Existence and uniqueness of $W_{loc}^{1,r}$-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 $W_{loc}^{1,r}({\mathbb{R}}^d))$.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 $W_{loc}^{1,p}({\mathbb{R}}^d)$), 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.

References in corpus (2)

• Noise Prevents Singularities in Linear Transport Equations
• An Ill Posed Cauchy Problem for a Hyperbolic System in Two Space Dimensions

Cited by in corpus (1)

• Strong solution for stochastic transport equations with irregular drift: existence and non-existence