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2D Free Boundary Value Problems

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Advances in Applied Analysis

Part of the book series: Trends in Mathematics ((TM))

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Abstract

Two-dimensional free boundary value problems are considered. Different models and their connections are discussed. Main attention is paid to the celebrated Hele-Shaw model. Complex-analytic methods are applied to its study.

Mathematics Subject Classification (2010). 76D27, 46E15, 35A10, 47G10, 30E25.

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References

  1. M.V. Dubatovskaya, E.V. Pesetskaya, S.V. Rogosin, On Hele-Shaw flow in a bounded domain, Proceedings of N.I.Lobachevsky Mathematical Center. Kazan, 14, 2002, 113-129.

    Google Scholar 

  2. J. Duchon, R. Robert, Estimation d’opérateur intégraux du type de Cauchy dans les eschelles d’Ovsjannikov et application, Ann. Inst. Fourier, Grenoble, 36, No. 1,1986, 83-95.

    Google Scholar 

  3. F.D. Gakhov, Boundary Value Problems, Nauka, Moscow, (3rd ed.)1977 (in Russian).

    Google Scholar 

  4. L.A. Galin, Unsteady filtration with the free boundary, Dokl. Akad. Nauk SSSR, 47, No. 4, 1945, 246-249 (in Russian). 236

    Google Scholar 

  5. R.P. Gilbert, Zhenyuan Xu, Several Hele-Shaw flows associated with injection mold-ing, in Complex Analysis and its Applications, (C.C. Yang, G.C. Wen, K.Y. Li and Y.M. Chiang (eds.)), Longman, Harlow, 1994, pp. 26-37.

    Google Scholar 

  6. I. Gohberg, N. Krupnik, Introduction to the Theory of One-Dimensional SingularIntegral Equations, vol. I, II, Birkhäuser Verlag, Basel-Boston-Berlin, 1992.

    Google Scholar 

  7. G.M. Goluzin, Geometric Theory of Functions of Complex Variables, Amer. Math.Soc., Providence, RI, 1969.

    Google Scholar 

  8. B. Gustafsson, On a differential equation arising in a Hele-Shaw flow moving bound-ary problem, Arkiv för Matematik, 22, 1984, 251-268. 234

    Google Scholar 

  9. B. Gustafsson, D. Prokhorov, A. Vasil’ev, Infinite lifetime for the starlike dynamicsin Hele-Shaw cells, Proc. Amer. Math. Soc., 132, No. 9, 2004, 2661-2669. 228

    Google Scholar 

  10. B. Gustafsson, A. Vasil’ev, Conformal and Potential Analysis in Hele-Shaw Cells, Birkhäuser Verlag, Basel-Stockholm, 2006. 227, 228, 231, 234

    Google Scholar 

  11. H.S. Hele-Shaw, The flow of water, Nature. 58 (1489), 1898, 33-36. 225

    Google Scholar 

  12. Yu.E. Hohlov, S.D. Howison, On the classification of solutions to the zero-surface-tension model for the Hele-Shaw free boundary flows, Quart. J. Appl. Math., 51, No. 4,1993, 777-789. 226, 227, 228, 236

    Google Scholar 

  13. Yu.E. Hohlov, S.D. Howison, C. Huntingford, J.R. Ockendon, A.A. Lacey, A model for non-smooth free boundaries in Hele-Shaw flows, Quart. J. Mech. Math., 47, No. 1,1994, 107-128. 236

    Google Scholar 

  14. S.D. Howison, Complex variable methods in Hele-Shaw moving boundary problems. Euro. J. Appl. Math., 3, 1992, 209-224. 236

    Google Scholar 

  15. O.A. Ladyzhenskaya, N.N. Uraletseva, Linear and Quasilinear Elliptic Equations, Nauka, Moscow, 1964 (in Russian; Engl. transl. by Academic Press, New York, 1968). 243

