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

TL;DR

This paper studies the differentiability of solutions to a drift-diffusion equation with nonlocal Lévy-type diffusion, showing under certain conditions that solutions become differentiable with Hölder continuous derivatives over time.

Contribution

It establishes differentiability results for solutions of nonlocal drift-diffusion equations in supercritical and critical cases, under specific regularity conditions.

Findings

Solutions are differentiable with Hölder continuous derivatives for positive times.

Differentiability holds under conditions on drift velocity and forcing term.

Results apply to supercritical and critical Lévy-type diffusion cases.

Abstract

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.