## Arkiv för Matematik

- Ark. Mat.
- Volume 49, Number 2 (2011), 357-382.

### Perturbation theorems for Hele-Shaw flows and their applications

#### 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, the so called Polubarinova–Galin equation. This theorem enables us to explore properties of solutions with initial functions close to polynomials. Applications of this theorem are given in the suction and injection cases. In the former case, we show that if the initial domain is close to a disk, most of the fluid will be sucked before the strong solution blows up. In the latter 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.

#### Note

The author is indebted to her adviser, Govind Menon, for many things, including his constant guidance and important opinions. This material is based upon work supported by the National Science Foundation under grant nos. DMS 06-05006 and DMS 07-48482.

#### Article information

**Source**

Ark. Mat., Volume 49, Number 2 (2011), 357-382.

**Dates**

Received: 4 August 2009

Revised: 21 August 2010

First available in Project Euclid: 31 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.afm/1485907148

**Digital Object Identifier**

doi:10.1007/s11512-010-0138-9

**Mathematical Reviews number (MathSciNet)**

MR2826949

**Zentralblatt MATH identifier**

1252.35221

**Rights**

2010 © Institut Mittag-Leffler

#### Citation

Lin, Yu-Lin. Perturbation theorems for Hele-Shaw flows and their applications. Ark. Mat. 49 (2011), no. 2, 357--382. doi:10.1007/s11512-010-0138-9. https://projecteuclid.org/euclid.afm/1485907148