A potential theory approach to the capillarity-driven Hele-Shaw problem

TL;DR

This paper applies potential theory to analyze the Hele-Shaw problem with surface tension, establishing well-posedness, smoothing, and exponential stability of stationary solutions in a rigorous mathematical framework.

Contribution

It introduces a potential theory approach to the Hele-Shaw problem, deriving local well-posedness, smoothing, and stability results with optimal function space considerations.

Findings

Derived local well-posedness and parabolic smoothing for the Hele-Shaw problem.

Established exponential stability of stationary solutions.

Developed a generalized linearized stability principle for quasilinear parabolic problems.

Abstract

In this paper, we demonstrate that potential theory provides a powerful framework for analyzing quasistationary fluid flows in bounded geometries, where the bulk dynamics are governed by elliptic equations with constant coefficients. This approach is illustrated by the two-dimensional Hele-Shaw problem with surface tension, for which we derive local well-posedness and parabolic smoothing in (almost) optimal function spaces. In addition, we establish a generalized principle of linearized stability for a particular class of abstract quasilinear parabolic problems, which enables us to show that the stationary solutions to the Hele-Shaw problem are exponentially stable.