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arXiv · 1711.05072

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

Jinlong Wei·Jinqiao Duan·Hongjun Gao·Guangying Lv
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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.

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Jinlong Wei, Jinqiao Duan, Hongjun Gao, Guangying Lv. 2017-11-14. Strong solution for stochastic transport equations with irregular drift: existence and non-existence. https://arxiv.org/abs/1711.05072

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