Phase separation patterns from directional quenching

TL;DR

This paper investigates how directional quenching influences pattern formation in bistable systems like Allen-Cahn and Cahn-Hilliard equations, providing existence results for interfaces and striped patterns as the bistable region expands.

Contribution

It introduces a model for directional quenching in bistable systems and analyzes the conditions for the emergence of interfaces and striped patterns.

Findings

Existence of single interfaces under certain conditions

Non-existence of some striped patterns

Patterns depend on the progression of the bistable region

Abstract

We study the effect of directional quenching on patterns formed in simple bistable systems such as the Allen-Cahn and the Cahn-Hilliard equation on the plane. We model directional quenching as an externally triggered change in system parameters, changing the system from monostable to bistable across an interface. We are then interested in patterns forming in the bistable region, in particular as the trigger progresses and increases the bistable region. We find existence and non-existence results of single interfaces and striped patterns.