Uniform measure attractors of the distribution-dependent 2D stochastic Navier-Stokes equations driven by nonlinear noise
Jiangwei Zhang, Juntao Wu
TL;DR
The paper develops a rigorous framework for uniform measure attractors of distribution-dependent 2D stochastic Navier–Stokes equations with time-dependent almost periodic forcing and nonlinear noise. It establishes well-posedness, derives robust long-time uniform estimates, and proves the existence and uniqueness of a uniform measure attractor, even in the absence of the Feller property, by leveraging a skew-product measure framework and compactness arguments. The attractor has a kernel-based structure that links the global dynamics to the family of hull-induced forcing terms, providing a comprehensive description of the system's asymptotic measure behavior. This work extends uniform attractor theory to mean-field, nonautonomous SPDEs, with potential implications for mean-field fluid models and related stochastic systems.
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 $\{P^{(g,h)}{(τ,t)}\}_{τ\leq t}$.