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

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.