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.