Large-time rescaling behaviors of some rational type solutions to the Polubarinova-Galin equation with injection

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Publication date: 16 October 2008

Abstract: The main goal of this paper is to give a precise description of rescaling behaviors of rational type global strong solutions to the Polubarinova-Galin equation. The Polubarinova-Galin equation is the reformulation of the zero surface tension Hele-Shaw problem with a single source at the origin by considering the moving domain as the Riemann mapping of the unit disk centered at the origin. The coefficients

{a_{k} (t)}_{k g e q 2}

of the polynomial strong solution

f_{k_{0}} (x i, t) = s u m_{i = 1}^{k_{0}} a_{i} (t) x i^{i}

decay to zero algebraically as

t^{- l a m b d a_{k}}

(

l a m b d a_{k} = k / 2

) and the decay is even faster if the low Richardson moments vanish. The dynamics for global solutions are discussed as well.

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