Some remarks on a viscous regularization of the nonlinear diffusion equation

TL;DR

This paper presents an alternative derivation and proof of existence for a viscous regularization of the nonlinear diffusion equation, and proposes a non-smooth variant potentially leading to hysteresis effects.

Contribution

It offers a new derivation and proof approach for viscous regularization of the diffusion equation and introduces a non-smooth variant with possible hysteretic behavior.

Findings

Alternative derivation of viscous regularization

Proof of existence using Galerkin and compactness

Proposal of a non-smooth regularization variant

Abstract

We illustrate an alternative derivation of the viscous regularization of the diffusion equation which was studied in [A. Novick-Cohen and R. L. Pego. {\em Trans. Amer. Math. Soc.}, 324:331--351]. We provide an alternative proof of existence of solutions, based on the Galerkin method and on compactness arguments. In addition, we propose a "non-smooth" variant of the viscous regularization which we believe may result in interesting hysteretic effects.