98-640 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) Oct 13, 98
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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.

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