Self-Similar Solutions to the Hele-Shaw Problem with Surface Tension

TL;DR

This paper proves the existence of self-similar solutions to the Hele-Shaw problem with surface tension, showing initial corner singularities evolve into smooth shapes over time.

Contribution

It establishes the existence of a family of self-similar solutions with initial corners, advancing understanding of surface tension effects in Hele-Shaw flows.

Findings

Existence of self-similar solutions with initial corners

Solutions become smooth for positive times

Initial corner angle close to π

Abstract

We consider the Hele-Shaw problem with surface tension in an infinite domain. We prove the existence of a family of self-similar solutions. At $t=0$, these solutions have a corner of angle $\theta$ with $ 0 < |\theta - \pi| \ll 1$, and for $t>0$, the solutions are smooth.