Control problems for stationary models of magnetic hydrodynamics of a viscous incompressible fluid |
G. V. Alekseev |
2005, issue 1-2, P. 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 |
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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, № 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, № 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 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 № 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 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. |