Large-time rescaling behaviors of Stokes and Hele-Shaw flows driven by injection

TL;DR

This paper analyzes the long-term rescaling behaviors of Hele-Shaw and Stokes flows driven by injection, using complex moments and conserved quantities to describe the evolution of solutions over time.

Contribution

It provides a detailed description of the rescaling behaviors of solutions to Hele-Shaw and Stokes flows driven by injection, extending previous results with a new analytical approach.

Findings

Rescaling behaviors are characterized in terms of Richardson complex moments.

The method applies to both Hele-Shaw and Stokes flows driven by injection.

Conserved quantities play a key role in describing flow evolution.

Abstract

In this paper, we give a precise description of the rescaling behaviors of global strong polynomial solutions to the reformulation of zero surface tension Hele-Shaw problem driven by injection, the Polubarinova-Galin equation, in terms of Richardson complex moments. From past results, we know that this set of solutions is large. This method can also be applied to zero surface tension Stokes flow driven by injection and a rescaling behavior is given in terms of many conserved quantities as well.