Stability estimates for advection-diffusion equations with rough vector fields

We develop stability estimates for advection-diffusion equations in the DiPerna-Lions setting, i.e. with a velocity field that is Sobolev regular in the spatial variable. The estimates are formulated in terms of the distances from the optimal transport theory with logarithmic cost functions, and provide thus a bound for the rate of weak convergence. First, we obtain an estimate for distributional solutions given by different data. Second, we study the implicit finite volume discretization and we prove an error estimate for the rate of convergence of approximate solutions towards the unique exact solution. Both estimates are optimal invarious respects.