Rough linear transport equation with an irregular drift

We study the linear transport equation partial derivative/partial derivative t u(t, x) + b(t, x) . del u(t, x)+ del u(t, x) . partial derivative/partial derivative t X(t) = 0, u(0, x) = u(0)(x) where b is a vector field of limited regularity and X a vector-valued Hlder continuous driving term. Using the theory of controlled rough paths we give a meaning to the weak formulation of the partial differential equation and solve this equation for smooth vector fields b. In the case of the fractional Brownian motion a phenomenon of regularization by noise is displayed.