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

Abstract

We prove some existence, uniqueness and non-existence results of stochastic strong solutions for a class of stochastic transport equations with a $q$-integrable (in time), bounded and $\alpha$-Hölder continuous (in space) drift coefficient. More precisely, we show that for a Sobolev differentiable initial condition, there exists a unique stochastic strong solution when $\alpha >2/q$, while for $\alpha +1<2/q$ with spatial dimension higher than one, we can choose proper initial data and drift coefficients so that there is no stochastic strong solutions.

References in corpus (4)

• Well-posedness of the transport equation by stochastic perturbation
• Noise Prevents Singularities in Linear Transport Equations
• Pathwise uniqueness and continuous dependence for SDEs with nonregular drift
• Existence and uniqueness of $W_{loc}^{1,r}$-solutions for stochastic transport equations