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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: [email protected]
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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 

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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