zbMATH — the first resource for mathematics

Attractors-repellers in the space of contours in the Stokes-Leibenson problem for Hele-Shaw flows. (English. Russian original) Zbl 1338.37128
J. Math. Sci., New York 189, No. 4, 568-581 (2013); translation from Probl. Mat. Anal. 69, 23-34 (2013).
Summary: It is shown that, in the space of quasicontours playing a role of free boundaries in the Stokes-Leibenson problem, there is a manifold of codimension 1 such that some points of this manifold are attractors in the case of sink and repellers in the case of source, whereas, on the contrary, other points are repellers in the case of sink and attractor in the case of source.
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
76D27 Other free boundary flows; Hele-Shaw flows
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI
[1] G. G. Stokes, ”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,” Brit. Ass. Rep. 143–144 (1898). · JFM 29.0645.04
[2] L. S. Leibenson, Oil Producing Mechanics. Part II [in Russian], Nefteizdat, Moscow (1934). · JFM 60.1387.03
[3] L. A. Galin, ”Unsteady filtration with a free surface,” C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 47, 246–249 (1945). · Zbl 0061.46202
[4] P. J. Polubarinova–Kotschina, ”On the displacement of the oilbearing contour,” C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 47, 250–254 (1945). · Zbl 0061.46112
[5] P. Ya. Polubarinova-Kochina [P. J. Polubarinova–Kotschina], ”On unsteady motions in filtration theory: on the displacement of the oilbearing contour” [in Russian], Prikl. Mat. Mekh. 9, No. 1, 79–90 (1945). · Zbl 0061.46112
[6] P. P. Kufarev, ”Solution of a problem on the oilbearing contour for a circle” [in Russian], Dokl. AN SSSR 60, No. 8, 1333–1334 (1948).
[7] S. Richardson, ”Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel,” J. Fluid Mech. 56, 609–618 (1972). · Zbl 0256.76024 · doi:10.1017/S0022112072002551
[8] Yu. P. Vinogradov and P. P. Kufarev, ”On a problem of filtration” [in Russian], Prikl. Mat. Mekh. 12, No. 2, 181–198 (1948). · Zbl 0032.27901
[9] P. Ya. Kochina [P. Ya. Polubarinova-Kochina], Selected Works. Hydrodynamics and Filtration Theory [in Russian], Nauka, Moscow (1991).
[10] J. R. Ockendon and S. D. Howison, ”Kochina and Hele-Shaw in modern mathematics, natural sciences and industry” [in Russian], Prikl. Mat. Mekh. 66, No. 3, 515–524 (2002); English transl.: J. Appl. Math. Mech. 66, No. 3, 505–512 (2002). · Zbl 1066.01513 · doi:10.1016/S0021-8928(02)00060-6
[11] B. Gustafsson, ”Applications of variational inequalities to a moving boundary problem for Hele-Shaw flows,” SIAM J. Math. Anal. 16, 279–300 (1985). · Zbl 0605.76043 · doi:10.1137/0516021
[12] B. Gustafsson and A. Vasil’ev, Conformal and Potential Analysis in Hele-Shaw cells, Birkäuser, Basel (2006). · Zbl 1122.76002
[13] S. D. Howison and J. R. Ockendon, ”Papers from the conference held in Oxford,” Euro. J. Appl. Math. 10, 511–709 (1999). · Zbl 0951.00024 · doi:10.1017/S0956792599009870
[14] A. M. Meirmanov and B. Zaltzman, ”Global in time solution to the Hele-Shaw problem with a change of topology,” Euro. J. Appl. Math. 13, 431–447 (2002). · Zbl 1068.76022
[15] L. N. Aleksandrov, Kinetics of Crystallization and Recrystallization of Semiconducting Films [in Russian], Nauka, Novosibirsk (1985).
[16] E. N. Kablov, The Casting Blades of Gas Turbine Engines [in Russian], MISIS, Moscow (2001).
[17] B. G. Thomas and Ch. Beckermann, Modeling of Casting, Welding Advanced Solidification Processes, San Diego, California (1998).
[18] P. I. Plotnikov and V. N. Starovojtov, ”The Stefan problem with surface tension as the limit of a phase field model” [in Russian], Differ. Uravn. 29, No. 3, 461–471 (1993); English transl.: Differ. Equations 29, No. 3, 395–404 (1993).
[19] G. Caginalp, ”Stefan and Hele-Shaw type problems as asymptotics limits of the phase field equations,” Phys. Rev. A (3) 39, No. 11, 5887–5896 (1989). · Zbl 1027.80505 · doi:10.1103/PhysRevA.39.5887
[20] G. Caginalp and X. Chen, ”Convergence of the phase field model to its sharp interface limits,” Euro. J. Appl. Math. 12, 20–42 (2000).
