Perturbation theorems for Hele-Shaw flows and their applications

TL;DR

This paper establishes a perturbation theorem for Hele-Shaw flows driven by injection or suction, enabling analysis of solutions close to polynomial form and revealing large-time behaviors and solution stability.

Contribution

It introduces a new perturbation theorem for Hele-Shaw solutions, extending understanding of flow behaviors near polynomial solutions and providing simplified proofs of existence and uniqueness.

Findings

Most fluid is sucked before blow-up when initial domain is near a disk

Large-time rescaling behaviors are characterized in terms of invariant moments

The theorem generalizes previous results for small data to larger data regimes

Abstract

In this work, we give a perturbation theorem for strong polynomial solutions to the zero surface tension Hele-Shaw equation driven by injection or suction, so called the Polubarinova-Galin equation. This theorem enables us to explore properties of solutions with initial functions close to but are not polynomial. Applications of this theorem are given in the suction or injection case. In the former case, we show that if the initial domain is close to a disk, most of fluid will be sucked before the strong solution blows up. In the later case, we obtain precise large-time rescaling behaviors for large data to Hele-Shaw flows in terms of invariant Richardson complex moments. This rescaling behavior result generalizes a recent result regarding large-time rescaling behavior for small data in terms of moments. As a byproduct of a theorem in this paper, a short proof of existence and uniqueness of strong solutions to the Polubarinova-Galin equation is given.