Far Eastern Mathematical Journal

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Variational inequalities, boundary – value problems and optimal control for the Navier – Stokes equations


A. Yu. Chebotarev, A. A. Illarionov, E. V. Amosova

2008, issue 1, Ñ. 121–140


Abstract
The survey of results in the mathematical hydrodynamics by mathematical modeling laboratory of IAM FEB RAS is considered. The statements of the some open problems for the Navier – Stokes equations is discussed.

Keywords:
Navier – Stokes equations, variational inequalities, the solvability of the boundary – value problems, optimality conditions

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References

[1] J. Leray, “Etude de diverses equations, integrales non lineaire et de queques problemes que posent l'Hydrodynamique”, J. Math. Pures Appl., 35:12 (1933), 1–82.
[2] O. A. Ladyzhenskaya, Matematicheskie voprosy dinamiki vyazkoj neszhimaemoj zhidkosti, Nauka, M., 1970.
[3] C. J. Amick, “Existence of solutions to the nonhomogeneous steady Navier – Stokes equations”, Indiana Univ. Math. J., 33:6 (1984), 817–830.
[4] L. I. Sazonov, “O sushhestvovanie stacionarnogo simmetrichnogo resheniya dvumernoj zadachi o protekanii zhidkosti”, Matem. zametki, 54:6 (1993), 138–141.
[5] H. Fujita, “On the stationary solutions to Navier – Stokes equations in simmetric plane domains under general outflow conditions”, Theory and Numerical methods, Proceeding of International Conference on Navier – Stokes equations (Varenna, Italy, 1997), Pitman Research Notes in Math., 388, Longman, Harlow, 1998, 16–30.
[6] H. Fujita, H. Morimoto, “A remark on the existence of steady Navier – Stokes flow with non-vanishing outflow conditions”, Nonlinear Waves, Gakuto International Series in Math. Science and Appl., 10, 1997, 53–61.
[7] A. A. Illarionov, “Nekotorye zamechaniya o lemme Xopfa”, Vestnik TOGU, 2007, ¹ 4(7), 81–88.
[8] V. V. Ragulin, “K zadache o protekanii vyazkoj zhidkosti skvoz' ogranichennuyu oblast' pri zadannom perepade davleniya ili napora”, Dinamika sploshnoj sredy, 27, 1976, 78–92.
[9] C. Begue, C. Conca, F. Murat and O. Pironneau, “A nouveau sur les equations de Stokes et de Navier – Stokes avec des conditions aux limites sur la pression”, C. R. Acad. Sc. Paris. Serie I, 304:2 (1987), 23–28.
[10] A. Yu. Chebotarev, “Subdifferencial'nye kraevye zadachi dlya stacionarnyx uravnenij Nav'e – Stoksa”, Differenc. uravneniya, 28:8 (1992), 1443–1450.
[11] C. Conca, F. Murat, O. Pironneau, “The Stokes and Navier – Stokes equation with boundary conditions involving the pressure”, Japan J. Math., 20:2 (1994), 279–318.
[12] A. A. Illarionov, A. Yu. Chebotarev, “O razreshimosti smeshannoj kraevoj zadachi dlya stacionarnyx uravnenij Nav'e – Stoksa”, Differenc. uravneniya, 37:5 (2001), 689–695.
[13] A. A. Illarionov, “O razreshimosti kraevyx zadach dlya stacionarnyx uravnenij Nav'e – Stoksa”, Dal'nevostochnyj matem. zhurn., 2:1 (2001), 16–36.
[14] R. Temam, Uravneniya Nav'e – Stoksa. Teoriya i chislennyj analiz, M., 1981.
[15] V. Girault, “Curl-conforming finite element methods for Navier – Stokes equations with non-standard boundary conditions in $R^3$”, Proc. of the Oberwolfash Meeting on Navier – Stokes Equations and Numerical Methods, Lecture Notes in Mathematics, ed. R. Rautmann, Springer, 1990, 201–218.
