# Well-Posedness and qualitative behaviour of the Mullins-Sekerka problem with ninety-degree angle boundary contact

## TL;DR

This paper establishes local well-posedness and stability results for the Mullins-Sekerka problem with ninety-degree boundary contact, using maximal regularity and linearization techniques to analyze interface motion.

## Contribution

It introduces a novel approach to prove well-posedness and stability for the Mullins-Sekerka problem with boundary contact at ninety degrees, including the handling of Neumann trace spaces.

## Key findings

- Proved local well-posedness of the problem.
- Established stability and exponential convergence to equilibrium.
- Developed a framework for analyzing interface motion with boundary contact.

## Abstract

We show local well-posedness for the Mullins-Sekerka system with ninety degree angle boundary contact. We will describe the motion of the moving interface by a height function over a fixed reference surface. Using the theory of maximal regularity together with a linearization of the equations and a localization argument we will prove well-posedness of the full nonlinear problem via the contraction mapping principle. Here one difficulty lies in choosing the right space for the Neumann trace of the height function and showing maximal $L_p-L_q$-regularity for the linear problem. In the second part we show that solutions starting close to certain equilibria exist globally in time, are stable, and converge to an equilibrium solution at an exponential rate.