From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation

TL;DR

This paper rigorously proves the convergence of viscous Cahn-Hilliard solutions to a regularized forward-backward parabolic equation as diffusion vanishes, including boundary conditions and error estimates.

Contribution

It provides a rigorous convergence proof, boundary condition handling, and an error estimate for the transition from viscous Cahn-Hilliard to forward-backward equations.

Findings

Convergence of solutions as diffusive coefficient approaches zero.

Error estimates with specified convergence rates.

Handling of non-homogeneous boundary conditions.

Abstract

A rigorous proof is given for the convergence of the solutions of a viscous Cahn-Hilliard system to the solution of the regularized version of the forward-backward parabolic equation, as the coefficient of the diffusive term goes to 0. Non-homogenous Neumann boundary condition are handled for the chemical potential and the subdifferential of a possible non-smooth double-well functional is considered in the equation. An error estimate for the difference of solutions is also proved in a suitable norm and with a specified rate of convergence.