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

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.