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 formula_1 is the concentration of the fluid, with formula_2 indicating domains, then the equation is written as  where formula_4 is a diffusion coefficient with units of formula_5 and formula_6 gives the length of the transition regions between the domains. Here formula_7 is the partial time derivative and formula_8 is the Laplacian in formula_9 dimensions. Additionally, the quantity formula_10 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  as a free energy functional, then  so that the free energy does not grow in time. This also indicates segregation into domains is the asymptotic outcome of the evolution of this equation. In real experiments, the segregation of an initially mixed binary fluid into domains is observed. The segregation is characterized by the following facts. The Cahn–Hilliard equations finds applications in diverse fields: in complex fluids and soft matter (interfacial fluid flow, polymer science and in industrial applications). The solution of the Cahn–Hilliard equation for a binary mixture demonstrated to coincide well with the solution of a Stefan problem and the model of Thomas and Windle. Of interest to researchers at present is the coupling of the phase separation of the Cahn–Hilliard equation to the Navier–Stokes equations of fluid flow.