C¹-Regularity of the Free Boundary for Hele-Shaw Flow with Source and Drift

Yuming Paul Zhang

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Pith reviewed 2026-05-07 12:04 UTC · model grok-4.3

classification 🧮 math.AP

keywords boundaryfreehele-shawadvectiondriftflowlipschitzsmall

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The pith

Lipschitz free boundaries with small constant in Hele-Shaw flow with source and drift are C^1 locally, with a smoothing corollary for the vertical Muskat problem under small data and fast propagation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Hele-Shaw flow describes fluid moving between two parallel plates, with a free boundary separating regions of different fluid presence. The model here includes a source term for fluid addition and a drift term for external forcing. The authors show that if this boundary starts with only mild steepness (Lipschitz with small constant), it must develop a continuous tangent, upgrading to C^1 regularity nearby. In the special 2D vertical setup with an added advection term, small starting shapes and weak advection combined with high flow speed cause the boundary to smooth out to uniform C^1 after some time.

Core claim

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}.

Load-bearing premise

The free boundary is assumed Lipschitz with a sufficiently small Lipschitz constant in a local neighborhood; this smallness is required for the regularity bootstrap to close.

read the original 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.

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Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result relies on standard tools from elliptic and parabolic regularity theory applied to the Hele-Shaw equation with added terms; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)

standard math Standard local regularity theory for free boundary problems in parabolic equations
Invoked to upgrade Lipschitz regularity to C^1 when the Lipschitz constant is small.
domain assumption The Hele-Shaw flow with source and drift satisfies the expected weak formulation or viscosity solution properties
Assumed from the model setup and prior work to apply the regularity bootstrap.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Global well-posedness for the Hele-Shaw problem with point injection
math.AP 2026-05 unverdicted novelty 6.0

Global well-posedness is established for the nonlocal interface equation arising from the Hele-Shaw problem with point injection in star-shaped domains with Lipschitz initial data.