Energy estimates and convergence of weak solutions of the porous medium equation

TL;DR

This paper investigates the convergence of weak solutions to the porous medium equation with Robin boundary conditions, showing strong $L^2$ convergence to solutions with Neumann or Dirichlet boundary conditions as a parameter varies.

Contribution

It introduces a microscopic dynamics approach to derive energy estimates that prove convergence of solutions with different boundary conditions.

Findings

Strong $L^2$ convergence to Neumann and Dirichlet solutions.

Energy estimates are key to establishing convergence.

The method links microscopic dynamics to macroscopic PDE behavior.

Abstract

We study the convergence of the weak solution of the porous medium equation with a type of Robin boundary conditions, by tuning a parameter either to zero or to infinity. The convergence is in the strong sense, with respect to the $L^2$-norm, and the limiting function solves the same equation with Neumann (resp. Dirichlet) boundary conditions when the parameter is taken to zero (resp. infinity). Our approach is to consider an underlying microscopic dynamics whose space-time evolution of the density is ruled by the solution of those equations and from this, we derive sufficiently strong energy estimates which are the keystone to the proof of our convergence result.