Stochastic regularization for transport equations

We investigate a stochastic transport equation driven by a multiplicative noise. For drift coefficients in Lq(0,T;Cbα(Rd))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^q(0,T;{\mathcal {C}}^\alpha _b({\mathbb {R}}^d))$$\end{document} (α>2/q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >2/q$$\end{document}) and initial data in W1,r(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,r}({\mathbb {R}}^d)$$\end{document}, we show the existence and uniqueness of stochastic strong solutions. Opposite to the deterministic case where the same assumptions on drift coefficients and initial data induce nonexistence of strong solutions, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. However, for α+1<2/q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha +1<2/q$$\end{document} with spatial dimension higher than one, we can choose suitable initial data and drift coefficients so that the stochastic strong solutions do not exist.