Convergence of the one-dimensional Cahn-Hilliard equation

TL;DR

This paper proves that the one-dimensional Cahn-Hilliard equation converges to a Stefan problem as the small parameter approaches zero, using variational methods and gradient flow structure under well-prepared initial data.

Contribution

It establishes the convergence of the one-dimensional Cahn-Hilliard equation to a Stefan problem in the sharp interface limit with a novel variational approach.

Findings

Convergence to Stefan problem as epsilon approaches zero

Validation of variational methods for analyzing phase transitions

Demonstration of gradient flow structure in the convergence process

Abstract

We consider the Cahn-Hilliard equation in one space dimension with scaling a small parameter \epsilon and a non-convex potential W. In the limit \espilon \to 0, under the assumption that the initial data are energetically well-prepared, we show the convergence to a Stefan problem. The proof is based on variational methods and exploits the gradient flow structure of the Cahn-Hilliard equation.