Stability of planar fronts for a non--local phase kinetics equation with a conservation law in $D \le 3$

TL;DR

This paper proves the nonlinear stability of planar interface fronts in a non-local phase kinetics equation with conservation law in dimensions up to three, showing solutions relax to equilibrium profiles at explicit algebraic rates.

Contribution

It establishes local nonlinear stability and explicit relaxation rates for planar fronts in a non-local phase kinetics model with conservation law in low dimensions.

Findings

Perturbations decay at algebraic rates in L^1 norm.

Solutions relax to equilibrium profiles with explicit rate estimates.

The stability result applies to sub-critical temperature regimes.

Abstract

We consider, in a $D-$dimensional cylinder, a non--local evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider sub--critical temperatures, for which there are two local spatially homogeneous equilibria, and show a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibrium; i.e., the planar fronts: We show that an initial perturbation of a front that is sufficiently small in $L^2$ norm, and sufficiently localized yields a solution that relaxes to another front, selected by a conservation law, in the $L^1$ norm at an algebraic rate that we explicitly estimate. We also obtain rates for the relaxation in the $L^2$ norm and the rate of decrease of the excess free energy.