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

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 $\{a_k(t){\}}_{k\ge 2}$ of the polynomial strong solution $f_{k_0}(\hat{I}¾,t)={\sum }_{i=1}^{k_0}{a}_i(t)\hat{I}{¾}^i$ decay to zero algebraically as $t^{-\hat{I}{»}_k}$ ($\hat{I}{»}_k=k/2$) and the decay is even faster if the low Richardson moments vanish. The dynamics for global solutions are discussed as well.

This paper has been withdrawn by the author. 34 pages