[21] V. G. Danilov, G. A. Omel’yanov, and E. V. Radkevich, ”Asymptotics of the solution to the phase field system and the modified Stefan problem” [in Russian], Differ. Uravn. 31, No. 3, 483–491 (1995); English transl.: Differ. Equations 31, No. 3, 446–454 (1995). · Zbl 0855.35134
[22] J. Escher and G. Simonett, ”Classical solutions of multidimensional Hele-Shaw models,” SIAM J. Math. Anal. 28, No. 5, 1028–1047 (1997). · Zbl 0888.35142 · doi:10.1137/S0036141095291919
[23] G. Prokert, ”On evolution equations for moving domains,” Z. Anal. Anwend. 18, No. 1, 67–95 (1999). · Zbl 0920.35174 · doi:10.4171/ZAA/870
[24] A. S. Demidov, ”Functional geometric method for solving free boundary problems for harmonic functions” [in Russian], Usp. Mat. Nauk 65 (1), 3–96 (2010); English transl.: Russ. Math. Surv. 65 (1), 1–94 (2010). · Zbl 1205.35343 · doi:10.1070/RM2010v065n01ABEH004661
[25] A. S. Demidov, ”Evolution of the perturbation of a circle in the Stokes–Leibenson problem for the Hele-Shaw flow” [in Russian], Sovrem. Mat. Prilozh. 2, 3–24 (2003); English transl.: J. Math. Sci., New York 123, No. 5, 4381–4403 (2004). · Zbl 1072.76025 · doi:10.1023/B:JOTH.0000040301.53259.05
[26] A. S. Demidov, ”Evolution of the perturbation of a circle in the Stokes–Leibenson problem for the Hele-Shaw flow. Part II” [in Russian], Sovrem. Mat. Prilozh. 24, 51–65 (2005); English transl.: J. Math. Sci., New York, 139, No. 6, 7064–7078 (2006). · Zbl 1127.76021 · doi:10.1007/s10958-006-0406-1
[27] A. S. Demidov, ”On the evolution of a weak perturbation of a circle in the problem of a Hele-shaw flow” [in Russian], Usp. Mat. Nauk 57, No. 6, 177–178 (2002); English transl.: Russ. Math. Surv. 57, No. 6, 1212–1214 (2002). · Zbl 1062.35081 · doi:10.1070/RM2002v057n06ABEH000580
[28] V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential Equations [in Russian], IRT, Izhevsk (2000).
[29] A. Antontsev, A. M. Meirmanov, and V. Yurinsky, Hele-Shaw Flow in Two Dimensions: Global-in-Time Classical Solutions, Preprint No. 6, Universidade da Beira Interior, Portugal (1999).
[30] A. S. Demidov and J.-P. Lohéac, A Quasicontour Model of Stokes–Leibenson Problem for Hele-Shaw Flows, Preprint No. 328, CNRS UMR 5585 (2001).
[31] A. S. Demidov and J.-P. Lohéac, ”On the evolution near some attractive manifold in a problem for the Hele-Shaw flows,” In: Abstracts of International Conference ”Mathematical Ideas of P. L. Chebyshev and their Application for Modern Problems of Natural Sciences” (Obninsk, Russia, May 2002, 14–18), pp. 37–38 (2002).
[32] A. S. Demidov and J.-P. Lohéac, ’The Stokes–Leibenson problem for Hele-Shaw flows,” In: Patterns and Waves. Papers from the Conference on Patterns and Waves: Theory and Applications, St. Petersburg, Russia, July 8–13, 2002, pp. 103–124, AkademPrint, St. Petersburg (2003).
[33] O. A. Vasil’eva and A. S. Demidov, ”The finite point model of the Stokes–Leibenson problem for Hele-Shaw flow” [in Russian], Fundam. Prikl. Mat. 5, No. 1, 67–84 (1999).
[34] A. S. Demidov, ”Polygonal model of Hele-Shaw flow” [in Russian], Usp. Mat. Nauk 53, No. 4, 195–196 (1998). · doi:10.4213/rm20
[35] P. Ya. Kochina and A. R. Shirich, ”On the displacement of the oilbearing contour (experiment)” [in Russian], Izv. AN SSSR, Otd. Tekhn. Nauk 11, 105–107 (1954).
[36] P. G. Saffman and G. I. Taylor, ”The penetration of a fluid into a porous medium of Hele-Shaw cell containing a more viscous liquid,” Proc. Royal Soc. A 245, 312–329 (1958). · Zbl 0086.41603 · doi:10.1098/rspa.1958.0085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.