Coarsening Mechanism for Systems Governed by the Cahn-Hilliard Equation with Degenerate Diffusion Mobility
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We study a Cahn--Hilliard equation with a diffusion mobility that is degenerate in both phases and a double-well potential that is continuously differentiable. Using asymptotic analysis, we show that the interface separating the two phases does not move in the $t=O(1)$ or $O(\varepsilon^{-1})$ time scales, although in the latter regime there is a nontrivial porous medium diffusion process in both phases. Interface motion occurs in the $t=O(\varepsilon^{-2})$ time scale and is determined by quasi-stationary porous medium diffusion processes in both bulk phases, together with a surface diffusion process along the interface itself. In addition, in off-critical systems where one phase---the minor phase---occupies only a small fraction of the system and consists of many disjoint components, it is the quasi-stationary porous medium diffusion process that provides communications between the disjoint components and accounts for the occurrence of coarsening.