Univalent Functions in Two-Dimensional Free Boundary Problems

Acta Applicandae Mathematicae

Volumn 79, Issue 3, 2003, Pages 249-280

  Vasil'ev, Alexander ^a  

^a DEPARTAMENTO DE FÍSICA   (Chile)

Author keywords

Complex analysis; Conformal map; Free boundary problem; Special univalent functions

Indexed keywords

CONFORMAL MAPPING; CRYSTALLIZATION; FUNCTION EVALUATION; GEOMETRY; MATHEMATICAL MODELS; MELTING; SOLIDIFICATION; SURFACE TENSION; VISCOUS FLOW;

FREE BOUNDARY FLOWS; PLANAR FLOWS; UNIVALENT FUNCTIONS;

BOUNDARY LAYERS;

EID: 0345415624     PISSN: 01678019     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:ACAP.0000003674.35389.bb     Document Type: Article

References (83)

1. Aleksandrov, I. A.: Parametric Continuations in the Theory of Univalent Functions, Nauka, Moscow, 1976 (in Russian).
2. Alimov, M., Kornev, K. and Mukhamadullina, G.: Hysteretic effects in the problems of artificial freezing, SIAM J. Appl. Math. 59(2) (1998), 387-410.
3. Avkhadiev, F. G., Aksent'ev, L. A. and Elizarov, M. A.: Sufficient conditions for the finite-valence of analytic functions, and their applications, Itogi Nauki i Tekhniki, Mathematical Analysis, Vol. 25, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987, 3-121; English Transl.: J. Soviet Math. 49(1) (1990), 715-799.
4. Avkhadiev, F. G., Aksent'ev, L. A. and Elizarov, M. A.: Sufficient conditions for the finite-valence of analytic functions, and their applications, Itogi Nauki i Tekhniki, Mathematical Analysis, Vol. 25, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987, 3-121; English Transl.: J. Soviet Math. 49(1) (1990), 715-799.
5. Billingham, J.: Surface-tension-driven flow in fat fluit wedges and cones, J. Fluid Mech. 397 (1999), 45-71.
6. Boussinesq, M. J.: Calcul du pouvoir refroidissant des fluides, J. Math. Pures Appl. Sér. 6 1 (1905), 285-332.
7. Brannan, D. A. and Kirwan, W. E.: On some classes of bounded univalent functions, J. London Math. Soc. (2) 1 (1969), 431-443.
8. Carleson, L. and Makarov, N.: Aggregation in the plane and Loewner's equation, Comm. Math. Phys. 216(3) (2001), 583-607.
9. Ceniceros, H. G., Hou, T. Y. and Si, H.: Numerical study of Hele-Shaw flow with suction, Physics of Fluids 11(9) (1999), 2471-2486.
10. Cummings, L. M., Hohlov, Yu. E., Howison, S. D. and Kornev, K.: Two-dimensional solidification and melting in potential flows, J. Fluid Mech. 378 (1999), 1-18.
11. Davis, S. H.: Theory of Solidification, Cambridge Univ. Press, 2001.
12. Di Benedetto, E. and Friedmann, A.: The ill-posed Hele-Shaw model and the Stefan problem for supercooled water, Trans. Amer. Math. Soc. 282(1) (1984), 183-204.
13. Duchon, J. and Robert, R.: Evolution d'une interface par capillarité et diffusion de volume, 1: existence locale en temps, Ann. Inst. H. Poincaré 1 (1984), 361-378.
14. Duren, P.: Univalent Functions, Springer, New York, 1983.
15. Elliott, C. M. and Janovsky, V.: A variational inequality approach to Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh A 88 (1981), 93-107.
16. Elliott, C. M. and Ockendon, J. R.: Weak and Variational Methods for Moving Boundary Problem, Pitman, London, 1992.
17. Entov, V. M. and Ètingof, P. I.: Bubble contraction in Hele-Shaw cells, Quart. J. Mech. Appl. Math. 44(4) (1991), 507-535.
18. Escher, J. and Simonett, G.: On Hele-Shaw models with surface tension, Math. Res. Lett. 3(4) (1996), 467-474.
19. Escher, J. and Simonett, G.: Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations 2(4) (1997), 619-642.
20. Escher, J. and Simonett, G.: A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations 143(2) (1998), 267-292.
21. Escher, J. and Simonett, G.: Moving surfaces and abstract parabolic evolution equations, In: Topics in Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl. 35, Birkhäuser, Basel, 1999, pp. 183-212.
22. Fejér, L.: Mechanische Quadraturen mit positiven Cotesschen Zahlen, Math. Z. 37(2) (1933), 287-309.
23. Fejér, L. and Szegö, G.: Special conformal mappings, Duke Math. J. 18 (1951), 535-548.
