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A variational inequality approach to Hele-Shaw flow with a moving boundary

Published online by Cambridge University Press:  14 November 2011

C. M. Elliott  and
V. Janovský
C. M. Elliott
Affiliation:
Imperial College, London
V. Janovský
Affiliation:
Charles University, Prague
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Synopsis

This paper is concerned with the study of a mathematical model of the injection of fluid into a finite Hele–Shaw cell. The mathematical problem is one of solving Laplace's equation in an unknown region whose boundary changes with time. By a transformation of the dependent variable, an elliptic variational inequality formulation of the moving boundary problem is obtained. The variational inequality is shown to have a unique solution up to the time at which the cell is filled. Regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 88 , Issue 1-2 , 1981 , pp. 93 - 107
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Agmon, S., Douglis, A., and Nirenberg, L.. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I. Comm. Pure Appl. Math. 12 (1959), 623–727.CrossRefGoogle Scholar
2Bramble, J. H. and Hilbert, S. R.. Estimates of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970), 112–124.CrossRefGoogle Scholar
3Brezis, H.. Problems unilateraux. J. Math. Pures Appl. 51 (1972), 1–168.Google Scholar
4Duvaut, G.. Resolution d'un probleme de Stefan. C.R. Acad. Sci. Paris Ser. A 276 (1973), 1461–1463.Google Scholar
5Elliott, C. M.. The numerical solution of an electrochemical machining moving boundary problem via a variational inequality formulation. J. Inst. Math. Appl. 25 (1980), 121–131.CrossRefGoogle Scholar
6Elliott, C. M. and Janovsky, V.. Proceedings of MAFELAP 1978, ed. Whiteman, J. R. (London: Academic, 1979).Google Scholar
7Lions, J. L.. Optimal Control of Systems Governed by Partial Differential Equations (Berlin: Springer, 1971).CrossRefGoogle Scholar
8Lions, J. L. and Stampacchia, G.. Variational inequalities. Comm. Pure Appl. Math. 20 (1967), 493–519.CrossRefGoogle Scholar
9Richardson, S.. Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56 (1972), 609–618.CrossRefGoogle Scholar