Far Eastern Mathematical Journal

To content of the issue


Control problems for stationary models of magnetic hydrodynamics of a viscous incompressible fluid


G. V. Alekseev

2005, issue 1-2, Ñ. 117–145


Abstract
A general technique is proposed for analysing control problems for stationary magnetohydrodynamic models of a viscous incompressible fluid. The review of results of study of boundary and control problems for stationary MHD models is given.

Keywords:
magnetic hydrodynamics, viscous fluid, boundary value problems, control problems, optimality system, solvability, local uniqueness

Download the article (PDF-file)

References

[1] A. G. Kulikovskij, G. A. Lyubimov, Magnitnaya gidrodinamika, Fizmatgiz, M., 1962, 248 s.
[2] Dzh. Sherklif, Kurs magnitnoj gidrodinamiki, Mir, M., 1967, 320 s.
[3] L. D. Landau, E. M. Lifshic, Teoreticheskaya fizika, t. 8, E'lektrodinamika sploshnyx sred, Nauka, M., 1982, 624 s.
[4] V. V. Nikol'skij, E'lektrodinamika i rasprostranenie voln, Nauka, M., 1978, 544 s.
[5] O. A. Ladyzhenskaya, V. A. Solonnikov, “O razreshimosti nestacionarnyx zadach v magnitnoj gidrodinamike”, Dokl. AN SSSR, 124 (1959), 26–28.
[6] O. A. Ladyzhenskaya, V. A. Solonnikov, “Reshenie nekotoryx nestacionarnyx zadach magnitnoj gidrodinamiki dlya vyazkoj neszhimaemoj zhidkosti”, Tr. Matem. in-ta im. V. A. Steklova, 59, 1960, 115–174.
[7] V. A. Solonnikov, “O nekotoryx stacionarnyx kraevyx zadachax magnitnoj gidrodinamiki”, Tr. Matem. in-ta im. V. A. Steklova, 59, 1960, 174–187.
[8] R. H. Dyer, D. E. Edmunds, “A uniqueness theorem in magnetohydrodynamics”, Arch. Rat. Mech. Anal., 8 (1961), 254–262.
[9] R. H. Dyer, D. E. Edmunds, “On the existence of solutions of the equations of magnetohydro-dynamics”, Arch. Rat. Mech. Anal., 9 (1962), 403–410.
[10] D. E. Edmunds, “Sur I'unicite des solutions des equations de la magnetohydrodynamique”, C.r. Acad. Sci. Paris, 254 (1962), 1377–1379.
[11] D. E. Edmunds, “Sur les equations differentielles de la magnetohydrodynamique”, C.r. Acad. Sci. Paris, 254 (1962), 4248–4250.
[12] J. Forste, “Ein Existenzsatz fur stationare Stromungen in der Magnetohydrodynamik”, Mon. Deutsch. Wiss. Berlin, 6 (1964), 886–894.
[13] J. Forste, “Ein Einzigkeitssatz fur stationare Stromungen in der Magnetohydrodynamik”, Mon. Deutsch. Wiss. Berlin, 9 (1967), 241–247.
[14] G. Lassner, “Uber ein Rand-Anfangswertproblem der Magnetohydrodynamik”, Arch. Rat. Mech. Anal., 30 (1967), 388–405.
[15] E. Sanchez-Palencia, “Existence des solutions de certains problemes aux limites en magnetohydrodynamique”, J. Mec, 7:3 (1968), 405–426.
[16] E. Sanchez-Palencia, “Quelques resultats d'existence et d'unicite pour des ecoulements magnetohydrodynamique non stationnaires”, J. Me?c, 8:4 (1969), 509–541.
[17] G. V. Alekseev, “O sushhestvovanii techeniya provodyashhej zhidkosti v slabo iskrivlennom kanale”, Dinamika sploshnoj sredy, 3, Izd-vo IG SO RAN, Novosibirsk, 1969, 7–16.
[18] G. G. Branover, A. B. Cinober, Magnitnaya gidrodinamika neszhimaemyx sred, Nauka, M., 1970, 380 s.
[19] G. Duvaut, J.-L. Lions, “Inequations en thermoelasticite et magnetohydrodynamique”, Arch. Rat. Mech. Anal., 46 (1972), 241–279.
