The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain

Abstract

In this paper we consider the local existence and global existence with probability $1-\sigma $ $(0<\sigma <1)$ of pathwise solutions to the three-dimensional stochastic Navier-Stokes equation perturbed by a cylindrical Wiener processe $W(t)$ in an exteriour domain: \begin{equation*} dX(t)=[-AX(t)+B\left( X(t)\right) +f_{\ast }(t)]dt+\Phi (t)dW(t), \end{equation*} where $A=-P\Delta $ is the Stokes operator, and $f_{\ast }(t)$ and $\Phi (t)$ satisfy some conditions. We also consider the decay of pathwise solutions.