This paper is devoted to investigating mean dynamics and stability analysis for stochastic 3D Lagrangian-averaged Navier–Stokes (LANS) equations driven by infinite delay on unbounded domains. We first prove the existence of a unique solution to stochastic 3D LANS equations with infinite delay when the non-delayed external force is locally integrable, the delay term is globally Lipschitz continuous and the nonlinear diffusion term is locally Lipschitz continuous. This enables us to define a mean random dynamical system. Besides, we find that such a dynamical system possesses a unique weak pullback mean random attractor, which is a minimal, weakly compact and weakly pullback attracting set. Furthermore, we prove the existence and uniqueness of stationary solutions (equilibrium solutions) to the corresponding deterministic equation via the classical Galerkin method, the Lax–Milgram and the Brouwer fixed theorems. The stability properties of stationary solutions are also considered. By a direct approach, we first show the local stability of stationary solutions when the delay term has a general form and then apply the abstract results to two kinds of infinite delays. Second, we establish the exponential stability of stationary solutions in the case of unbounded distributed delay. Third, we investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing appropriate Lyapunov functionals. Eventually, we discuss the polynomial asymptotic stability in the particular case of proportional delay.

本文致力于研究无界域上由无限延迟驱动的随机 3D 拉格朗日平均纳维-斯托克斯 (LANS) 方程的平均动力学和稳定性分析。我们首先证明当非延迟外力局部可积、延迟项全局 Lipschitz 连续且非线性扩散项局部 Lipschitz 连续时，具有无限延迟的随机 3D LANS 方程存在唯一解的存在。这使我们能够定义平均随机动力系统。此外，我们发现这样的动力系统具有独特的弱回调均值随机吸引子，它是一个最小的、弱紧致的、弱回调的吸引集。此外，我们通过经典的伽辽金方法、Lax-Milgram和Brouwer固定定理证明了相应确定性方程的平稳解（平衡解）的存在性和唯一性。还考虑了平稳解的稳定性。通过直接方法，我们首先展示当延迟项具有一般形式时平稳解的局部稳定性，然后将抽象结果应用于两种无限延迟。其次，我们建立了无界分布延迟情况下平稳解的指数稳定性。第三，我们通过构造适当的李雅普诺夫泛函来研究无界变量时滞情况下平稳解的渐近稳定性。最后，我们讨论比例延迟特定情况下的多项式渐近稳定性。