Backward Uniqueness and the existence of the spectral limit for some parabolic SPDEs

TL;DR

This paper investigates the long-term behavior of solutions to certain parabolic stochastic partial differential equations, establishing backward uniqueness and spectral limits, with applications to linear and nonlinear SPDEs including stochastic Navier-Stokes equations.

Contribution

It extends deterministic PDE results to stochastic cases by proving backward uniqueness and spectral limits for a broad class of SPDEs.

Findings

Proved backward uniqueness for abstract SPDEs.

Established existence of spectral limits for solutions.

Applied results to specific linear and nonlinear SPDEs, including stochastic Navier-Stokes equations.

Abstract

The aim of this article is to study the asymptotic behaviour for large times of solutions to a certain class of stochastic partial differential equations of parabolic type. In particular, we will prove the backward uniqueness result and the existence of the spectral limit for abstract SPDEs and then show how these results can be applied to some concrete linear and nonlinear SPDEs. For example, we will consider linear parabolic SPDEs with gradient noise and stochastic NSEs with multiplicative noise. Our results generalize the results proved in \cite{[Ghidaglia-1986]} for deterministic PDEs.