Interface dynamics in discrete forward-backward diffusion equations

TL;DR

This paper investigates the macroscopic behavior of phase interfaces in a discrete lattice model with bistable nonlinearity, deriving a free boundary problem with hysteresis and rigorously justifying the limit dynamics.

Contribution

It introduces a new free boundary problem with hysteresis for interface motion and provides a rigorous derivation for specific nonlinearities, linking lattice solutions to macroscopic limits.

Findings

Numerical evidence for pinning, depinning, and annihilation of interfaces.

Identification of a generalized Stefan condition and hysteretic flow rule.

Proof of the limit dynamics for certain initial data and nonlinearities.

Abstract

We study the motion of phase interfaces in a diffusive lattice equation with bistable nonlinearity and derive a free boundary problem with hysteresis to describe the macroscopic evolution in the parabolic scaling limit. The first part of the paper deals with general bistable nonlinearities and is restricted to numerical experiments and heuristic arguments. We discuss the formation of macroscopic data and present numerical evidence for pinning, depinning, and annihilation of interfaces. Afterwards we identify a generalized Stefan condition along with a hysteretic flow rule that characterize the dynamics of both standing and moving interfaces. In the second part, we rigorously justify the limit dynamics for single-interface data and a special piecewise affine nonlinearity. We prove persistence of such data, derive upper bounds for the macroscopic interface speed, and show that the macroscopic limit can indeed be described by the free boundary problem. The fundamental ingredient to our proofs is a representation formula that links the solutions of the nonlinear lattice to the discrete heat kernel and enables us to derive macroscopic compactness results in the space of continuous functions.