Existence and uniqueness of the Hele-Shaw problem with injection

TL;DR

This paper provides a concise proof of existence and uniqueness for the Hele-Shaw problem with injection, demonstrating that strong solutions can be approximated by polynomial solutions over time.

Contribution

It offers a new, streamlined proof of the Polubarinova-Galin equation's well-posedness and shows approximation of solutions by polynomial solutions.

Findings

Existence and uniqueness of solutions established.

Strong solutions can be approximated by polynomial solutions.

The proof leverages Lin's main theorem.

Abstract

This paper gives a new and short proof of existence and uniqueness of the Polubarinova-Galin equation. The existence proof is an application of the main theorem in Lin's paper. Furthermore, we can conclude that every strong solution can be approximated by many strong polynomial solutions locally in time.