Global existence of the stochastic Navier-Stokes equations in $L^3$ with small data

TL;DR

This paper proves the global existence and uniqueness of solutions for the stochastic 3D Navier-Stokes equations with small initial data in $L^3$, under certain small noise conditions, extending understanding of stochastic fluid dynamics.

Contribution

It establishes the almost global existence of strong solutions to stochastic Navier-Stokes equations with small $L^3$ data under natural noise smallness assumptions.

Findings

Global-in-time strong solutions exist for small $L^3$ data.

Solutions are unique and pathwise in the stochastic setting.

Existence holds under natural smallness conditions on the noise.

Abstract

We address the global-in-time existence and pathwise uniqueness of solutions for the stochastic incompressible Navier-Stokes equations with a multiplicative noise on the three-dimensional torus. Under natural smallness conditions on the noise, we prove the almost global existence result for small $L^{3}$ data. Namely, we show that for data sufficiently small, there exists a global-in-time strong $L^{3}$ solution in a space of probability arbitrarily close to~$1$.