Hysteretic dynamics of phase interfaces in bilinear forward-backward diffusion equations

TL;DR

This paper investigates the complex hysteretic behavior of phase interfaces in bilinear diffusion equations, introducing a convergent approximation scheme and analyzing microscopic oscillations with numerical insights.

Contribution

It presents a novel time-discrete approximation method for phase interfaces in bilinear diffusion equations and addresses microscopic oscillations in the convergence analysis.

Findings

Convergence of the approximation scheme as step size vanishes

Control of microscopic oscillations caused by propagating phase interfaces

Numerical simulations illustrating the hysteretic dynamics

Abstract

We study single-interface solutions to a free boundary problem that couples bilinear bulk diffusion to the Stefan condition and a hysteretic flow rule for phase boundaries. We introduce a time-discrete approximation scheme and establish its convergence in the limit of vanishing step size. The main difficulty in our proof are strong microscopic oscillations which are produced by a propagating phase interface and need to be controlled on the macroscopic scale. We also present numerical simulations and discuss the link to the viscous regularization of ill-posed diffusion equations.