Entropy Solution Theory for Fractional Degenerate Convection-Diffusion Equations
Provides a theoretical and numerical framework for a broad class of equations, benefiting researchers in PDEs and numerical analysis, but the results are incremental extensions of existing entropy solution theory.
The paper defines weak entropy solutions for a class of fractional degenerate convection-diffusion equations, proves well-posedness under weak regularity assumptions, and introduces a monotone conservative numerical scheme with proven convergence. Numerical experiments illustrate solution behavior.
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 Lévy 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.