Random attractors for the stochastic Nernst-Planck-Navier-Stokes system with multiplicative white noise

TL;DR

This paper studies the long-term behavior of a 2D stochastic Nernst-Planck-Navier-Stokes system with multiplicative noise, proving the existence of a compact random attractor and its stability as noise diminishes.

Contribution

It transforms the stochastic system into a deterministic form, establishes global well-posedness, and proves the existence and upper semicontinuity of a random attractor.

Findings

Existence of a compact random attractor for the stochastic system

Global well-posedness of the transformed deterministic system

Upper semicontinuity of the attractor as noise intensity approaches zero

Abstract

In this paper, we consider the 2D periodic stochastic Nernst-Planck-Navier-Stokes equations with body forces perturbed by multiplicative white noise. We first transform the stochastic Nernst-Planck-Navier-Stokes system into the deterministic system and address the problem of global well-posedness of the solution. Then, we generate a corresponding random dynamical system and dedicate to proving the existence of a compact random attractor for such random dynamical system. Furthermore, upper semicontinuity of the random attractor is established when the noise intensity approaches to zero.