Hysteresis and phase transitions in a lattice regularization of an ill-posed forward-backward diffusion equation

TL;DR

This paper investigates the dynamics of phase interfaces in a lattice model of an ill-posed diffusion equation, revealing hysteresis and phase transition behaviors in the parabolic scaling limit.

Contribution

It introduces a lattice regularization approach for an ill-posed diffusion equation and proves that solutions exhibit hysteretic Stefan conditions asymptotically.

Findings

Lattice solutions approximate a free boundary problem with hysteresis.

Microscopic fluctuations are controlled in the analysis.

Phase interface dynamics follow a hysteretic Stefan condition.

Abstract

We consider a lattice regularization for an ill-posed diffusion equation with trilinear constitutive law and study the dynamics of phase interfaces in the parabolic scaling limit. Our main result guarantees for a certain class of single-interface initial data that the lattice solutions satisfy asymptotically a free boundary problem with hysteretic Stefan condition. The key challenge in the proof is to control the microscopic fluctuations that are inevitably produced by the backward diffusion when a particle passes the spinodal region.