Periodic Random Attractors for Stochastic Navier-Stokes Equations on Unbounded Domains

TL;DR

This paper investigates the long-term behavior of solutions to stochastic 2D Navier-Stokes equations on unbounded domains, establishing the existence of unique, possibly periodic, random attractors despite non-compact Sobolev embeddings.

Contribution

It introduces a continuous cocycle framework and proves the existence and uniqueness of tempered random attractors, including their periodicity under deterministic forcing.

Findings

Existence and uniqueness of tempered random attractors for stochastic Navier-Stokes equations.

Characterization of attractor structures via complete solutions.

Establishment of pullback asymptotic compactness despite non-compact Sobolev embeddings.

Abstract

This paper is concerned with the asymptotic behavior of solutions of the two-dimensional Navier-Stokes equations with both non-autonomous deterministic and stochastic terms defined on unbounded domains. We first introduce a continuous cocycle for the equations and then prove the existence and uniqueness of tempered random attractors. We also characterize the structures of the random attractors by complete solutions. When deterministic forcing terms are periodic, we show that the tempered random attractors are also periodic. Since the Sobolev embeddings on unbounded domains are not compact, we establish the pullback asymptotic compactness of solutions by Ball's idea of energy equations.