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.
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.
"Perturbation theorems for Hele-Shaw flows and their applications." Ark. Mat. 49 (2) 357 - 382, October 2011. https://doi.org/10.1007/s11512-010-0138-9