Paper detail

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, $\{Ω(t)\}_{t\geq 0}$. Here $Ω(0)$ is a small perturbation of $f(B_{1}(0),0)$ if $f(ξ,t)$ is a global strong polynomial solution to the Polubarinova-Galin equation with injection at the origin and we prove the solution $Ω(t)$ is global as well. We rescale the domain $Ω(t)$ so that the new domain $Ω^{'}(t)$ always has area $π$ and we consider $\partialΩ^{'}(t)$ 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 $t^{-λ}$. This decay also implies that the curvature of $\partialΩ^{'}(t)$ decays to 1 algebraically as $t^{-λ}$. 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 $f(ξ,t)=a_{1}(t)ξ$.