Well-posedness of the vector advection equations by stochastic perturbation

TL;DR

This paper demonstrates that adding stochastic noise to a linear vector advection equation ensures its well-posedness, even when deterministic versions may be ill-posed, by analyzing a model of passive magnetic fields in random fluids.

Contribution

It establishes the well-posedness of stochastic vector advection equations with infinite-dimensional noise, highlighting the regularizing effect of stochastic perturbations.

Findings

Noise induces well-posedness in the model

The equation remains well-posed under certain integrability conditions

Stochastic perturbation counteracts potential ill-posedness

Abstract

A linear stochastic vector advection equation is considered. The equation may model a passive magnetic field in a random fluid. The driving velocity field is a integrable to a certain power and the noise is infinite dimensional. We prove that, thanks to the noise, the equation is well posed in a suitable sense, opposite to what may happen without noise.