Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Energy methods for the Cahn-Hilliard equation


Author: Amy Novick-Cohen
Journal: Quart. Appl. Math. 46 (1988), 681-690
MSC: Primary 82A25; Secondary 35K55, 80A30, 82A70
DOI: https://doi.org/10.1090/qam/973383
MathSciNet review: 973383
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Abstract: The Cahn-Hilliard equation, which is important in the context of first-order phase transition, has frequently been studied in its simplified form,

$\displaystyle {c_t} = \Delta \left[ {h\left( c \right) - K\Delta c} \right],$

where $ c\left( {x, t} \right)$ is a concentration, $ h\left( c \right)$ is a nonmonotone chemical potential, and $ K$, the coefficient of gradient energy, is a positive constant. In this paper we consider the Cahn-Hilliard equation with nonconstant mobility and gradient energy coefficients,

$\displaystyle {c_t} = \nabla \cdot \left[ {M\left( c \right)\nabla \left\{ {h\left( c \right) - K\left( c \right)\Delta c} \right\}} \right],$

where $ M\left( c \right)$ and $ K\left( c \right)$ are assumed to be positive. When $ K$ is constant, the free energy functional

$\displaystyle F\left( t \right) = \int_\Omega {\left\{ {\int ^{c} {h\left( {\ba... ...ht)d\bar c + \frac{1}{2}K{{\left\vert {\nabla c} \right\vert}^2}} } \right\}dx}$

acts as a Liapounov functional for the Cahn-Hilliard equation. However, when $ K$ is nonconstant $ F\left( t \right)$ no longer acts as a Liapounov functional, and it becomes relevant to examine an alternative energy. In this paper the stability of spatially homogeneous states is studied in terms of the energy

$\displaystyle E\left( t \right) = \int_\Omega {\int_0^{c - {c_0}} {\int_0^{\tilde c} {{M^{ - 1}}\left( {\bar c + {c_0}} \right)d\bar c} d\tilde c} } dx.$

The possibility of dependence of $ h\left( c \right)$, $ M\left( c \right)$, and $ K\left( c \right)$ on a spatially uniform temperature is also considered and the physical implications of the location of the limit of monotonic global stability in the average concentration-temperature plane is discussed. In particular, this limit is shown to lie below the critical temperature.

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DOI: https://doi.org/10.1090/qam/973383
Article copyright: © Copyright 1988 American Mathematical Society


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