  We investigate a stochastic transport equation driven by a multiplicative
noise. For $L^q(0,T;W^{1,p}({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient
and $W^{1,r}({\mathbb R}^d)$ initial data, we obtain the existence and
uniqueness of stochastic strong solutions (in $W^{1,r}_{loc}({\mathbb
R}^d))$.In particular, when $r=\infty$, we establish a Lipschitz estimate for
solutions and this question is opened by Fedrizzi and Flandoli in case of
$L^q(0,T;L^p({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient. Moreover,
opposite to the deterministic case where $L^q(0,T;W^{1,p}({\mathbb
R}^d;{\mathbb R}^d))$ drift coefficient and $W^{1,p}({\mathbb R}^d)$ initial
data may induce non-existence for strong solutions (in $W^{1,p}_{loc}({\mathbb
R}^d)$), we prove that a multiplicative stochastic perturbation of Brownian
type is enough to render the equation well-posed. It is an interesting example
of a deterministic PDE that becomes well-posed under the influence of a
multiplicative Brownian type noise. We extend the existing results
\cite{FF2,FGP1} partially.
