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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
Published electronically: April 8, 2004
MathSciNet review: 2054792
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Abstract | References | Similar Articles | Additional Information

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 $\Omega_0$, the Hele-Shaw chain of subordinating domains $\Omega(t)$, $\Omega_0=\Omega(0)$, exists for an infinite time under injection at the point of starlikeness.


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  • 1. D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, J. London Math. Soc. (2) 1 (1969), 431–443. MR 0251208
  • 2. Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
  • 3. C. M. Elliott and V. Janovský, A variational inequality approach to Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), no. 1-2, 93–107. MR 611303, 10.1017/S0308210500017315
  • 4. L. A. Galin, Unsteady filtration with a free surface, C. R. (Doklady) Acad. Sci. URSS (N.S.) 47 (1945), 246–249. MR 0014004
  • 5. G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
  • 6. Björn Gustafsson, Applications of variational inequalities to a moving boundary problem for Hele-Shaw flows, SIAM J. Math. Anal. 16 (1985), no. 2, 279–300. MR 777468, 10.1137/0516021
  • 7. A. W. Goodman, Univalent functions, Vols. I, II, Mariner Publishing Company, Inc., U. South Florida, Tampa, FL, 1983.
  • 8. Yu. E. Hohlov, D. V. Prokhorov, and A. Ju. Vasil′ev, On geometrical properties of free boundaries in the Hele-Shaw flows moving boundary problem, Lobachevskii J. Math. 1 (1998), 3–12 (electronic). MR 1660267
  • 9. S. D. Howison, Complex variable methods in Hele-Shaw moving boundary problems, European J. Appl. Math. 3 (1992), no. 3, 209–224. MR 1182213, 10.1017/S0956792500000802
  • 10. O. Kuznetsova, Invariant families in the Hele-Shaw problem, Preprint TRITA-MAT-2003-07, Royal Institute of Technology, Stockholm, Sweden, 2003.
  • 11. P. J. Polubarinova-Kotschina, On the displacement of the oil-bearing contour, C. R. (Doklady) Acad. Sci. URSS (N. S.) 47 (1945), 250–254. MR 0013336
  • 12. P. J. Poloubarinova-Kochina, Concerning unsteady motions in the theory of filtration, Appl. Math. Mech. [Akad. Nauk SSSR. Prikl. Mat. Mech.] 9 (1945), 79–90 (Russian., with English summary). MR 0013009
  • 13. Christian Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen; Studia Mathematica/Mathematische Lehrbücher, Band XXV. MR 0507768
  • 14. Michael Reissig and Lothar von Wolfersdorf, A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane, Ark. Mat. 31 (1993), no. 1, 101–116. MR 1230268, 10.1007/BF02559501
  • 15. S. Richardson, Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, J. Fluid Mech. 56 (1972), 609-618.
  • 16. Makoto Sakai, Quadrature domains, Lecture Notes in Mathematics, vol. 934, Springer-Verlag, Berlin-New York, 1982. MR 663007
  • 17. Makoto Sakai, Regularity of boundaries of quadrature domains in two dimensions, SIAM J. Math. Anal. 24 (1993), no. 2, 341–364. MR 1205531, 10.1137/0524023
  • 18. Harold S. Shapiro, The Schwarz function and its generalization to higher dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, 9, John Wiley & Sons, Inc., New York, 1992. A Wiley-Interscience Publication. MR 1160990
  • 19. J. Stankiewicz, Some remarks concerning starlike functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 143–146 (English, with Loose Russian summary). MR 0264053
  • 20. A. Vasil'ev, Univalent functions in the dynamics of viscous flows, Comput. Methods and Function Theory 1 (2001), no. 2, 311-337.
  • 21. Yu. P. Vinogradov and P. P. Kufarev, On a problem of filtration, Akad. Nauk SSSR. Prikl. Mat. Meh. 12 (1948), 181–198 (Russian). MR 0024727

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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: http://dx.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