Infinite lifetime for the starlike dynamics in Hele-Shaw cells

Authors:
Björn Gustafsson, Dmitri Prokhorov and Alexander Vasil'ev

Journal:
Proc. Amer. Math. Soc. **132** (2004), 2661-2669

MSC (2000):
Primary 30C45, 76D27, 76S05; Secondary 35Q35, 30C35

DOI:
https://doi.org/10.1090/S0002-9939-04-07419-2

Published electronically:
April 8, 2004

MathSciNet review:
2054792

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Abstract: One of the ``folklore" questions in the theory of free boundary problems is the lifetime of the starlike dynamics in a Hele-Shaw cell. We prove precisely that, starting with a starlike analytic phase domain , the Hele-Shaw chain of subordinating domains , , exists for an infinite time under injection at the point of starlikeness.

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Additional Information

**Björn Gustafsson**

Affiliation:
Department of Mathematics, Royal Institute of Technology, Stockholm 100 44, Sweden

Email:
gbjorn@math.kth.se

**Dmitri Prokhorov**

Affiliation:
Department of Mathematics and Mechanics, Saratov State University, Saratov 410012, Russia

Email:
ProkhorovDV@info.sgu.ru

**Alexander Vasil'ev**

Affiliation:
Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile

Email:
alexander.vasiliev@mat.utfsm.cl

DOI:
https://doi.org/10.1090/S0002-9939-04-07419-2

Keywords:
Free boundary problem,
Hele-Shaw flow,
univalent function,
starlike function,
L\"owner-Kufarev equation

Received by editor(s):
January 3, 2003

Received by editor(s) in revised form:
June 10, 2003

Published electronically:
April 8, 2004

Additional Notes:
The first author was partially supported by the Swedish Research Council, the Göran Gustafsson Foundation, and Fondecyt (Chile) # 7030011. The second author was supported by Fondecyt (Chile) # 7010093, and the third author was partially supported by Projects Fondecyt (Chile) # 1030373, 1020067, and UTFSM 12.03.23.

Communicated by:
Juha M. Heinonen

Article copyright:
© Copyright 2004
American Mathematical Society