Large-time rescaling behaviors for large data to the Hele-Shaw problem

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 $f(xi,t)$ is 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 $f(xi,t)=a_{1}(t)xi$.