Hele-Shaw flow in thin threads: A rigorous limit result

TL;DR

This paper rigorously proves that solutions of the 2D Hele-Shaw problem with surface tension converge to the Thin Film equation as the thread becomes thin, using scaled estimates for the nonlinear evolution equations.

Contribution

It provides a rigorous mathematical proof of the limit transition from Hele-Shaw flow to the Thin Film equation in the thin thread regime.

Findings

Convergence of Hele-Shaw solutions to Thin Film equation established.

Use of scaled parabolic estimates for nonlinear evolution equations.

Mathematical framework for thin thread limit in Hele-Shaw flow.

Abstract

We rigorously prove the convergence of appropriately scaled solutions of the 2D Hele-Shaw moving boundary problem with surface tension in the limit of thin threads to the solution of the formally corresponding Thin Film equation. The proof is based on scaled parabolic estimates for the nonlocal, nonlinear evolution equations that arise from these problems.