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arXiv · 2508.15491

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

Bogdan-Vasile Matioc·Christoph Walker
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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.

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Bogdan-Vasile Matioc, Christoph Walker. 2026-05-20. A potential theory approach to the capillarity-driven Hele-Shaw problem. https://arxiv.org/abs/2508.15491

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