[16] A. A. Illarionov, “Nelokal'naya kraevaya zadacha s pereopredeleniem dlya stacionarnyx uravnenij Nav'e – Stoksa”, Zhurn. vych. matem i matem. fiziki, 48:6 (2008), 1056–1061.
[17] A. A. Illarionov, “Nelokal'naya kraevaya zadacha s pereopredeleniem dlya e'llipticheskogo uravneniya”, Sib. zhurn. industr. matem., 10:2(30) (2007), 64–69.
[18] A. Yu. Chebotarev, “Ob odnostoronnix e'kstremal'nyx zadachax, svyazannyx s sistemoj Stoksa”, Dinamika sploshnoj sredy, 102, IG SO AN SSSR, Novosibirsk, 1991, 133–147.
[19] G. V. Alekseev, A. Yu. Chebotarev, “Some extremum and unilateral boundary value problems in viscous hydrodynamics”, International Series of Numerical Mathematics, 106, Birkha?user Verlag, Basel, 1992, 1–11.
[20] A. Yu. Chebotarev, “Subdifferential inverse problems for stationary systems of Navier – Stokes type”, J. Inverse and Ill Posed Problems, 3:4 (1995), 268–279.
[21] A. Yu. Chebotarev, “Korrektnost' zadachi ob e'lektromagnitnyx kolebaniyax v polyarizuemoj srede”, Dinamika sploshnoj sredy, 107, IG SO RAN, Novosibirsk, 1993, 98–105.
[22] A. Yu. Chebotarev, “Granichnye obratnye zadachi dlya uravnenij Nav'e – Stoksa s subdifferencial'nym pereopredeleniem”, Differenc. uravneniya, 31:5 (1995), 677–683.
[23] C. Foias, R. Temam, “Structure of the set of stationary solutions of the Navier – Stokes equations”, Comm. Pure Appl. Math., 30 (1977), 149–164.
[24] A. Yu. Chebotarev, “Variacionnye neravenstva dlya operatora tipa Nav'e – Stoksa i odnostoronnie zadachi dlya uravnenij vyazkoj teploprovodnoj zhidkosti”, Matem. zametki, 70:2 (2001), 296–307.
[25] A. Yu. Chebotarev, “Predel'nyj perexod po vyazkosti v variacionnyx neravenstvax dlya operatora Nav'e – Stoksa”, Dal'nevostochnyj matem. sb., 1996, ¹ 2, 193–197.
[26] A. Yu. Chebotarev, “Modelirovanie stacionarnyx techenij v kanale variacionnymi neravenstvami Nav'e – Stoksa”, Prikladnaya mexanika i texnicheskaya fizika, 2003, ¹ 6, 123–129.
[27] A. V. Kazhixov, “Razreshimost' nachal'no-kraevoj zadachi dlya uravnenij dvizheniya neodnorodnoj vyazkoj neszhimaemoj zhidkosti”, Dokl. AN SSSR, 216:5 (1974), 1008–1010.
[28] A. V. Kazhixov, “Razreshimost' nekotoryx odnostoronnix kraevyx zadach dlya uravnenij Nav'e – Stoksa”, Nestacionarnye problemy gidrodinamiki, Dinamika sploshnoj sredy, 16, In-t gidrodinamiki SO AN SSSR, Novosibirsk, 1974, 5–34.
[29] N. N. Frolov, “Kraevaya zadacha, opisyvayushhaya dvizhenie neodnorodnoj zhidkosti”, Sib. matem. zhurnal, 37:2 (1996), 433–451.
[30] A. Yu. Chebotarev, “Stacionarnye variacionnye neravenstva v modeli neodnorodnoj neszhimaemoj zhidkosti”, Sib. matem. zhurn., 38:5 (1997), 1184–1193.
[31] A. Yu. Chebotarev, “Obratnye zadachi dlya nelinejnyx e'volyucionnyx uravnenij tipa Nav'e – Stoksa”, Differenc. uravneniya, 31:3 (1995), 517–524.
[32] A. Yu. Chebotarev, “Subdifferential inverse problems for evolution Navier – Stokes systems”, J. Inv. and Ill Posed Problems, 8 (2000), 275–287.
[33] A. Yu. Chebotarev, A. S. Savenkova, “Variacionnye neravenstva v magnitnoj gidrodinamike”, Matem. zametki, 82:1 (2007), 135–149.
[34] A. Yu. Chebotarev, “Subdifferencial'nye kraevye zadachi magnitnoj gidrodinamiki”, Differenc. uravneniya, 43:12 (2007), 1700–1709.
[35] A. V. Fursikov, “Zadachi upravleniya i teoremy,kasayushhiesya odnoznachnoj razreshimosti smeshannoj kraevoj zadachi dlya trexmernyx uravnenij Nav'e – Stoksa i E'jlera”, Matem. sb., 115:2 (1981), 281–306.
