How do degenerate mobilities determine singularity formation in Cahn-Hilliard equations?

TL;DR

This paper investigates how degenerate mobilities influence the formation of singularities in Cahn-Hilliard equations, providing a theoretical framework and numerical methods for understanding phase separation with vanishing mobility.

Contribution

It develops a singular perturbation theory to identify degeneracies leading to infinite-time singularities in Cahn-Hilliard models, advancing both theory and numerical approaches.

Findings

Identifies degeneracy ranges causing singularities in finite or infinite time.

Establishes a rigorous sharp interface theory for degenerate mobilities.

Develops robust numerical methods for simulating these phenomena.

Abstract

Cahn-Hilliard models are central for describing the evolution of interfaces in phase separation processes and free boundary problems. In general, they have non-constant and often degenerate mobilities. However, in the latter case, the spontaneous appearance of points of vanishing mobility and their impact on the solution are not well understood. In this paper we develop a singular perturbation theory to identify a range of degeneracies for which the solution of the Cahn-Hilliard equation forms a singularity in infinite time. This analysis forms the basis for a rigorous sharp interface theory and enables the systematic development of robust numerical methods for this family of model equations.