Stability of stochastic 2D Navier-Stokes equations with memory and Poisson jumps

Diem Dang Huan

doi:10.30538/oms2020.0131

Abstract

The objective of this paper is to study the stability of the weak solutions of stochastic 2D Navier-Stokes equations with memory and Poisson jumps. The asymptotic stability of the stochastic Navier-Stoke equation as a semilinear stochastic evolution equation in Hilbert spaces is obtained in both mean square and almost sure senses. Our results can extend and improve some existing ones.

Highlights

N this paper, we will investigate the stability of the weak solutions of stochastic 2D Navier-Stokes equations with memory and Poisson jumps of the form:
The theory of stochastic Navier-Stokes equations apparently has its roots in the 1959 edition of the Landau and Lifshitz [5] and the first work on the stochastic Navier-Stokes equations written from the mathematical point of view is the paper [6]
The current paper can be regarded as the extension of the work of Caraballo and Real [20] to the stochastic settings, simultaneously extend and improve the one of Taniguchi [35] (where the random external force field does not include delays and the external force field contains delay but the memory function ρ(t) is a differentiable function) as well as the asymptotic behavior results published in [7] and the papers announced by Chen [22], Wan and Zhou [23]

Summary

Introduction

The current paper can be regarded as the extension of the work of Caraballo and Real [20] to the stochastic settings, simultaneously extend and improve the one of Taniguchi [35] (where the random external force field does not include delays and the external force field contains delay but the memory function ρ(t) is a differentiable function) as well as the asymptotic behavior results published in [7] (where the external force fields does not contain delays and the stochastic Navier-Stokes equation without non-Gaussian Lévy noise perturbation) and the papers announced by Chen [22], Wan and Zhou [23] (where the random external force field does not contain discontinuous multiplicative noise).

Preliminaries

Main results