Large-time rescaling behaviors of some rational type solutions to the Polubarinova-Galin equation with injection

TL;DR

This paper investigates the long-term rescaling behaviors of rational solutions to the Polubarinova-Galin equation, revealing algebraic decay rates of solution coefficients and discussing the dynamics of global solutions.

Contribution

It provides a detailed description of the asymptotic rescaling behaviors of rational solutions to the Polubarinova-Galin equation, including decay rates and effects of Richardson moments.

Findings

Coefficients decay algebraically as t^{-k/2}

Faster decay occurs when low Richardson moments vanish

Discussion of global solution dynamics

Abstract

The main goal of this paper is to give a precise description of rescaling behaviors of rational type global strong solutions to the Polubarinova-Galin equation. The Polubarinova-Galin equation is the reformulation of the zero surface tension Hele-Shaw problem with a single source at the origin by considering the moving domain as the Riemann mapping of the unit disk centered at the origin. The coefficients $\{a_{k}(t)\}_{k\geq 2}$ of the polynomial strong solution $f_{k_{0}}(\xi,t)=\sum_{i=1}^{k_{0}}a_{i}(t)\xi^{i}$ decay to zero algebraically as $t^{-\lambda_{k}}$ ($\lambda_{k}=k/2$) and the decay is even faster if the low Richardson moments vanish. The dynamics for global solutions are discussed as well.