Update 2015-10-20-math.md

_posts/2015-10-20-math.md

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 ## The Cahn-Hilliard equation
 The Cahn–Hilliard equation describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. It was originally proposed in 1958 to model phase separation in binary alloys \cite{Cahn-Hilliard, Cahn-Hilliard2}. Since then it has been used to describe various phenomena, such as spinodal decomposition \cite{MARALDI201231}, diblock copolymer \cite{Choksi,JEONG20141263}, image inpainting \cite{4032803}, tumor growth simulation \cite{tumor,WISE2008524} or multiphase fluid flows \cite{BADALASSI2003371, Kotschote}. 
 
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 There are two main approaches to describing phase transition phenomena: sharp-interface models and phase-field models. Sharp-interface models involve the resolution of a moving boundary problem, meaning that partial differential equations have to be solved for each phase. This can lead to physical \cite{Anderson} and computational complications \cite{Barrett}, such as jump discontinuities across the interface. Phase-field models replace sharp-interfaces by thin transitions regions where the interfacial forces are smoothly distributed (diffuse interfaces). For this reason, phase-field models are also referred to as \emph{diffuse interface} models, and they are emerging as a promising tool to treat problems with interfaces.