On the dynamics of roots and poles for solutions of the Polubarinova-Galin equation

TL;DR

This paper investigates the movement of roots and poles in solutions to the Polubarinova-Galin equation, revealing complex dynamics but ultimately simple long-term behavior, with broader implications for rational functions and solution univalence.

Contribution

It provides new insights into root and pole dynamics, extends analysis to rational functions, and proves univalence of global solutions in the context of Hele-Shaw flow.

Findings

Roots exhibit complex motion but asymptotically simplify.

Sharp estimates for pole motion and coefficient decay are established.

Global solutions must be univalent over time.

Abstract

We study the dynamics of roots of f'(z,t), where f(z,t) is a locally univalent polynomial solution of the Polubarinova-Galin equation for the evolution of the conformal map onto a Hele-Shaw blob subject to injection at one point. We give examples of the sometimes complicated motion of roots, but show also that the asymptotic behavior is simple. More generally we allow f'(z,t) to be a rational function and give sharp estimates for the motion of poles and for the decay of the Taylor coefficients. We also prove that any global in time locally univalent solution actually has to be univalent.