Existence and uniqueness of $W^{1,r}_{loc}$-solutions for stochastic transport equations (1711.05067v1)
Abstract: We investigate a stochastic transport equation driven by a multiplicative noise. For $Lq(0,T;W{1,p}({\mathbb R}d;{\mathbb R}d))$ 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 $Lq(0,T;Lp({\mathbb R}d;{\mathbb R}d))$ drift coefficient. Moreover, opposite to the deterministic case where $Lq(0,T;W{1,p}({\mathbb R}d;{\mathbb R}d))$ 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.