Global solutions for two-phase Hele-Shaw bubble for a near-circular initial shape

TL;DR

This paper proves the global existence and asymptotic stability of near-circular bubbles in a Hele-Shaw cell with surface tension and finite viscosity ratio, using a vortex sheet method.

Contribution

It establishes the global existence and stability of solutions for near-circular initial shapes without requiring analyticity, extending previous results.

Findings

Global existence of solutions for near-circular bubbles

Asymptotic stability of the circular shape

Applicability to non-analytic initial conditions

Abstract

Using a vortex sheet method we prove global existence of a near circular initial bubble in a Hele-Shaw cell with surface tension and generally finite nonzero viscosity ratio between fluids inside and outside the bubble. The circular shape is shown to be asymptotically stable for all sufficiently smooth small perturbation. The initial condition in this case, while smooth, need not be analytic.