Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems

TL;DR

This paper investigates how certain Cahn-Hilliard systems can asymptotically approximate a broad class of nonlinear diffusion equations, including classical models like Stefan, porous media, and Hele-Shaw, with convergence and error estimates.

Contribution

It demonstrates that by choosing appropriate potentials, Cahn-Hilliard systems can serve as a unifying framework for various nonlinear diffusion equations, providing convergence results.

Findings

Cahn-Hilliard systems can approximate multiple nonlinear diffusion models.

Convergence and error estimates are established for the asymptotic limits.

The approach unifies different diffusion equations under a common framework.

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 of these problems can be obtained as limits of the Cahn-Hilliard related problems. Convergence results and error estimates are proved.