E.A. Carlen, M.C. Carvalho, E. Orlandi
Algebraic rate of decay for the excess free energy
and stability of fronts for a non-local phase kinetics
equation with a conservation law, II
(249K, PostScript)
ABSTRACT. We continue our study of 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 equilibria, and complete the
proof of a local nonlinear stability result for the
minimum free energy profile for the magnetization at the interface
between regions of these two different local equilibrium; i.e., the fronts.
We show that an initial perturbation $v_0$ of a front
that is sufficiently small in $L^2$
norm, and sufficiently localized that
$\int x^2v_0(x)^2{\rm d}x < \infty$,
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.