Abstract
This is the first of two papers devoted to the study of a nonlocal 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 subcritical 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 equilibria; 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 ∫ x 2 v 0(x)2 dx<∞, 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 suffice to establish the main result in an important special case.
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Carlen, E.A., Carvalho, M.C. & Orlandi, E. Algebraic Rate of Decay for the Excess Free Energy and Stability of Fronts for a Nonlocal Phase Kinetics Equation with a Conservation Law. I. Journal of Statistical Physics 95, 1069–1117 (1999). https://doi.org/10.1023/A:1004562719554
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DOI: https://doi.org/10.1023/A:1004562719554