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

TL;DR

This paper investigates the existence and non-existence of strong solutions for stochastic transport equations with irregular drift, establishing conditions based on regularity and integrability that determine when solutions exist or not.

Contribution

It provides new criteria for existence and non-existence of stochastic strong solutions under irregular drift conditions, extending understanding in stochastic PDE theory.

Findings

Existence of unique solutions when lpha > 2/q for Sobolev initial data.

Non-existence of solutions when lpha + 1 < 2/q in higher dimensions with specific initial data.

Conditions linking drift regularity and solution existence in stochastic transport equations.

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 $\alpha$-H\"{o}lder 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.