Stability analysis of stochastic 3D Lagrangian-averaged Navier-Stokes equations with infinite delay

Abstract

The asymptotic behaviour of stochastic three-dimensional Lagrangian-averaged Navier-Stokes equations with infinite delay and nonlinear hereditary noise is analysed. First, using Galerkin’s approximations and the monotonicity method, we prove the existence and uniqueness of solutions when the non-delayed external force is locally integrable and the delay terms are globally Lipschitz continuous with an additional assumption. Next, we show the existence and uniqueness of stationary solutions to the corresponding deterministic equation via the Lax-Milgram and the Schauder theorems. Later, we focus on the stability properties of stationary solutions. To begin with, we discuss the local stability of stationary solutions for general delay terms by using a direct method and then apply the abstract results to two kinds of infinite delays. Besides, the exponential stability of stationary solutions is also established in the case of unbounded distributed delay. Moreover, we investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing appropriate Lyapunov functionals. Eventually, we establish criteria on the polynomial asymptotic stability of stationary solutions for the special case of proportional delay.