Rough linear transport equation with an irregular drift

We study the linear transport equation $\frac{\partial}{\partial t} u ( t,x ) +b ( t,x ) \cdot \nabla u ( t,x ) + \nabla u ( t,x ) \cdot \frac{\partial}{\partial t} X ( t ) =0, \hspace{2em} u ( 0,x ) =u_{0} ( x )$ where $b$ is a vectorfield of limited regularity and $X$ a vector-valued Hölder continuous driving term. Using the theory of controlled rough paths we give a meaning to the weak formulation of the PDE and solve that equation for smooth vectorfields $b$. In the case of the fractional Brownian motion a phenomenon of regularization by noise is displayed.

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