[20] Sh. Saxaev, V. A. Solonnikov, “Ocenki resheniya odnoj kraevoj zadachi magnitnoj gidrodinamiki”, Tr. Matem. in-ta im. V. A. Steklova, 127, 1975, 87–108.
[21] L. I. Stupyalis, “Nestacionarnaya zadacha magnitnoj gidrodinamiki”, Zap. nauch. seminarov LOMI, 52, 1975, 175–217.
[22] L. I. Stupyalis, “O razreshimosti nachal'no-kraevoj zadachi magnitnoj gidrodinamiki”, Zap. nauch. seminarov LOMI, 69, 1977, 219–239.
[23] O. A. Ladyzhenskaya, V. A. Solonnikov, “The linearization principle and invariant manifold problems of magnetohydrodynamics”, J. Sov. Math., 8 (1977), 384–422.
[24] J. Forste, “Uber die Grundgleichungen der Plasmadynamick auf der Basis der Zweiflussigke- itstheorie”, Z. Angew. Math. Mech., 59 (1979), 553–558.
[25] L. I. Stupyalis, “Nestacionarnaya zadacha magnitnoj gidrodinamiki dlya sluchaya dvux prostranstvennyx peremennyx”, Kraevye zadachi matematicheskoj fiziki, Tr. Matem. in-ta im. V. A. Steklova, 147, ¹ 10, 1980, 156–168.
[26] L. I. Stupyalis, “Ob odnoj kraevoj zadache dlya stacionarnoj sistemy uravnenij magnitnoj gidrodinamiki”, Kraevye zadachi matematicheskoj fiziki, Tr. Matem. in-ta imeni V. A. Steklova, 147, ¹ 10, 1980, 169–193.
[27] G. V. Alekseev, “O razreshimosti odnorodnoj kraevoj zadachi dlya uravnenij magnitnoj gidrodinamiki ideal'noj zhidkosti”, Dinamika sploshnoj sredy, 57, Izd-vo IG SO RAN, Novosibirsk, 1982, 6–24.
[28] M. Sermange, R. Temam, “Some mathematical questions related to the MHD equations”, Comm. Pure. Appl. Math., 36 (1983), 635–664.
[29] Z. Yoshida, Y. Giga, “On the Ohm-Navier-Stokes system in magnetohydrodynamics”, J. Math. Phys., 1983, 2860–2864.
[30] S. V. Chizhonkov, “Ob odnoj sisteme uravnenij tipa magnitnoj gidrodinamiki”, Dokl. AN SSSR, 278:5 (1984), 1074–1077.
[31] Y. Giga, Z. Yoshida, “On the equations of the two-component theory in magnetohydro-dynamics”, Communs. Partial Diff. Eqns., 9 (1984), 503–522.
[32] D. Ebel, M. C. Shen, “On the linear stability of a toroidal plasma with resistivity, viscosity, and Hall effect”, J. Math. Anal. Appl., 125 (1987), 81–103.
[33] D. Ebel, M. C. Shen, “Linearization principle for a toroidal Hall current plasma with viscosity and resistivity”, Anal. Mat. Pura Appl., 150 (1988), 39–65.
[34] J. Blum, Numerical simulation and optimal control in plasma physics: with applications in tokamaks, Gauthier-Villars, Paris, 1989.
[35] Y. Giga, Z. Yoshida, “A dynamic free-boundary problem in physics”, SIAM J. Math. Anal., 21:5 (1990), 1118–1138.
[36] V. L. Pospelov, “Ob ustojchivosti stacionarnogo resheniya odnoj zadachi magnitnoj gidrodinamiki”, Differenc. ur-ya, 27:5 (1991), 875–886.
[37] V. N. Samoxin, “O sisteme uravnenij magnitnoj gidrodinamiki nelinejno vyazkix sred”, Differenc. ur-ya, 27:5 (1991), 886–896.
[38] V. N. Samoxin, “Sushhestvovanie resheniya odnoj modifikacii sistemy uravnenij magnitnoj gidrodinamiki”, Matem. sb., 182:3 (1991), 395–407.
[39] M. D. Gunzburger, A. J. Meir, J. S. Peterson, “On the existence, uniqueness, and finite element approximation of solution of the equations of stationary, incompressible magneto-hydrodynamics”, Math. Comp., 56:194 (1991), 523–563.
[40] O. Besson, J. Bourgeois, P.-A. Chevalier, J. Rappaz, R. Touzani, “Numerical model of electromagnetic casting processes”, J. Comput. Phys., 92 (1991), 482–507.
