Local and global strong solutions to the stochastic incompressible Navier-Stokes equations in critical Besov space

Abstract

Considering the stochastic Navier-Stokes system in $\mathbb{R}^d$ forced by a multiplicative white noise, we establish the local existence and uniqueness of the strong solution when the initial data take values in the critical space $\dot{B}_{p,r}^{\frac{d}{p}-1}(\mathbb{R}^d)$. The proof is based on the contraction mapping principle, stopping time and stochastic estimates. Then we prove the global existence of strong solutions in probability if the initial data are sufficiently small, which contain a class of highly oscillating "large" data.