The Hele-Shaw flow as the sharp interface limit of the Cahn-Hilliard equation with disparate mobilities

TL;DR

This paper investigates the connection between the Cahn-Hilliard equation with varying mobilities and the Hele-Shaw flow, establishing the sharp interface limit and analyzing solution behavior under certain conditions.

Contribution

It constructs weak solutions for the Cahn-Hilliard equation with disparate mobilities and proves their convergence to Hele-Shaw flow in the sharp interface limit.

Findings

Weak solutions are constructed for the Cahn-Hilliard equation with degenerate mobility.

Precompactness of solutions is established under natural initial data assumptions.

The sharp interface limit corresponds to Hele-Shaw flow with optimal energy dissipation.

Abstract

In this paper, we study the sharp interface limit for solutions of the Cahn-Hilliard equation with disparate mobilities. This means that the mobility function degenerates in one of the two energetically favorable configurations, suppressing the diffusion in that phase. First, we construct suitable weak solutions to this Cahn-Hilliard equation. Second, we prove precompactness of these solutions under natural assumptions on the initial data. Third, under an additional energy convergence assumption, we show that the sharp interface limit is a distributional solution to the Hele-Shaw flow with optimal energy-dissipation rate.