Nonlinear diffusion equations as asymptotic limits of Cahn–Hilliard systems

Abstract

An asymptotic limit of a class of Cahn–Hilliard systems is investigated to obtain a general nonlinear diffusion equation. The target diffusion equation may reproduce a number of well-known model equations: Stefan problem, porous media equation, Hele-Shaw profile, nonlinear diffusion of singular logarithmic type, nonlinear diffusion of Penrose–Fife type, fast diffusion equation and so on. Namely, by setting the suitable potential of the Cahn–Hilliard systems, all these problems can be obtained as limits of the Cahn–Hilliard related problems. Convergence results and error estimates are proved.