    Google Scholar 

  16. L.S. Leibenzon, Oil producing mechanics, in two volumes, Nefteizdat, Moscow, 1934 (in Russian). 232

    Google Scholar 

  17. J.A. McGeough, H. Rasmussen, On the derivation of the quasi-steady model in electro-chamical machining, J. Inst. Math. and Appl.., 13, No. 1, 1974, 13-21. 226

    Google Scholar 

  18. A.M. Meirmanov, Stefan Problem, Nauka, Moscow, 1986 (in Russian). 226

    Google Scholar 

  19. A.M. Meirmanov, B. Zaltzman, Global in time solution to the Hele-Shaw problem with a change of topology. Euro. J. Appl. Math., 13, No. 4, 2002, 431-447.

    Google Scholar 

  20. M. Muskat, The flow of the homogeneous fluids through porous media. McGraw-Hill, N.Y., 1937. 225

    Google Scholar 

  21. N.I. Muskhelishvili, Singular Integral Equations, P. Noordhoff N.V., Groningen, 1953.

    Google Scholar 

  22. L. Nirenberg, An abstract form of the nonlinear Cauchy-Kowalevski theorem, J. Differential Geom., 6, 1972, 561-576. 235

    Google Scholar 

  23. T. Nishida, A note on a theorem of Nirenberg, J. Differential Geom., 12, 1977, 629-235

    Google Scholar 

  24. J.R. Ockendon, S.D. Howison, P.Ja. Kochina and Hele-Shaw in modern mathematics, natural sciences and technology, Prikl. Mat. i Mech., 66, vyp. 3, 2002, 515-524 (in Russian). 225

    Google Scholar 

  25. H. Oertel (ed.), Prandtl’s essentials of fluid mechanics, Springer Verlag, New York, 2004.

    Google Scholar 

  26. L.V. Ovsjannikov, A singular operator in a scale of Banach spaces, Dokl. AN SSSR, 163, No. 4, 1965, 819-822 (in Russian). 235

    Google Scholar 

  27. C.P.Please, G. Pettet, D.L.S. McElwain, A new approach to modelling the formation of necrotic regions in tumours, Appl. Math. Letter, 11, No. 3, 1998, 89-94. 226

    Google Scholar 

  28. P.Ya. Polubarinova-Kochina, On the motion of the oil contour, Dokl. Akad. Nauk SSSR, 47, No. 4, 1945, 254-257 (in Russian). 225, 226, 227, 236

    Google Scholar 

  29. M. Reissig, About a nonstationary mixed problem for holomorphic functions arising by the study of a potential flow past a circular cylinder with permeable surface, Math. Nachr., 164, 1993, 283-297.

    Google Scholar 

  30. M. Reissig, L. von Wolfersdorf, A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane, Arkiv för Math., 31, No. 1, 1993, 101-110. 228, 237, 238

    Google Scholar 

  31. M. Reissig, The existence and uniqueness of analytic solutions for moving boundary value problem for Hele-Shaw flows in the plane, Nonlinear Anal. Theory, Methods & Appl., 23, No. 5, 1994, 565-576. 228

    Google Scholar 

  32. M. Reissig, A generalized theorem of Peano in scale of Banach spaces with completely continuous imbedding, Funkc. Ekvacioj, 37, 1994, 521-530. 228

    Google Scholar 

  33. M. Reissig, Leray-Volevich conditions for systems of abstract evolution equations of Nirenberg/ Nishida type, Tsukuba J. Math., 18, No. 1, 1994, 193-202. 228

    Google Scholar 

  34. M. Reissig, F. Hübner, Analytical and numerical treatment of Hele-Shaw models with and without regularization, in Generalized Analytic Functions (H. Florian et al., eds.), Kluwer AP, Amsterdam, 1998, pp. 271-187. 236

