A remark on global solutions to random 3D vorticity equations for small initial data

TL;DR

This paper establishes the existence and uniqueness of global solutions to 3D stochastic vorticity equations with small initial data, extending classical results to a stochastic setting using rough path theory.

Contribution

It introduces a stochastic framework for 3D vorticity equations and proves global well-posedness for small initial data, utilizing controlled rough path integration.

Findings

Global solutions exist and are unique for small initial data

Solutions satisfy stochastic vorticity equations in the rough path sense

Extension of classical deterministic results to stochastic equations

Abstract

In this paper, we prove that the solution constructed in \cite{BR16} satisfies the stochastic vorticity equations with the stochastic integration being understood in the sense of the integration of controlled rough path introduced in \cite{G04}. As a result, we obtain the existence and uniqueness of the global solutions to the stochastic vorticity equations in 3D case for the small initial data independent of time, which can be viewed as a stochastic version of the Kato-Fujita result (see \cite{KF62}).