Existence and uniqueness of $W^{1,r}_{loc}$-solutions for stochastic transport equations

TL;DR

This paper proves the existence and uniqueness of solutions to stochastic transport equations with certain regularity conditions on the drift and initial data, showing stochastic noise can ensure well-posedness where deterministic cases may fail.

Contribution

It establishes well-posedness of stochastic transport equations with $W^{1,r}_{loc}$ solutions, extending previous results and demonstrating noise-induced regularization.

Findings

Existence and uniqueness of solutions under specified conditions.

Lipschitz estimate for solutions when $r= abla ext{infty}$.

Stochastic noise ensures well-posedness where deterministic equations may not.

Abstract

We investigate a stochastic transport equation driven by a multiplicative noise. For $L^q(0,T;W^{1,p}({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient and $W^{1,r}({\mathbb R}^d)$ initial data, we obtain the existence and uniqueness of stochastic strong solutions (in $W^{1,r}_{loc}({\mathbb R}^d))$.In particular, when $r=\infty$, we establish a Lipschitz estimate for solutions and this question is opened by Fedrizzi and Flandoli in case of $L^q(0,T;L^p({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient. Moreover, opposite to the deterministic case where $L^q(0,T;W^{1,p}({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient and $W^{1,p}({\mathbb R}^d)$ initial data may induce non-existence for strong solutions (in $W^{1,p}_{loc}({\mathbb R}^d)$), we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. It is an interesting example of a deterministic PDE that becomes well-posed under the influence of a multiplicative Brownian type noise. We extend the existing results \cite{FF2,FGP1} partially.