Stochastic transport equations with unbounded divergence

Abstract

We study in this article the existence and uniqueness of solutions to a class of stochastic transport equations with irregular coefficients and unbounded divergence. In the first result we assume the drift is $L^2([0,T]\times {\mathbb{R}}^d)\cap L^{\infty }([0,T]\times {\mathbb{R}}^d)$ and the divergence is the locally integrable. In the second result we show that the smoothing acts as a selection criterion when the drift is in $L^2([0,T]\times {\mathbb{R}}^d)\cap L^{\infty }([0,T]\times {\mathbb{R}}^d)$ without any condition on the divergence.

References in corpus (3)

• Noise Prevents Singularities in Linear Transport Equations
• An Ill Posed Cauchy Problem for a Hyperbolic System in Two Space Dimensions
• On some recent results concerning non-uniqueness for the transport equation

Cited by in corpus (1)

• LDP and CLT for SPDEs with Transport Noise