02/12/2016, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Léonard Monsaingeon, Université de Lorraine, Nancy (France)
Numerical investigation of the free boundary regularity for a degenerate advection-diffusion problem
In this talk I will describe the free boundary regularity of the traveling wave solutions to a degenerate advection-diffusion problem of Porous Medium type, whose existence was proved in a previous work with A. Novikov (PennState Univ, USA) and J.-M. Roquejoffre (IMT Toulouse, France). I will set up a finite-difference scheme allowing to compute approximate solutions and capture the free boundaries, and will carry out a numerical investigation of their regularity. Based on some nondegeneracy assumptions supported by solid numerical evidence, I will show the Lipschitz regularity of the free boundaries. 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. I will also discuss the existence of corners in the more analytical framework of viscosity solutions to certain periodic Hamilton-Jacobi equations, whose validity will be again supported by numerical evidence.
