FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
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, P. 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 |
Download the article (PDF-file) |
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, № 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, № 2, 193–197. [26] A. Yu. Chebotarev, “Modelirovanie stacionarnyx techenij v kanale variacionnymi neravenstvami Nav'e – Stoksa”, Prikladnaya mexanika i texnicheskaya fizika, 2003, № 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, 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, № 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, № 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, № 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. |