The Cahn-Hilliard equation

The Cahn-Hilliard (CH) equation describes the phase separation taking
place in a binary alloy subject to a rapid cooling process. More
precisely, it governs the evolution of the (relative) concentration
difference between the two components of the material. This equation
was proposed by J.W. Cahn and J.E. Hilliard in the late Fifties. Since
then it has been analyzed by many authors. Also, it has been recently
used to model other phenomena arising, for instance, in modeling tumor
growth, image segmentation and population dynamics. However, the
original CH equation has a phenomenological nature and lacks of a
rigorous justification. At the beginning of the Nineties, G. Giacomin
and J. L. Lebowitz proved that the hydrodynamic limit of a phase
segregation (separation) process is a nonlocal variant of the CH
equation. More precisely, the so-called chemical potential depends on
the concentration through a nonlocal spatial operator characterized by
a suitable interaction kernel. Thus the physically well-justified CH
equation is actually an integro-differential equation, while the
original CH equation can be viewed as its local approximation. After
an overview on the CH equation, I intend to give a (hopefully) not too
technical presentation of some recent mathematical results on the
nonlocal case.