[41] J. Rappaz, R. Touzani, “Modelling of a two-dimensional magnetohydrodynamic problem”, Eur. J. Mech. B/Fluids, 10:5 (1991), 451–453.
[42] J. Rappaz, R. Touzani, “On a two-dimensional magnetohydrodynamic problem. 1. Modelling and Analysis”, Rairo Mode?l. Math. Anal. Numer, 26:2 (1992), 347–364.
[43] V. N. Samoxin, “O stacionarnyx zadachax magnitnoj gidrodinamiki nen'yutonovskix sred”, Sib. matem. zhurn., 33:4 (1992), 120–127.
[44] M. Spada, H. Wobig, “On the existence and uniqueness of dissipative plasma equilibria in a toroidal”, J. Phys. A., 25 (1992), 1575–1591.
[45] G. Strohmer, “About an initial-boundary value problem from magneto-hydrodynamics”, Math. Z., 209 (1992), 345–362.
[46] A. J. Meir, “The equations of stationary, incompressible magnetohydrodynamics with mixed boundary conditions”, Comp. Math. Applic., 25 (1993), 13–29.
[47] A. J. Meir, P. G. Schmidt, “A velocity-current formulation for stationary MHD flow”, Appl. Math. Comp., 65 (1994), 95–109.
[48] L. S. Hou, A. J. Meir, “Boundary optimal control of MHD flows”, Appl. Math. Optim., 32 (1995), 143–162.
[49] G. Milone, V. A. Solonnikov, “On an initial boundary-value problem for equations of magnetohydrodynamics with Hall and ion-sleep effect”, Zap. Nauchn. Sem. S. Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 221, 1995, 167–184.
[50] G. Milone, V. A. Solonnikov, “On the solvability of some initial boundary-value problems of magnetofluidmechanics with Hall and ion-sleep effects”, Atti. Accad. Naz. Lincei. Rend. Cl. Sci. Mat. Natur., 6 (1995), 117–132.
[51] M. P. Galanin, Yu. P. Popov, Kvazistacionarnye e'lektromagnitnye polya v neodnorodnyx sredax (matematicheskoe modelirovanie), Nauka, Fizmatlit, M., 1995.
[52] L. Hou, S. Ravindran, “Computations of boundary optimal control problems for an electrically conducting fluid”, J. Comput. Phys., 128 (1996), 319–330.
[53] A. J. Meir, P. G. Schmidt, “Variational methods for stationary MHD flow under natural interface conditions”, Nonlinear Analysis, 26:4 (1996), 659–689.
[54] J.-F. Gerbeau, C. Le Bris, “Existence of solution for a density-dependent magnetohydro- dynamic equation”, Adv. Differential Equations, 2:3 (1997), 427–452.
[55] J.-F. Gerbeau, C. Le Bris, “On a coupled system arising in magnetohydrodynamics”, Appl. Math. Lett., 12 (1999), 53–57.
[56] M. Wiedmer, “Finite element approximation for equations of magnetohydrodynamics”, Math. Comp., 69:229 (1999), 83–101.
[57] A. J. Meir, P. G. Schmidt, “Analysis and numerical approximation of a stationary MHD flow problem with nonideal boundary”, SIAM J. Numer. Anal., 36 (2000), 1304–1332.
[58] J.-F. Gerbeau, “A stabilized finite element method for the incompressible magnetohydro-dynamics equations”, Numer. Math., 87 (2000), 83–111.
[59] D. Schotzau, “Mixed finite element methods for stationary incompressible magnetohydro-dynamics”, Numer. Math., 96 (2004), 771–880.
[60] V. Girault, P. A. Raviart, “Finite element methods for Navier – Stokes equations”, Theory and algorithms, Springer-Verlag, Berlin, 1986, 376 pp.
[61] A. Valli, Orthogonal decompositions of $L^2(|Omega)^3$, Preprint UTM 493., Department of Mathematics. University of Trento, 1995.
