Global Existence and Stability of 3D Stochastic NSEs in Bounded and Unbounded Domains Driven by a Special Multiplicative Wiener Process

Abstract

The aim of this work is to extend the results from a recent paper by Hong, Li and Liu, from bounded domains to both bounded and unbounded domains. We show the global existence and uniqueness of 3D stochastic Navier-Stokes equations with nonlinear multiplicative noise for every initial data from the Sobolev space $H^1$. We do not use any tightness argument. Instead, we firstly show the existence of a local maximal solution and then we use Lyapunov function to prove the solution is global. Our approach is motivated by a recent paper of the first named author with Ferrario, Maurelli and Zanella about a similar result for stochastic nonlinear Schr\"odinger Equations. The main idea is that once the local existence of strong solutions is established, a very strong noise pushing toward the origin, will make blow-up impossible.