Exponential attractors for random dynamical systems and applications

TL;DR

This paper develops a method to construct exponential attractors for stochastic PDEs, demonstrating their stability and convergence to deterministic attractors as randomness diminishes.

Contribution

It introduces a framework for exponential attractors in random dynamical systems and applies it to stochastic reaction-diffusion equations, showing convergence properties.

Findings

Existence of exponential attractors for abstract random dynamical systems

Dependence of attractors on parameters and perturbation amplitude

Convergence of stochastic attractors to deterministic ones as noise decreases

Abstract

The paper is devoted to constructing a random exponential attractor for some classes of stochastic PDE's. We first prove the existence of an exponential attractor for abstract random dynamical systems and study its dependence on a parameter and then apply these results to a nonlinear reaction-diffusion system with a random perturbation. We show, in particular, that the attractors can be constructed in such a way that the symmetric distance between the attractors for stochastic and deterministic problems goes to zero with the amplitude of the random perturbation.