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Theorem on the existence of solutions of quasi-static moving boundary problems

Published online by Cambridge University Press:  26 September 2008

Bart Klein Obbink
Bart Klein Obbink
Affiliation:
Department of Mathematics and Computational Science, Eindhoven University of Technology, P. O. Box 513, Eindhoven, The Netherlands (e-mail: bartk@win.tue.nl)
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Abstract

Using the theory of conformal mappings, we show that two-dimensional quasi-static moving boundary problems can be described by a non-linear Löwner-Kufarev equation and a functional relation ℱ between the shape of the boundary and the velocity at the boundary. Together with the initial data, this leads to an initial value problem. Assuming that ℱ satisfies certain conditions, we prove a theorem stating that this initial value problem has a local solution in time. The proof is based on some straightforward estimates on solutions of Löwner-Kufarev equations and an iteration technique.

Information

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 5 , October 1995 , pp. 529 - 538
Copyright
Copyright © Cambridge University Press 1995

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References

[1]Antanovskii, L. K. 1992 Creeping thermocapillary motion. Euro. J. Mech. B 11(6), 741–758.Google Scholar
[2]Coddington, E. A. & Levinson, N. 1955 Theory of Ordinary Differential Equations. McGraw-Hill.Google Scholar
[3]Duren, P. L. 1983 Univalent Functions. Grundl. der math. Wissens. 259, Springer-Verlag.Google Scholar
[4]Goluzin, G. M. 1969 Geometric theories of functions of a complex variable. Am. Math. Soc. (English translation).Google Scholar
[5]Hopper, R. W. 1990 Plane Stokes flow driven by capillarity on a free surface. J. Fluid Mechanics 213, 349–375.CrossRefGoogle Scholar
[6]Howison, S. D. 1992 Complex variable methods in Hele-Shaw moving boundary problems. Euro. J. Appl. Math. 3, 209–224.CrossRefGoogle Scholar
[7]Kufarev, P. P. 1943 On one-parameter families of analytic functions. Mat. Sb. 13(55. Pt. 1), 87–118 (in Russian).Google Scholar
[8]Kufarev, P. P. 1947 A theorem on solutions of a differential equation. Uchenye Zapiski Toms. Gos. Un. (5), 20–21.Google Scholar
[9]Löwner, K. 1923 Untersuchungen über schlichte Konf. Abb. des Einheitskr.. Math. Ann. 89, 103–121.Google Scholar
[10]Pommerenke, Ch. 1965 Über die Subordination analytischer Funktione. J. reine und angewandte Math. 218, 159–173.Google Scholar
[11]Pommerenke, Ch. 1992 Boundary Behaviour of Conformal Maps. Grundl. der math. Wissens. Springer-Verlag.Google Scholar
[12]Vinogradov, U. P. & Kufarev, P. P. 1948 Prikl. Mat. Mech. 12 (in Russian).Google Scholar