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

arXiv0810.2975MaRDI QIDQ6211265

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Publication date: 16 October 2008

Abstract: 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, ${Omega(t)}_{tgeq0}$. 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 $Omeg{a}^{\text{'}}(t)$ always has area $pi$ and we consider $partialOmeg{a}^{\text{'}}(t)$ 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 $partialOmeg{a}^{\text{'}}(t)$ 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 $f(xi,t)=a_1(t)xi$.

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