    Google Scholar 

  35. M. Reissig and S.V. Rogosin, with an appendix of F. Huebner, Analytical and numer-ical treatment of a complex model for Hele-Shaw moving boundary value problems with kinetic undercooling regularization, Euro J. Appl. Math., 10, 1999, 561-579. 235, 236

    Google Scholar 

  36. S D. Richardson, Hele-Shaw flows with a free boundary produced by injection of fluid into a narrow channel. J. Fluid Mech., 56, 1972, 609-618. 233

    Google Scholar 

  37. S.V. Rogosin, On a scale of Banach spaces. Proceedings of Institute of Mathematics. Minsk, 12, No. 1, 2004, 126-133.

    Google Scholar 

  38. S.V. Rogosin, Hele-Shaw Moving Boundary Value Problem in a Bounded Domain. Local in Time Solvability, Complex Variables, 50, No. 7-11, 2005, 745-764.

    Google Scholar 

  39. L.A. Romero, The Fingering Problem in a Hele-Shaw Cell. PhD Thesis, California Institute of Technology, 1981. 236

    Google Scholar 

  40. P.G. Saffman, Viscous fingering in Hele-Shaw cells. J. Fluid Mech., 173, 1986, 73-94. 227

    Google Scholar 

  41. P.G. Saffman, G.I. Taylor, The penetration of a fluid into porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. London. Ser. A, 245, No.1242, 1958, 312-329. 227

    Google Scholar 

  42. P.G. Saffman, Viscous fingering in Hele-Shaw cells, J. Fluid Mech., 173, 1986, 73-94. 236

    Google Scholar 

  43. V.A. Solonnikov, Boundary and initial-boundary value problems for the Navier-Stokes equations in domains with non-compact boundaries / In: Mathematical Topics in Fluid Mechanics (J.-F. Rodrigues, A. Sequeira (eds.)). Longman, Harlow, 1993, pp. 117-162.

    Google Scholar 

  44. S. Tanveer, Evolution of Hele-Shaw interface for small surface tension, Phil. Trans.R. Soc. Lond., A 343, 1993, 155-204. 236

    Google Scholar 

  45. W. Tutschke, Solution of Initial Value Problems in Classes of Generalized Analytic Functions, Teubner, Leipzig, 1989. 235, 239

    Google Scholar 

  46. A.N. Varchenko, P.I. Etingof, Why the Boundary of a Round Drop Becomes an Inverse Image of an Ellipse, Nauka, Moscow, 1995 (in Russian). 227

    Google Scholar 

  47. A. Vasil’ev, From the Hele-Shaw Experiment to Integrable Systems: A Historical Overview. Coml. Anal. Oper. Theory, 3,2009, 551-585. 225

    Google Scholar 

  48. Yu.P. Vinogradov, P.P. Kufarev, On a filtration problem. Prikl. Mat. Mech. 12, 1948, 181-198 (in Russian)(English translation: University of Delaware, Applied Mathematics Institute, Technical Report 182A, 1984). 233, 234

    Google Scholar 

  49. P. Willmott, S.D. Howison, J. Dewynne, The Mathematics of Financial Derivatives: A Student Introduction, Cambridge University Press, Cambridge, 1996. 226

    Google Scholar 

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Authors and Affiliations

  1. Department of Economics, Belarusian State University, Nezavisimosti ave, 4, 220030, Minsk, Belarus

    Sergei V. Rogosin

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  1. Sergei V. Rogosin
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Correspondence to Sergei V. Rogosin .

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  1. , Department of Economics, Belarusian State University, Nezavisimosti ave 4, Minsk, 220030, Belarus

    Sergei V. Rogosin

  2. , Department of Economics, Belarus State University, Nezavisimosti ave 4, Minsk, 220030, Belarus

    Anna A. Koroleva

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Rogosin, S.V. (2012). 2D Free Boundary Value Problems. In: Rogosin, S., Koroleva, A. (eds) Advances in Applied Analysis. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0417-2_6

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