Stability analysis for stationary solutions of the Mullins-Sekerka flow with boundary contact

TL;DR

This paper conducts a comprehensive linearized stability analysis of stationary solutions in the Mullins-Sekerka flow with boundary contact, revealing how stability depends on geometric parameters and providing initial nonlinear stability results.

Contribution

It offers the first complete linearized stability analysis for stationary solutions of the Mullins-Sekerka flow with boundary contact and explores nonlinear stability for curved boundaries.

Findings

Stability varies from exponential to unstable depending on curvature and boundary parameters.

Stationary solutions include flat interfaces and circular arcs.

Nonlinear stability results are established for curved boundaries.

Abstract

We first give a complete linearized stability analysis around stationary solutions of the Mullins-Sekerka flow with $90^\circ$ contact angle in two space dimensions. The stationary solutions include flat interfaces, as well as arcs of circles. We investigate the different stability behaviour in dependence of properties of the stationary solution, such as its curvature and length, as well as the curvature of the boundary of the domain at the two contact points. We show that the behaviour changes in terms of these parameters, ranging from exponential stability to instability. We also give a first result on nonlinear stability for curved boundaries.