[62] A. Alonso, A. Valli, “Some remarks on the characterization of the space of tangential traces of $H(\rot ; \Omega)$ and the construction of the extension operator”, Manuscr. Math., 89 (1996), 159–178.
[63] M. Cessenat, “Mathematical methods in electromagnetism”, Linear theory and applications, 41 (1996), Word Scientific Publishing.
[64] A. Buffa, M. Costabel, D. Sheen, “On traces for $H(\curl, \Omega)$ in Lipschitz domains”, J. Math. Anal. Appl., 276:2 (2002), 845–876.
[65] A. Alonso, A. Valli, “An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations”, Math. Comp., 68 (1999), 607–631.
[66] J. Saranen, “Degeneralized harmonic fields in domain with anisotropic nonhomogeneous media”, J. Math. Anal. Appl., 88 (1982), 104–115.
[67] J. Saranen, “On electric and magnetic problems for vector fields in anisotropic nonhomog-eneous media”, J. Math. Anal. Appl., 91 (1983), 254–275.
[68] M. E. Bogovskij, “Reshenie nekotoryx zadach vektornogo analiza, svyazannyx s operatorami $\div$ i $\grad$”, Teoriya kubaturnyx formul i prilozheniya funkcional'nogo analiza k zadacham matematicheskoj fiziki, IM SO AN SSSR, Novosibirsk, 1980, 5–40.
[69] R. Picard, “On the boundary value problems of electro- and magnetostatics”, Proc. Royal Soc. Edinburgh., 92A (1982), 165–174.
[70] P. R. Kotiuga, P. P. Silvester, “Vector potential formulation for three-dimensional magnetostatics”, J. Appl. Phys., 53 (1982), 8399–8401.
[71] J. Bolik, W. von Wahl, “Estimating $\nabla u$ in terms of $\div u$, $\curl u$ and either $u \cdot \nu$ or $u \times \nu$ and the topology”, Math. Meth. Appl. Sci., 20 (1997), 737–744.
[72] D. Mitrea, M. Mitrea, J. Pipher, “Vector potential thory on nonsmooth domains in $R^3$ and applications to electromagnetic scattering”, J. Fourier Anal. Appl., 3:2 (1997), 131–192.
[73] P. Fernandes, G. Gilardi, “Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions”, Math. Mod. Meth. Appl. Sci., 7 (1997), 957–991.
[74] G. Auchmuty, “Reconstruction of the velocity in the three-demensional fluid flows”, Proc. Royal. Soc. Lond., 454A (1998), 607–630.
[75] C. Amrouche, C. Bernardi, M. Dauge, V. Girault, “Vector potentials in three-dimensional non-smooth domains”, Math. Meth. Appl. Sci., 21 (1998), 823–864.
[76] G. Auchmuty, J. C. Alexander, “$L^2$-well-posedness of planar div-curl systems”, Arch. Rat. Mech. Anal., 160 (2001), 91–134.
[77] G. Auchmuty, J. Alexander, $L^2$-well posedeness of div-rot systems in space, Preprint, 2002.
[78] G. V. Alekseev, Teoreticheskij analiz obratnyx e'kstremal'nyx zadach dlya stacionarnyx uravnenij magnitnoj gidrodinamiki vyazkoj neszhimaemoj zhidkosti, Preprint ¹ 1 IPM DVO RAN, Dal'nauka, Vladivostok, 2002, 78 s.
[79] G. V. Alekseev, “Razreshimost' stacionarnyx zadach granichnogo upravleniya dlya uravnenij teplovoj konvekcii”, Sib. mat. zhurn., 39:5 (1998), 982–998.
[80] G. V. Alekseev, “Stacionarnye zadachi granichnogo upravleniya dlya uravnenij teplovoj konvekcii”, Dokl. RAN, 362:2 (1998), 174–177.
[81] G. V. Alekseev, A. B. Smyshlyaev, D. A. Tereshko, Neodnorodnye kraevye zadachi dlya stacionarnyx uravnenij teplomassoperenosa, Preprint ¹ 19 IPM DVO RAN, Dal'nauka, Vladivostok, 2000, 60 s.
