Optimal Control by Multipoles in the Hele-Shaw Problem
The two-dimensional Hele-Shaw problem for a fluid spot with free boundary can be solved using the Polubarinova–Galin equation. The main condition of its applicability is the smoothness of the spot boundary. In the sink-case, this problem is not well-posed and the boundary loses smoothness within finite time—the only exception being the disk centred on the sink. An extensive literature deals with the study of the Hele-Shaw problem with non-smooth boundary or with surface tension, but the problem remains open. In our work, we propose to study this flow from a control point of view, by introducing an analogue of multipoles (term taken from the theory of electromagnetic fields). This allows us to control the shape of the spot and to avoid non-smoothness phenomenon on its border. For any polynomial contours, we demonstrate how all the fluid can be extracted, while the border remains smooth until the very end. We find, in particular, sufficient conditions for controllability and a link between Richardson’s moments and Polubarinova–Galin equation.
KeywordsHele-Shaw flow multipoles controllability optimal control
Mathematics Subject ClassificationPrimary 49K15 Secondary 76D27
Unable to display preview. Download preview PDF.
- 1.Darcy, H.: Les fontaines publiques de la ville de Dijon: exposition et application des principes suivre et des formules employer dans les questions de distribution d’eau. Victor Dalmont, Paris (1856)Google Scholar
- 6.Hörmander L.: Linear Partial Differential Operators, Fourth Printing. Springer, Berlin (1976)Google Scholar
- 7.Kuznetsova, O.S.: On polynomial solutions of the Hele-Shaw problem. Sib. Mat. Zh. 42(5), 1084–1093 (2001). [Engl. transl.: Sib. Math. J. 42(5), 907–915 (2001)]Google Scholar
- 8.Lamb, H.: Hydrodynamics, 6th edn.. Cambridge Univ. Press., Cambridge (1932)Google Scholar
- 9.Lavrentiev, M.A., Shabat, B.V.: Methods of Complex Function Theory. Nauka, Moscow (1987)Google Scholar
- 10.Leibenson, L.S.: Oil Producing Mechanics, Part II. Moscow, Neftizdat (1934)Google Scholar
- 12.Polubarinova-Kochina, P.Ya.: Concerning unsteady motions in the theory of filtration. Prik. Mat. Mech. 9(1), 79-90 (1945). (In Russian)Google Scholar
- 15.Stokes, G.G.: Mathematical proof of the identity of the stream lines obtained by means of viscous film with those of a perfect fluid moving in two dimensions. Br. Ass. Rep. 143 (Papers, V, 278) (1898)Google Scholar
- 16.Varchenko, A.N., Etingof, P.I.: Why the Boundary of a Round Drop Becomes a Curve of Order Four? University Lecture Series, vol. 3. AMS, New York (1992)Google Scholar
- 17.Zelikina, L.F.: K voprosu o reguliarnom sinteze (1982). (In Russian)Google Scholar