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, $\{\Omega(t)\}_{t\geq 0}$. 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 $\partial\Omega^{'}(t)$ 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 $\partial\Omega^{'}(t)$ 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 Vondenhoff's work which deals with the case that $f(\xi,t)=a_{1}(t)\xi$.