Entropy Solution Theory for Fractional Degenerate Convection-Diffusion 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 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.

References in corpus (5)

• Non-uniqueness of weak solutions for the fractal Burgers equation
• The discontinuous Galerkin method for fractal conservation laws
• The discontinuous Galerkin method for fractional degenerate convection-diffusion equations
• Stability of Entropy Solutions for Levy Mixed Hyperbolic-Parabolic Equations
• Barenblatt profiles for a nonlocal porous media equation

Cited by in corpus (10)

• Continuous dependence estimates for nonlinear fractional convection-diffusion equations
• Optimal continuous dependence estimates for fractional degenerate parabolic equations
• Asymptotic behaviour of solutions to fractional diffusion-convection equations
• Some Free Boundary Problems involving Nonlocal Diffusion and Aggregation
• A framework for non-local, non-linear initial value problems
• Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection
• A general fractional porous medium equation
• Uniform tail estimates and $L^p({\mathbb{R}}^N)$-convergence for finite-difference approximations of nonlinear diffusion equations
• Vanishing viscosity limit of a conservation law regularised by a Riesz-Feller operator
• Nonlocal degenerate parabolic hyperbolic equations on bounded domains