Algebraic Rate of Decay for the Excess Free Energy and Stability of Fronts for a Nonlocal Phase Kinetics Equation with a Conservation Law. I |
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Authors: | Carlen E A Carvalho M C Orlandi E |
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Institution: | (1) School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, 30332;(2) School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, 30332; on;(3) Departamento de Matemática da Faculdade de Ciencias de Lisboa, 1700 Lisboa codex, Portugal;(4) Dipartimento di Matematica, Universitá degli Studi di Roma Tre, 00146 Rome, Italy |
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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.on leave from |
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Keywords: | phase kinetics fronts nonlinear stability |
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