reduced stochastic system on exponentially attracting slow manifold reduced system is derived to errors of $\mathcal{O}(\epsilon)$. Second, averaging derives an...
Article
Full-text available
The macroscopic behaviour of dissipative stochastic partial differential equations usually can be described by a finite-dimensional system. This article proves that a macroscopic reduced model may be constructed for stochastic reaction–diffusion equations by artificially separating the system into two distinct slow and fast time parts. An averaging...
Article
An averaged system for the slow-fast stochastic FitzHugh-Nagumo system is derived in this paper. The rate of convergence in probability is obtained as a byproduct. Moreover the deviation between the original system and the averaged system is studied. A martingale approach proves that the deviation is described by a Gaussian process. The deviation g...
Article
In this paper, relations between the asymptotic behavior for a stochastic wave equation and a heat equation are considered. By introducing almost surely D–α-contracting property for random dynamical systems, we obtain a global random attractor of the stochastic wave equation νuttν+utν−Δuν+f(uν)=νW˙ endowed with Dirichlet boundary condition for any...
Article
The approximation in probability for a singular perturbed nonlinear stochastic heat equation is studied. First the approximation result in the sense of probability is obtained for solutions defined on any finite time interval. Furthermore it is proved that the long time behavior of the stochastic system is described by a global random attractor whi...
Article
Full-text available
An effective macroscopic model for a stochastic microscopic system is derived. The original microscopic system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes or heterogeneities. The homogenized effective model is still a stochastic partial differential equation but defined on a unified domai...
Article
Full-text available
In order to understand the impact of random influences at physical boundary on the evolution of multiscale systems, a stochastic partial differential equation model under a fast random dynamical boundary condition is investigated. The noises in the model and in the boundary condition are both additive. An effective equation is derived and justified...
Article
Full-text available
A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities)...
Article
Full-text available
Random invariant manifolds often provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete. The asymptotic dynamical behavior is thus...
Article
The ocean thermohaline circulation under uncertainty is investigated by a random dynamical systems approach. It is shown that the asymptotic dynamics of the thermohaline circulation is described by a random attractor and by a system with finite degrees of freedom.
Article
Stochastic partial differential equations arise as mathematical models of complex multiscale systems under random influences. Invariant manifolds often provide geometric structure for understanding stochastic dynamics. In this paper, a random invariant manifold reduction principle is proved for a class of stochastic \emph{partial} differential equa...
Article
The Swift–Hohenberg fluid convection system with both local and nonlocal nonlinearities under the influence of white noise is studied. The objective is to understand the difference in the dynamical behavior in both local and nonlocal cases. It is proved that when sufficiently many of its Fourier modes are forced, the system has a unique invariant m...
Article
Feng [Phys. Lett. A 293 (2002) 50] obtained a kind of explicit exact solutions to the Liénard equation, and applied these results to find some explicit exact solitary wave solutions to the nonlinear Schrödinger equation and the Pochhammer–Chree equation. In this Letter, more explicit exact solitary wave solutions for the generalized Pochhammer–Chre...

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