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

By Yu-lin Lin

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, {Ω(t)}t≥0. Here Ω(0) is a small perturbation of f(B1(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 ∂Ω ′ (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 ∂Ω ′ (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) = a1(t)ξ. Keywords: Hele-Shaw flows, starlike function, rescaling behavior.

Year: 2012
OAI identifier: oai:CiteSeerX.psu:10.1.1.244.4677
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