24. Fokas, A. S. and Tanveer, S.: A Hele-Shaw problem and the second Painlevé transcendent, Math. Proc. Cambridge Philos. Soc. 124 (1998), 169-191.
25. Galin, L. A.: Unsteady filtration with a free surface, Dokl. Akad. Nauk USSR 47 (1945), 246-249 (in Russian).
26. Gillow, K. A. and Howison, S. D.: A bibliography of free and moving boundary problems for Hele-Shaw and Stokes flow, Published electronically at URL〉http://www.maths.ox.ac.uk/̃howison/Hele-Shaw.
27. Goldstein, M. E. and Reid, R. I.: Effect of flow on freezing and thawing of saturated porous media, Proc. Roy. Soc. London Ser. A 364 (1978), 45-73.
28. Goluzin, G. M.: Geometric Theory of Functions of a Complex Variable, Transl. Math. Monogr., 26, Amer. Math. Soc., Providence, RI, 1969.
29. Goodman, A. W.: Univalent Functions, Vols I, II, Mariner Publishing Company, Inc., U. South Florida, 1983.
30. Goodman, A. W. and Saff, E. B.: On univalent functions convex in one direction, Proc. Amer. Math. Soc. 73(2) (1979), 183-187.
31. Gordeev, Y. N. and Kornev, K. G: Crystallization in forced flow: The Saffman-Taylor problem, Eur. J. Appl. Math. 10 (1999), 535-545.
32. Gustafsson, B.: On a differential equation arising in a Hele-Shaw flow moving boundary problem, Ark. Mat. 22(1) (1984), 251-268.
33. Gustafsson, B.: Applications of variational inequalities to a moving boundary problem for Hele-Shaw flows, SIAM J. Math. Anal. 16(2) (1985), 279-300.
34. Gustafsson, B.: Properties of some balayage operators, with applications to quadrature domains and moving boundary problems, Nonlinear Anal. 22(10) (1994), 1221-1245.
35. Gustafsson, B., Prokhorov, D. and Vasil'ev, A.: Infinite lifetime for the starlike dynamics in Hele-Shaw cells, Proc. Amer. Math. Soc. (to appear).
36. Guy, H. L.: H. S. Hele-Shaw 1854-1941, Obituary Notices of Fellows of The Royal Society 3(10) (1941), 790-811..
37. Hedenmalm, H. and Shimorin, S.: Hele-Shaw flow on hyperbolic surfaces, J. Math. Pures Appl. (2002) (to appear).
38. Hele-Shaw, H. S.: The flow of water, Nature 58(1489) (1898), 33-36.
39. Hele-Shaw, H. S.: On the motion of a viscous fluid between two parallel plates, Trans. Roy. Inst. Nov. Archit. London 40 (1898), 218.
40. Hengartner, W. and Schober, G.: A remark on level curves for domain? convex in one direction, Collection of articles dedicated to Eberhard Hopf on the occasion of his 70th birthday, Applicable Anal. 3 (1973), 101-106.
41. Hohlov, Yu. E. and Howison, S. D.: On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flows, Quart. Appl. Math. 54(4) (1994), 777-789.
42. Hohlov, Yu. E. and Reissig, M.: On classical solvability for the Hele-Shaw moving boundary problem with kinetic undercooling regularization, Eur. J. Appl. Math. 6 (1995), 421-439.
43. Hohlov, Yu. E., Prokhorov, D. V. and Vasil'ev, A.: On geometrical properties of free boundaries in the Hele-Shaw flow moving boundary problem, Lobachevskiǐ J. Math. 1 (1998), 3-13 (electronic).
44. Howison, S. D.: Fingering in Hele-Shaw cells, J. Fluid Mech. 167 (1986), 439-453.
45. Howison, S. D.: Bubble growth in porous media and Hele-Shaw cells, Proc. Roy. Soc. Edinburgh Ser. A 102(1-2) (1986), 141-148.
46. Howison, S.D.: Cusp development in Hele-Shaw flow with a free surface, SIAM J. Appl. Math. 46(1) (1986), 20-26.
47. Howison, S. D.: Complex variable methods in Hele-Shaw moving boundary problems, Eur. J. Appl. Math. 3(3) (1992), 209-224.
48. Howison, S. D., Morgan, J. D. and Ockendon, J. R.: A class of codimension-two free boundary broblems, Eur. J. Appl. Math. 3(3) (1992), 209-224.
49. Ivantzov, G. P.: The temperature field around a spherical, cylindrical, or point crystal growing in a cooling solution, Dokl. Akad. Nauk USSR 58 (1947), 567-569 (in Russian).
50. Kaplan, W.: Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169-185.
51. Kimura, M.: Time local existence of a moving boundary of the Hele-Shaw flow with suction, Eur. J. Appl. Math. 10(6) (1999), 581-605.
52. Kornev, K. and Mukhamadullina, G.: Mathematical theory of freezing in porous media, Proc. Roy. Soc. London Ser. A 447 (1994), 281-297.
