Entropy Solution Theory for Fractional Degenerate Convection-Diffusion Equations

TL;DR

This paper develops a theory for entropy solutions of fractional degenerate convection-diffusion equations, establishing well-posedness, convergence of numerical schemes, and exploring connections to non-local operators and nonlinear HJB equations.

Contribution

It introduces a new framework for entropy solutions of fractional degenerate equations, including existence, uniqueness, and numerical approximation methods.

Findings

Proved well-posedness of entropy solutions under weak regularity.

Designed a convergent monotone numerical scheme.

Extended results to non-local Lévy operators and nonlinear HJB equations.

Abstract

We study a class of degenerate convection diffusion equations with a fractional nonlinear diffusion term. These equations are natural generalizations of anomalous diffusion equations, fractional conservations laws, local convection diffusion equations, and some fractional Porous medium equations. In this paper we define weak entropy solutions for this class of equations and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable functions. Then we introduce a monotone conservative numerical scheme and prove convergence toward an Entropy solution in the class of bounded integrable functions of bounded variation. We then extend the well-posedness results to non-local terms based on general L\'evy type operators, and establish some connections to fully non-linear HJB equations. Finally, we present some numerical experiments to give the reader an idea about the qualitative behavior of solutions of these equations.