[36] A. V. Fursikov, “Svojstva reshenij nekotoryx e'kstremal'nyx zadach, svyazannyx s sistemoj Nav'e – Stoksa”, Matem. sb., 118:3 (1982), 323–349.
[37] A. Yu. Chebotarev, “Granichnye e'kstremal'nye zadachi dinamiki vyazkoj neszhimaemoj zhidkosti”, Sib. matem. zhurn., 34:5 (1993), 202–213.
[38] A. Yu. Chebotarev, “Princip maksimuma v zadache granichnogo upravleniya techeniem vyazkoj zhidkosti”, Sib. matem. zhurn., 34:6 (1993), 189–197.
[39] A. Yu. Chebotarev, “Princip maksimuma v obratnyx e'kstremal'nyx zadachax dlya stacionarnyx sistem tipa Nav'e – Stoksa”, Dal'nevostochnyj matem. sb., 1995, ¹ 1, 92–100.
[40] A. Yu. Chebotarev, “Normal'nye resheniya kraevyx zadach dlya stacionarnyx sistem tipa Nav'e – Stoksa”, Sib. matem. zhurn., 36:4 (1995), 934–942.
[41] A. Yu. Chebotarev, “Suboptimal controls in extremum problems of viscous hydrodynamics”, Dal'nevostochnyj matem. sb., 1997, ¹ 3, 1–5.
[42] A. A. Illarionov, “Asimptotika reshenij zadachi optimal'nogo upravleniya dlya uravnenij Nav'e – Stoksa”, Zhurn. vych. matem. i matem. fiziki, 40:7 (2000), 1061–1070.
[43] A. A. Illarionov, “Ob asimptotike reshenij zadachi optimal'nogo upravleniya dlya uravnenij Nav'e – Stoksa”, Zhurn. vych. matem. i matem. fiziki, 41:7 (2001), 1045–1056.
[44] A. A. Illarionov, “Optimal'noe granichnoe upravlenie stacionarnym techeniem vyazkoj neodnorodnoj neszhimaemoj zhidkosti”, Matem. zametki, 69:5 (2001), 666–678.
[45] D. S. Konovalova, A. Yu. Chebotarev, “Optimal'noe startovoe upravlenie techeniem vyazkoj zhidkosti”, Dal'nevostochnyj matem. sb., 1996, ¹ 2, 110–119.
[46] A. Yu. Chebotarev, “Optimal'noe upravlenie v nestacionarnyx zadachax magnitnoj gidrodinamiki”, Sib. zhurn. industr. matematiki, 10:3 (2007), 138–148.
[47] A. Yu. Chebotarev, “Optimal'noe upravlenie tormozheniem MGD techeniya”, Prikladnaya mexanika i texnicheskaya fizika, 49:4 (2008).
[48] E. V. Lukina, “Issledovanie korrektnosti zadachi optimal'nogo upravleniya dlya e'volyucionnogo uravneniya Byurgersa s pravoj chast'yu”, Dal'nevostochnyj matem. sb., 1999, ¹ 7, 59–73.
[49] D. S. Konovalova, E. V. Lukina, “Issledovanie zadachi optimal'nogo upravleniya techeniyami vyazkogo gaza”, Zhurn. vych. matem. i mat. fiz., 40:3 (2000), 429–449.
[50] E. V. Lukina, “Optimal'noe upravlenie techeniyami vyazkogo gaza s podvizhnoj granicej”, Zhurn. vych. matem. i mat. fiz., 41:7 (2001), 1026–1044.
[51] E. V. Lukina, “Razreshimost' nestacionarnoj kraevoj zadachi dlya model'noj sistemy dinamiki barotropnogo gaza”, Dal'nevostochnyj matem. zhurn., 2:1 (2001), 17–37.
[52] E. V. Lukina, “Optimal'noe startovoe upravlenie barotropnym dvizheniem vyazkogo gaza”, Sibirskij zhurn. industr. matem., 5:4(12) (2002), 71–91.
[53] E. V. Lukina, “Global'nye resheniya mnogomernyx priblizhennyx uravnenij Nav'e – Stoksa vyazkogo gaza”, Sib. matem. zhurn., 44:2 (2003), 389–401.
[54] E. V. Amosova, “Optimal'noe upravlenie techeniem vyazkogo teploprovodnogo gaza”, Sib. zhurn. industr. matem., 10:2(30) (2007), 5–22.
[55] E. V. Amosova, “Optimal'noe upravlenie MGD-techeniem vyazkogo teploprovodnogo gaza”, Zhurn. vych. matem. i mat. fiz., 48:4 (2008), 623–633.
[56] E. V. Amosova, “Optimal'noe upravlenie techeniem vyazkogo teploprovodnogo gaza”, Sib. zhurn. industr. Matem., 10:2 (2007), 5–22.

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