Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-25T21:55:27.667Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
Get access Rights & Permissions [Opens in a new window]

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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