[82] G. V. Alekseev, “Obratnye e'kstremal'nye zadachi dlya stacionarnyx uravnenij teplomassoperenosa”, Dokl. RAN, 375:3 (2000), 315–319.
[83] G. V. Alekseev, E. A. Adomavichus, “Theoretical analysis of inverse extremal problems of admixture diffusion in viscous fluids”, J. Inv. Ill-Posed Probl., 9 (2001), 435–468.
[84] G. V. Alekseev, “Razreshimost' obratnyx e'kstremal'nyx zadach dlya stacionarnyx uravnenij teplomassoperenosa”, Sib. mat. zhurn., 42:5 (2001), 971–991.
[85] G. V. Alekseev, E'. A. Adomavichyus, “Issledovanie obratnyx e'kstremal'nyx zadach dlya nelinejnyx stacionarnyx uravnenij perenosa veshhestva”, Dal'nevost. matem. zhurn., 3:1 (2002), 79–92.
[86] G. V. Alekseev, “Obratnye e'kstremal'nye zadachi dlya stacionarnyx uravnenij teorii massoperenosa”, Zh. vychisl. mat. i mat. fiz., 42:3 (2002), 380–394.
[87] A. D. Ioffe, V. M. Tixomirov, Teoriya e'kstremal'nyx zadach, Nauka, M., 1974.
[88] G. V. Alekseev, R. V. Brizickij, “Razreshimost' smeshannoj zadachi dlya stacionarnyx uravnenij magnitnoj gidrodinamiki vyazkoj zhidkosti”, Dal'nevost. mat. zhurn., 3:2 (2002), 285–301.
[89] G. V. Alekseev, R. V. Brizickij, “O razreshimosti smeshannoj kraevoj zadachi dlya stacionarnyx uravnenij magnitnoj gidrodinamiki vyazkoj neszhimaemoj zhidkosti”, Vychisl. texnol., 7:1, spec. vyp. (2002), 242–250.
[90] G. V. Alekseev, R. V. Brizickij, “Razreshimost' obratnyx e'kstremal'nyx zadach dlya stacionarnyx uravnenij magnitnoj gidrodinamiki vyazkoj zhidkosti so smeshannymi granichnymi usloviyami”, Dal'nevost. mat. zhurn., 4:1 (2003), 108–126.
[91] G. V. Alekseev, “Zadachi upravleniya dlya stacionarnyx uravnenij magnitnoj gidrodinamiki vyazkoj neszhimaemoj zhidkosti”, Prikl. mex. texn. fiz., 44:6 (2003), 170–179.
[92] G. V. Alekseev, “Zadachi upravleniya dlya stacionarnyx uravnenij magnitnoj gidrodinamiki”, Dokl. RAN, 395:3 (2004), 322–325.
[93] G. V. Alekseev, “Razreshimost' zadach upravleniya dlya stacionarnyx uravnenij magnitnoj gidrodinamiki vyazkoj zhidkosti”, Sib. mat. zhurn., 45:2 (2004), 243–262.
[94] G. V. Alekseev, “O edinstvennosti resheniya zadachi upravleniya dlya stacionarnoj modeli magnitnoj gidrodinamiki vyazkoj neszhimaemoj zhidkosti”, Dal'nevost. mat. zhurn., 5:1 (2004), 142–157.
[95] G. V. Alekseev, Teoreticheskij analiz zadach upravleniya dlya stacionarnyx uravnenij magnitnoj gidrodinamiki vyazkoj teploprovodnoj zhidkosti, Preprint ¹ 1 IPM DVO RAN, Dal'nauka, Vladivostok, 2004, 80 s.
[96] G. V. Alekseev, “Obratnye e'kstremal'nye zadachi dlya stacionarnoj modeli magnitnoj gidrodinamiki teploprovodnoj zhidkosti”, Vychisl. texnol., 9:1, spec. vyp. (2004), 158–166.
[97] R. V. Brizickij, “Zadachi upravleniya dlya modeli MGD vyazkoj teploprovodnoj zhidkosti so smeshannymi granichnymi usloviyami”, Dal'nevost. mat. zhurn., 5:2 (2004), 226–238.
[98] G. V. Alekseev, “Kraevye zadachi i zadachi upravleniya dlya stacionarnoj modeli magnitnoj gidrodinamiki vyazkoj teploprovodnoj zhidkosti”, Dokl. RAN, 405:6 (2005), 744–748.
[99] G. V. Alekseev, R. V. Brizickij, “Zadachi upravleniya dlya stacionarnyx uravnenij magnitnoj gidrodinamiki vyazkoj teploprovodnoj zhidkosti so smeshannymi granichnymi usloviyami”, Zh. vychisl. mat. i mat. Fiz., 45:12 (2005), 2131–2147.

To content of the issue