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

TL;DR

This paper proves local and global existence and uniqueness of strong solutions to the stochastic incompressible Navier-Stokes equations in critical Besov spaces, using contraction mapping and stochastic estimates.

Contribution

It establishes the first local well-posedness results in critical Besov spaces for stochastic Navier-Stokes equations and extends to global solutions for small initial data.

Findings

Local existence and uniqueness of strong solutions in critical Besov spaces.

Global existence of solutions for small initial data with high oscillations.

Solution framework based on contraction mapping and stochastic estimates.

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.