Energy methods for the Cahn-Hilliard equation

The Cahn-Hilliard equation, which is important in the context of first-order phase transition, has frequently been studied in its simplified form, \[ {c_t} = \Delta \left [ {h\left ( c \right ) - K\Delta c} \right ],\] 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, \[ {c_t} = \nabla \cdot \left [ {M\left ( c \right )\nabla \left \{ {h\left ( c \right ) - K\left ( c \right )\Delta c} \right \}} \right ],\] where $M\left ( c \right )$ and $K\left ( c \right )$ are assumed to be positive. When $K$ is constant, the free energy functional \[ F\left ( t \right ) = \int _\Omega {\left \{ {\int ^{c} {h\left ( {\bar c} \right )d\bar c + \frac {1}{2}K{{\left | {\nabla c} \right |}^2}} } \right \}dx}\] 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 \[ E\left ( t \right ) = \int _\Omega {\int _0^{c - {c_0}} {\int _0^{\tilde c} {{M^{ - 1}}\left ( {\bar c + {c_0}} \right )d\bar c} d\tilde c} } dx.\] 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.