Global solutions to random 3D vorticity equations for small initial data

TL;DR

This paper establishes the existence, uniqueness, and continuous dependence of global solutions to stochastic 3D vorticity equations with small initial data, extending deterministic results to a stochastic setting with Gaussian noise.

Contribution

It introduces a novel approach to solving stochastic 3D vorticity equations by reducing them to random nonlinear parabolic equations, providing new insights into their regularity and dependence on initial data.

Findings

Proves global existence and uniqueness of solutions for small initial vorticity.

Demonstrates maximal regularity and weak continuity of solutions.

Shows pathwise continuous dependence on initial data.

Abstract

One proves the existence and uniqueness in $(L^p(\mathbb{R}^3))^3$, $\frac{3}{2}<p<2$, of a global mild solution to random vorticity equations associated to stochastic $3D$ Navier-Stokes equations with linear multiplicative Gaussian noise of convolution type, for sufficiently small initial vorticity. This resembles some earlier deterministic results of T. Kato [15] and are obtained by treating the equation in vorticity form and reducing the latter to a random nonlinear parabolic equation. The solution has maximal regularity in the spatial variables and is weakly continuous in $(L^3\cap L^{\frac{3p}{4p-6}})^3$ with respect to the time variable. Furthermore, we obtain the pathwise continuous dependence of solutions with respect to the initial data.