Non-local diffusion equations with L\'evy-type operators and divergence free drift

TL;DR

This paper investigates properties of solutions to non-local diffusion equations with divergence-free drift, establishing existence, maximum principles, and regularity results using Hardy space techniques and Lévy-type operators.

Contribution

It introduces a novel approach combining Hardy-Hardy space duality and molecular characterisation to study regularity of solutions with Lévy-type diffusion.

Findings

Proved existence and maximum principles for solutions.

Established positivity principles for the equations.

Demonstrated Hölder regularity using Hardy space methods.

Abstract

We are interested in some properties related to the solutions of non-local diffusion equations with divergence free drift. Existence, maximum principle and a positivity principle are proved. In order to study Holder regularity, we apply a method that relies in the Holder-Hardy spaces duality and in the molecular characterisation of local Hardy spaces. In these equations, the diffusion is given by L\'evy-type operators with an associated L\'evy measure satisfying some upper and lower bounds.