$C^1$-Regularity of the Free Boundary for Hele-Shaw Flow with Source and Drift

TL;DR

This paper proves that under certain smallness conditions, the free boundary in Hele-Shaw flow with drift and source terms is $C^{1}$ smooth, extending previous work and applying to 2D cases with advection.

Contribution

It establishes $C^{1}$ regularity of the free boundary for Hele-Shaw flow with drift and source, under small Lipschitz conditions and initial data assumptions.

Findings

Free boundary is $C^{1}$ if Lipschitz constant is small.

In 2D, free boundary becomes $C^{1}$ after finite time with small initial data.

Results extend previous work to include advection and source effects.

Abstract

This paper is a continuation of the work in \cite{kimzhang2024} concerning Hele-Shaw flow with both drift and source terms. We prove that, in a local neighborhood, if the free boundary is Lipschitz continuous with a sufficiently small Lipschitz constant, then the free boundary is $C^{1}$. As a corollary, we also consider the 2D vertical Hele-Shaw (or one-phase Muskat) problem with an advection term. We show that, provided the initial data and the advection term are small and the propagation speed is large, the free boundary becomes uniformly $C^1$ after a finite time.