Numerical investigation of the Free Boundary regularity for a degenerate advection-diffusion problem

TL;DR

This paper numerically investigates the regularity of free boundaries in a degenerate advection-diffusion problem, demonstrating Lipschitz regularity and potential corners, supported by simulations and analytical insights.

Contribution

It introduces a finite difference scheme for approximating solutions and provides numerical evidence for free boundary regularity and corner formation in a porous medium type problem.

Findings

Lipschitz regularity of free boundaries is supported by numerical evidence.

Free boundaries may develop Lipschitz corners for certain diffusion exponents.

Numerical and analytical methods suggest corners in viscosity solutions of related Hamilton-Jacobi equations.

Abstract

We study the free boundary regularity of the traveling wave solutions to a degenerate advection-diffusion problem of Porous Medium type, whose existence was proved in \cite{MonsaingonNovikovRoquejoffre}. We set up a finite difference scheme allowing to compute approximate solutions and capture the free boundaries, and we carry out a numerical investigation of their regularity. Based on some nondegeneracy assumptions supported by solid numerical evidence, we prove the Lipschitz regularity of the free boundaries. Our simulations indicate that this regularity is optimal, and the free boundaries seem to develop Lipschitz corners at least for some values of the nonlinear diffusion exponent. We discuss analytically the existence of corners in the framework of viscosity solutions to certain periodic Hamilton-Jacobi equations, whose validity is again supported by numerical evidence.