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Article overview
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Large-time rescaling behaviors for large data to the Hele-Shaw problem | Yulin Lin
; | Date: |
16 Oct 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,
${Omega(t)}_{tgeq 0}$. 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 Vondenhoff’s work which deals
with the case that $f(xi,t)=a_{1}(t)xi$. | Source: | arXiv, 0810.2975 | Services: | Forum | Review | PDF | Favorites |
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