Infinite lifetime for the starlike dynamics in Hele-Shaw cells

Gustafsson, Björn; Prokhorov, Dmitri; Vasil’ev, Alexander

doi:10.1090/S0002-9939-04-07419-2

Infinite lifetime for the starlike dynamics in Hele-Shaw cells
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by Björn Gustafsson, Dmitri Prokhorov and Alexander Vasil’ev PDF

Proc. Amer. Math. Soc. 132 (2004), 2661-2669 Request permission

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.

References

D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, J. London Math. Soc. (2) 1 (1969), 431–443. MR 251208, DOI 10.1112/jlms/s2-1.1.431
Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
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, DOI 10.1017/S0308210500017315
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
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, DOI 10.1090/mmono/026
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, DOI 10.1137/0516021
A. W. Goodman, Univalent functions, Vols. I, II, Mariner Publishing Company, Inc., U. South Florida, Tampa, FL, 1983.
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. MR 1660267
S. D. Howison, Complex variable methods in Hele-Shaw moving boundary problems, European J. Appl. Math. 3 (1992), no. 3, 209–224. MR 1182213, DOI 10.1017/S0956792500000802
O. Kuznetsova, Invariant families in the Hele-Shaw problem, Preprint TRITA-MAT-2003-07, Royal Institute of Technology, Stockholm, Sweden, 2003.
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
Christian Pommerenke, Univalent functions, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen. MR 0507768
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, DOI 10.1007/BF02559501
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.
Makoto Sakai, Quadrature domains, Lecture Notes in Mathematics, vol. 934, Springer-Verlag, Berlin-New York, 1982. MR 663007, DOI 10.1007/BFb0095892
Makoto Sakai, Regularity of boundaries of quadrature domains in two dimensions, SIAM J. Math. Anal. 24 (1993), no. 2, 341–364. MR 1205531, DOI 10.1137/0524023
Harold S. Shapiro, The Schwarz function and its generalization to higher dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 9, John Wiley & Sons, Inc., New York, 1992. A Wiley-Interscience Publication. MR 1160990
J. Stankiewicz, Some remarks concerning starlike functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 143–146 (English, with Russian summary). MR 264053
A. Vasil’ev, Univalent functions in the dynamics of viscous flows, Comput. Methods and Function Theory 1 (2001), no. 2, 311–337.
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945