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Stability of self-similar extinction solutions for a 3D Darcy flow suction problem

Published online by Cambridge University Press:  01 August 2009

ERWIN VONDENHOFF  and
GEORG PROKERT
ERWIN VONDENHOFF
Affiliation:
Eindhoven University of Technology, Department of Mathematics and Computer Science P.O. Box 513, 5600 MB Eindhoven, The Netherlands emails: [email protected], [email protected]
GEORG PROKERT
Affiliation:
Eindhoven University of Technology, Department of Mathematics and Computer Science P.O. Box 513, 5600 MB Eindhoven, The Netherlands emails: [email protected], [email protected]
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Abstract

We present a stability result for a class of non-trivial self-similarly vanishing solutions to a 3D Hele-Shaw moving boundary problem with surface tension and single-point suction. These solutions are domains that bifurcate from the trivial spherical solution. The moving domains have a geometric centre located at the suction point and they are axially symmetric. We show stability with respect to perturbations that preserve these properties.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 20 , Issue 4 , August 2009 , pp. 343 - 362
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Chen, X. & Friedman, A. (2003) A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth. SIAM J. Math. Anal. 35 (4), 974–986 (electronic).Google Scholar
[2]Crandall, M. G. & Rabinowitz, P. H. (1973) Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rational Mech. Anal. 52, 161–180.CrossRefGoogle Scholar
[3]Dierkes, U., Hildebrandt, S., Küster, A. & Wohlrab, O. 1992 Minimal Surfaces. I. Boundary Value Problems, Vol. 295 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Artin, M. et al. (editors), Springer-Verlag, Berlin.Google Scholar
[4]Elliott, C. M. & Ockendon, J. R. (1982) Weak and Variational Methods for Moving Boundary Problems, Vol. 59 of Research Notes in Mathematics, Jeffrey, A. & Douglas, R. G. (editors), Pitman (Advanced Publishing Program), Boston, MA.Google Scholar
[5]Escher, J. & Simonett, G. (1995) Maximal regularity for a free boundary problem. NoDEA Nonlinear Differential Equations Appl. 2 (4), 463–510.Google Scholar
[6]Escher, J. & Simonett, G. (1996) Analyticity of the interface in a free boundary problem. Math. Ann. 305 (3), 439–459.Google Scholar
[7]Fontelos, M. A. & Friedman, A. (2004) Symmetry-breaking bifurcations of charged drops. Arch. Ration. Mech. Anal. 172 (2) (2004), 267–294.CrossRefGoogle Scholar
[8]Friedman, A. & Hu, B. (2007) Bifurcation for a free boundary problem modeling tumor growth by Stokes equation. SIAM J. Math. Anal. 39 (1), 174–194.CrossRefGoogle Scholar
[9]Friedman, A. & Reitich, F. (2002) Quasi-static motion of a capillary drop. II. The three-dimensional case. J. Differential Equations 186 (2), 509–557.Google Scholar
[10]Hele-Shaw, J. H. S. (1898) The flow of water. Nature 58, 34–36.Google Scholar
[11]Hill, E. L. (1954) The theory of vector spherical harmonics. Am. J. Phys. 22, 211–214.Google Scholar
[12]Howison, S. (1998, June 1) Hele-shaw articles, http://people.maths.ox.ac.uk/~howison/hele-shaw/.Google Scholar
[13]Lunardi, A. (1995) Analytic semigroups and optimal regularity in parabolic problems. In: Progress in Nonlinear Differential Equations and their Applications, Vol. 16, Birkhäuser Verlag, Basel.Google Scholar
[14]Messiah, A. (1962) Quantum Mechanics Vol. 2, North Holland, Amsterdam.Google Scholar
[15]Prokert, G. & Vondenhoff, E. (submitted) Non-trivial self-similar extinction solutions for a 3d hele-shaw suction problem (electronic), ftp://ftp.win.tue.nl/pub/rana/rana07-19.pdf.Google Scholar
[16]Renardy, M. & Rogers, R. C. (1993) An Introduction to Partial Differential Equations, Vol. 13 of Texts in Applied Mathematics, John, F., Marsden, J. E., Sirovich, L., Golubitsky, M., & Jäger, W. (editors), Springer-Verlag, New York.Google Scholar
[17]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.Google Scholar
[18]Vondenhoff, E. (submitted) Large-time asymptotics of stokes flow for perturbed balls with injection and suction (electronic), ftp://ftp.win.tue.nl/pub/rana/rana08-20.pdf.Google Scholar
[19]Vondenhoff, E. (submitted) Large-time behaviour of hele-shaw flow with injection or suction for perturbations of balls in n (electronic), ftp://ftp.win.tue.nl/pub/rana/rana08-04.pdf.Google Scholar
[20]Vondenhoff, E. (2008) Long-time asymptotics of hele-shaw flow for perturbed balls with injection and suction (electronic), Interfaces and Free Boundaries, 10 (4), 483–502. ftp://ftp.win.tue.nl/pub/rana/rana06-42.pdf.Google Scholar