On the Differentiability issue of the drift-diffusion equation with nonlocal L\'evy-type diffusion

We investigate the differentiability issue of the drift-diffusion equation with nonlocal L'evy-type diffusion at either supercritical or critical type cases. Under the suitable conditions on the drift velocity and the forcing term in terms of the spatial H"older regularity, we prove that the vanishing viscosity solution is differentiable with some H"older continuous derivatives for any positive time.