Energy methods for the Cahn-Hilliard equation

Abstract

The Cahn-Hilliard equation, which is important in the context of first-order phase transition, has frequently been studied in its simplified form, \[ c t = Δ [ h ( c ) − K Δ c ] , {c_t} = \Delta \left [ {h\left ( c \right ) - K\Delta c} \right ], \] where c ( x , t ) c\left ( {x, t} \right ) is a concentration, h ( c ) h\left ( c \right ) is a nonmonotone chemical potential, and K 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, \[ c t = ∇ ⋅ [ M ( c ) ∇ { h ( c ) − K ( c ) Δ c } ] , {c_t} = abla \cdot \left [ {M\left ( c \right )abla \left \{ {h\left ( c \right ) - K\left ( c \right )\Delta c} \right \}} \right ], \] where M ( c ) M\left ( c \right ) and K ( c ) K\left ( c \right ) are assumed to be positive. When K K is constant, the free energy functional \[ F ( t ) = ∫ Ω { ∫ c h ( c ¯ ) d c ¯ + 1 2 K | ∇ c | 2 } d x F\left ( t \right ) = \int _\Omega {\left \{ {\int ^{c} {h\left ( {\bar c} \right )d\bar c + \frac {1}{2}K{{\left | {abla c} \right |}^2}} } \right \}dx} \] acts as a Liapounov functional for the Cahn-Hilliard equation. However, when K K is nonconstant F ( t ) 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 \[ E ( t ) = ∫ Ω ∫ 0 c − c 0 ∫ 0 c ~ M − 1 ( c ¯ + c 0 ) d c ¯ d c ~ d x . E\left ( t \right ) = \int _\Omega {\int _0^{c - {c_0}} {\int _0^{\tilde c} {{M^{ - 1}}\left ( {\bar c + {c_0}} \right )d\bar c} d\tilde c} } dx. \] The possibility of dependence of h ( c ) h\left ( c \right ) , M ( c ) M\left ( c \right ) , and K ( c ) 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.