Algebraic rate of decay for the excess free energy and stability of frontsfor a nonlocal phase kinetics equation with a conservation law. I

This is the first of two papers devoted to the study of a nonlocal evolutio n equation that describes the evolution of the local magnetization in a con tinuum limit of an Ising spin system with Kawasaki dynamics and Kac potenti als. We consider subcritical temperatures, for which there are two local eq uilibria, and begin the proof of a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface betwee n regions of these two different local equilibria; i.e., the fronts. We sha ll show in the second paper that an initial perturbation v(0), of a front t hat is sufficiently small in L-2 norm, and sufficiently localized that inte gral x(2)v(0)(x(2)) dx < infinity, yields a solution that relaxes to anothe r front, selected by a conservation law, in the L-1 norm at an algebraic ra te that we explicitly estimate. There we also obtain rates for the relaxati on in the L-2 norm and the rate of decrease of the excess free energy, Here we prove a number of estimates essential for this result. Moreover, the es timates proved here suffice to establish the main result in an important sp ecial case.