Global pathwise solutions of an abstract stochastic equation

TL;DR

This paper proves the existence and uniqueness of global pathwise solutions for a class of nonlinear stochastic evolution equations, including the 2D and 3D stochastic Navier-Stokes equations, under small initial data.

Contribution

It establishes the maximal pathwise solution framework for abstract stochastic equations and demonstrates global existence for stochastic Navier-Stokes equations on a torus with small initial data.

Findings

Existence and uniqueness of maximal pathwise solutions.

Global solutions for stochastic Navier-Stokes on torus with small data.

Solutions exist globally with positive probability for small initial data.

Abstract

We establish the existence and uniqueness of the maximal pathwise solution for an abstract nonlinear stochastic evolutional equation, which takes the two and three dimensional stochastic Navier-Stokes equations as a typical model, forced by a multiplicative white noise, and show that the pathwise solution exists globally in time in a positive probability when the initial data is sufficiently small. Moreover, a global pathwise solution is obtained for the stochastic Navier-Stokes equations defined on torus when the data is properly regular and small.