 98629 E.A.Carlen, M.C.Carvalho, E.Orlandi
 Algebraic rate of decay for the excess free energy
and stability of fronts for a nonlocal 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 nonlocal
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 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 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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