Global well-posedness for the Hele-Shaw problem with point injection

TL;DR

This paper proves global well-posedness for the Hele-Shaw problem with point injection in star-shaped domains, introducing a viscosity-solution framework and analyzing interface angle dynamics.

Contribution

It establishes global well-posedness for the interface evolution and connects viscosity solutions with classical theory in the Hele-Shaw problem.

Findings

Acute corners have positive waiting time before movement.

Obtuse corners move immediately upon injection.

The interface evolution is well-posed for Lipschitz initial data.

Abstract

We study the two-dimensional Hele-Shaw problem with point injection for star-shaped domains. We reduce the system to a nonlocal parabolic equation of the interface, and for arbitrary Lipschitz initial interface away from the source, we prove global well-posedness of the interface equation in a strong sense. We also introduce a viscosity-solution framework for the interface equation and relate it to the classical viscosity theory for the Hele-Shaw problem. As an application, we recover angle dynamics of Lipschitz initial interfaces: acute corners exhibit positive waiting time, while obtuse corners move immediately.