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-Sha
w 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 ${a_{k}(t)}_{kgeq 2}$ 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.
This paper addresses a rescaling behavior of some classes of global solutions to the zero surface tension Hele-Shaw problem with injection at the origin, ${Omega(t)}_{tgeq 0}$. Here $Omega(0)$ is a small perturbation of $f(B_{1}(0),0)$ if $f(xi,t)$ i
s a global strong polynomial solution to the Polubarinova-Galin equation with injection at the origin and we prove the solution $Omega(t)$ is global as well. We rescale the domain $Omega(t)$ so that the new domain $Omega^{}(t)$ always has area $pi$ and we consider $partialOmega^{}(t)$ as the radial perturbation of the unit circle centered at the origin for $t$ large enough. It is shown that the radial perturbation decays algebraically as $t^{-lambda}$. This decay also implies that the curvature of $partialOmega^{}(t)$ decays to 1 algebraically as $t^{-lambda}$. The decay is faster if the low Richardson moments vanish. We also explain this work as the generalization of Vondenhoffs work which deals with the case that $f(xi,t)=a_{1}(t)xi$.
This paper gives a new and short proof of existence and uniqueness of the Polubarinova-Galin equation. The existence proof is an application of the main theorem in Lins paper. Furthermore, we can conclude that every strong solution can be approximate
d by many strong polynomial solutions locally in time.
