On the differentiability of the solution to an equation with drift and fractional diffusion

TL;DR

This paper proves that solutions to certain equations with drift and fractional diffusion become immediately differentiable with Holder continuous derivatives under mild regularity assumptions, making them classical solutions.

Contribution

It establishes the immediate differentiability and classical nature of solutions for equations with critical or supercritical fractional diffusion under near minimal regularity conditions.

Findings

Solutions become immediately differentiable with Holder continuous derivatives.

Solutions are classical under mild regularity assumptions.

Applicable to equations with critical or supercritical fractional diffusion.

Abstract

We consider an equation with drift and either critical or supercritical fractional diffusion. Under a regularity assumption for the vector field that is marginally stronger than what is required for Holder continuity of the solutions, we prove that the solution becomes immediately differentiable with Holder continuous derivatives. Therefore, the solutions to the equation are classical.