Spectral analysis for periodic solutions of the Cahn�CHilliard equation on {\mathbb{R}} |
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Authors: | Peter Howard |
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Institution: | 1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA
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Abstract: | We consider the spectrum associated with the linear operator obtained when the Cahn–Hilliard equation on
\mathbbR{\mathbb{R}} is linearized about a stationary periodic solution. Our analysis is particularly motivated by the study of spinodal decomposition,
a phenomenon in which the rapid cooling (quenching) of a homogeneously mixed binary alloy causes separation to occur, resolving
the mixture into regions of different crystalline structure, separated by steep transition layers. In this context, a natural
problem regards the evolution of solutions initialized by small, random (in some sense) perturbations of the pre-quenching
homogeneous state. Solutions initialized in this way appear to evolve transiently toward certain unstable periodic solutions,
with the rate of evolution described by the spectrum associated with these periodic solutions. In the current paper, we use
Evans function methods and a perturbation argument to locate the spectrum associated with such periodic solutions. We also
briefly discuss a heuristic method due to Langer for relating our spectral information to coarsening rates. |
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