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

2017/11/14 by Wei, Jinlong, Duan, Jinqiao, Gao, Hongjun +1
#60H15 (35A01 35L02) #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.05072

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 α-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 α>2/q, while for α+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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