Cahn–Hilliard equation

The Cahn–Hilliard equation (after John W. Cahn and John E. Hilliard) is an equation of mathematical physics which describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. If c {displaystyle c} is the concentration of the fluid, with c = ± 1 {displaystyle c=pm 1} indicating domains, then the equation is written as The Cahn–Hilliard equation (after John W. Cahn and John E. Hilliard) is an equation of mathematical physics which describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. If c {displaystyle c} is the concentration of the fluid, with c = ± 1 {displaystyle c=pm 1} indicating domains, then the equation is written as where D {displaystyle D} is a diffusion coefficient with units of Length 2 / Time {displaystyle { ext{Length}}^{2}/{ ext{Time}}} and γ {displaystyle {sqrt {gamma }}} gives the length of the transition regions between the domains. Here ∂ / ∂ t {displaystyle partial /{partial t}} is the partial time derivative and ∇ 2 {displaystyle abla ^{2}} is the Laplacian in n {displaystyle n} dimensions. Additionally, the quantity μ = c 3 − c − γ ∇ 2 c {displaystyle mu =c^{3}-c-gamma abla ^{2}c} is identified as a chemical potential. Related to it is the Allen–Cahn equation, as well as the Stochastic Cahn–Hilliard Equation and the Stochastic Allen–Cahn equation. Of interest to mathematicians is the existence of a unique solution of the Cahn–Hilliard equation, given by smooth initial data. The proof relies essentially on the existence of a Lyapunov functional. Specifically, if we identify