98-629 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, I (356K, PostScript) Oct 6, 98

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Abstract. This is the first of two papers devoted to the study of a non-local evolution equation that describes the evolution of the local magnetization in a continuum limit of Ising spin systems with Kawasaki dynamics and Kac potentials. We consider sub--critical temperatures, for which there are two local equilibria, and begin the proof of 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 fronts. We shall show in the second paper 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. There we also obtain rates for the relaxation 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 estimates proved here suffices to establish the main result in an important special case.

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