Energy methods for the Cahn-Hilliard equation
Authors: Novick-Cohen Amy 
Published: December 2016

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,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript t Baseline equals nabla dot left-bracket upper M left-parenthesis c right-parenthesis nabla left-brace h left-parenthesis c right-parenthesis minus upper K left-parenthesis c right-parenthesis normal upper Delta c right-brace right-bracket comma">

c
t

=
∇
⋅

[

M

(
c
)

∇

{

h

(
c
)

−
K

(
c
)

Δ
c

}

]

,

{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

(
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis t right-parenthesis equals integral Underscript normal upper Omega Endscripts left-brace integral Overscript c Endscripts h left-parenthesis c overbar right-parenthesis d c overbar plus one half upper K StartAbsoluteValue nabla c EndAbsoluteValue squared right-brace d x">

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 | {\nabla 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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis t right-parenthesis equals integral Underscript normal upper Omega Endscripts integral Subscript 0 Superscript c minus c 0 Baseline integral Subscript 0 Superscript c overTilde Baseline upper M Superscript negative 1 Baseline left-parenthesis c overbar plus c 0 right-parenthesis d c overbar d c overTilde d x period">

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.