Uniform measure attractors of the distribution-dependent 2D stochastic Navier-Stokes equations driven by nonlinear noise

TL;DR

This paper establishes the existence and uniqueness of uniform measure attractors for a complex 2D stochastic Navier-Stokes system influenced by nonlinear noise and almost periodic external forces, overcoming significant analytical challenges.

Contribution

It introduces new conditions and estimates to prove attractor properties for distribution-dependent stochastic Navier-Stokes equations with almost periodic forcing.

Findings

Proved existence of uniform measure attractors.

Established uniqueness of these attractors.

Achieved joint continuity without Feller property reliance.

Abstract

In this paper, we investigate the uniform measure attractors of the distribution-dependent nonautonomous 2D stochastic Navier-Stokes equations driven by nonlinear noise and subject to almost periodic external forcing. Owing to the distribution-dependent structure and the almost periodicity of the external forcing, the resulting solution process becomes an inhomogeneous Markov process, presenting significant analytical challenges. To overcome these difficulties, we propose sufficient conditions on the time-dependent external forcing and distribution-dependent nonlinear terms, and develop novel analytical estimates. As a result, we establish the existence and uniqueness of uniform measure attractors for the system. Notably, the joint continuity of the family of processes is achieved without relying on the Feller property of the distribution law operators.