Global existence and decay for solutions of the Hele-Shaw flow with injection

TL;DR

This paper investigates the global existence and decay behavior of Hele-Shaw flows with surface tension, showing exponential decay without injection and algebraic decay with injection, depending on the injection rate.

Contribution

It establishes new decay rates for Hele-Shaw flows under different injection conditions, extending understanding of stability and long-term behavior.

Findings

Perturbations decay exponentially without fluid injection.

With fluid injection, the boundary approaches an expanding sphere at an algebraic rate.

Decay rates depend on the fluid injection rate.

Abstract

We study the global existence and decay to spherical equilibrium of Hele-Shaw flows with surface tension. We prove that without injection of fluid, perturbations of the sphere decay to zero exponentially fast. On the other hand, with a time-dependent rate of fluid injection into the Hele-Shaw cell, the distance from the moving boundary to an expanding sphere (with time-dependent radius) also decays to zero but with an algebraic rate, which depends on the injection rate of the fluid.