Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T05:22:40.022Z Has data issue: false hasContentIssue false
  • English
  • Français

Classification of degree three polynomial solutions to the Polubarinova–Galin equation

Published online by Cambridge University Press:  04 May 2015

YU-LIN LIN
YU-LIN LIN*
Affiliation:
Royal Institute of Technology, Stockholm, Sweden email: ylli@kth.se
Get access Rights & Permissions [Opens in a new window]

Abstract

The Polubarinova–Galin equation describes a parametric conformal map for the zero-surface-tension Hele-Shaw flow driven by injection or suction. In this paper, we classify degree three polynomial solutions to the Polubarinova–Galin equation into three categories: global solutions, solutions which can be continued after blow-up and solutions which cannot be continued after blow-up. The coefficient region of the initial functions in each category is obtained.

Keywords

Hele-Shawclassification of solutionsPolubarinova-Galin equation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 6 , December 2015 , pp. 849 - 861
Copyright
Copyright © Cambridge University Press 2015 

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]Brannan, D. A. (1967) Coefficient regions for univalent polynomials of small degree. Mathematika 14, 165–169.CrossRefGoogle Scholar
[2]Galin, L. (1945) Unsteady filtration with a free surface. Dokl. Akad. Nauk. SSSR. 47, 246–249.Google Scholar
[3]Gustafsson, B. (1984) On a differential equation arising in a Hele-Shaw flow moving boundary problem. Ark. Mat. 22, 251–268.CrossRefGoogle Scholar
[4]Gustafsson, B. (1988) Singular and special points on quadrature domains from an algebraic geometric point of view. J. Anal. Math. 51, 91–117.CrossRefGoogle Scholar
[5]Gustafsson, B. & Lin, Y.-L. (2013) On the dynamics of roots and poles for solutions of the Polubarinova–Galin equation. Ann. Acad. Sci. Fenn. Math. 38, 259–286.CrossRefGoogle Scholar
[6]Gustafsson, B., Prokhorov, D. & Vasil'ev, A. (2004) Infinite lifetime for the starlike dynamics in Hele-Shaw cells. Proc. Am. Math. Soc. 132, 2661–2669 (electronic).CrossRefGoogle Scholar
[7]Hohlov, Y. E. & Howison, S. D. (1993) On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flows. Quart. Appl. Math. 51, 777–789.CrossRefGoogle Scholar
[8]Howison, S. D. (1986) Cusp development in Hele-Shaw flow with a free surface. SIAM J. Appl. Math. 46, 20–26.CrossRefGoogle Scholar
[9]Huntingford, C. (1995) An exact solution to the one-phase zero-surface-tension Hele-Shaw free-boundary problem. Comput. Math. Appl. 29, 45–50.CrossRefGoogle Scholar
[10]Kuznetsova, O. S. & Tkachev, V. G. (2004) Ullemar's formula for the Jacobian of the complex moment mapping. Complex Var. Theory Appl. 49, 55–72.Google Scholar
[11]Polubarinova-Kochina, P. Y. (1945) On the motion of the oil contour. Dokl. Akad. Nauk. SSSR. 47, 254–257.Google Scholar
[12]Reissig, M. & von Wolfersdorf, L. (1993) A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane. Ark. Mat. 31, 101–116.CrossRefGoogle Scholar
[13]Richardson, S. (1972) Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56, 609–618.CrossRefGoogle Scholar
[14]Virogradov, Y. & Kufarev, P. (1948) On a problem of filtration. Akad. Nauk. SSSR. Prikl. Mat. Mech. 12, 181–198.Google Scholar