53. Kornev, K. and Vasil'ev, A.: Geometric properties of the solutions of a Hele-Shaw type equation, Proc. Amer. Math. Soc. 128(9) (2000), 2683-2685.
54. Maksimov, V. A.: On the stable shape of bodies solidified around a could sourse in a stream fluid, Izv. Akad. Nauk USSR, Mekhanika 4(41) (1965), 41-45 (in Russian).
55. Maksimov, V. A.: On the determination of the shape of bodies formed by solidification of the fluid phase of the stream, Prikl. Mat. Mekh. (J. Appl. Math. Mech.) 40(264) (1976), 264-272.
56. Marx, A.: Untersuchungen über schlichte Abbildungen, Math. Ann. 107 (1932/33), 40-67.
57. McLean, J. W. and Saffman, P. G.: The effect of surface tension on the shape of fingers in a Hele Shaw cell, J. Fluid Mech. 102 (1981), 455-469.
58. Mullins, W. W.: Grain boundary grooving by volume diffusion, Trans. Metallurgical Society of AIME 218 (1960), 354-361.
59. Muskhelishvili, N. I.: Singular Integral Equations, P. Noordhoff, Groningen, The Netherlands, 1953.
60. Ockendon, H. and Ockendon, J. R.: Viscous Flow, Cambridge Univ. Press, 1995.
61. Park, C.-W. and Homsy, G. M.: Two-phase displacement in Hele-Shaw cells: Theory, J. Fluid Mech. 139 (1984), 291-308.
62. Pólya, G. and Schiffer, M.: Sur la représentation conforme de l'extérieur d'une courbe fermée convexe, C. R. Acad. Sci. Paris 248 (1959), 2837-2839.
63. Polubarinova-Kochina, P. Ya.: On a problem of the motion of the contour of a petroleum shell, Dokl. Akad. Nauk USSR 47(4) (1945), 254-257 (in Russian).
64. Polubarinova-Kochina, P. Ya.: Concerning unsteady motions in the theory of filtration, Prikl. Mat. Mech. 9(1) (1945), 79-90 (in Russian).
65. Pommerenke, Ch.: Univalent Functions, with a chapter on quadratic differentials by G. Jensen, Vandenhoeck & Ruprecht, Göttingen, 1975.
66. Pommerenke, Ch.: Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992.
67. Prokert, G.: Existence results for Hele-Shaw flow driven by surface tension, Eur. J. Appl. Math. 9(2) (1998), 195-221.
68. Prokhorov, D. V.: Level lines of functions that are convex in the direction of an axis, Mat. Zametki (Math. Notes) 44(4) (1988), 523-527 (in Russian).
69. Prokhorov, D. and Vasil'ev, A.: Convex dynamics in Hele-Shaw cells, Internat. J. Math. Math. Sci. 31(11) (2002), 639-650.
70. Qing Nie and Fei Ran Tian: Singularities in Hele-Shaw flows, SIAM J. Appl. Math. 58(1) (1998), 34-54.
71. Reissig, M. and Von Wolfersdorf, L.: A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane, Ark. Mat. 31(1) (1993), 101-116.
72. Richardson, S.: Hele-Shaw flows with a free boundary produced by the injecton of fluid into a narrow channel, J. Fluid Mech. 56(4) (1972), 609-618.
73. Richardson, S.: On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flows, Quart. Appl. Math. 55(2) (1997), 313-319.
74. Robertson, M. S.: Analitic functions starlike in one direction, Amer. J. Math. 58 (1936), 465-472.
75. Ruscheweyh, S. and Salinas, L. C.: On the preservation of direction-convexity and the Goodman-Saff conjecture, Ann. Acad. Sci. Fenn. Ser. A-I Math. 14(1) (1989), 63-73.
76. Saffman, P. G. and Taylor, G. I.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. London Ser. A 245(281) (1958), 312-329.
77. Siegel, M., Tanveer, S. and Dai, W. S.: Singular effects of surface tension in evolving Hele-Shaw flows, J. Fluid Mech. 323 (1996), 201-236.
78. Stankiewicz, J.: Some remarks concerning starlike functions, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 18 (1970), 143-146.
79. Tanveer, S.: Evolution of Hele-Shaw interface for small surface tension, Philos. Trans. Roy. Soc. London Ser. A 343(1668) (1993), 155-204.
80. Tian, F.-R.: Hele-Shaw problems in multidimensional spaces, J. Nonlinear Sci. 10(2) (2000), 275-290.
81. Vasil'ev, A.: Univalent functions in the dynamics of viscows flows, Comp. Methods Funct. Theory 1(2) (2001), 311-337.
82. Vasil'ev, A. and Markina, I.: On the geometry of Hele-Shaw flows with small surface tension, Interfaces and Free Boundaries 5(2) (2003), 183-192.
83. Vinogradov, Yu. P. and Kufarev, P. P.: On a problem of filtration, Akad. Nauk SSSR. Prikl. Mat. Meh. 12 (1948), 181-198 (in Russian).