Hamiltonian Systems in the~Theory of Small Oscillations of a~Rotating Ideal Fluid.~I

M. V. Fokin

Matematičeskie trudy, Tome 4 (2001) no. 2, pp. 155-206.

Résumé

This article describes the behavior of solutions to two-dimensional Hamiltonian systems arising in the theory of small oscillations of a rotating ideal fluid. Representation is established for a class of exact solutions to the linearized Euler equations (the Poincaré–Sobolev system), with the help of which a mathematical model is constructed for the process of origination and development of vortex structures in a cylindric domain. The first part of the article deals with the general properties of fluid oscillations and describes those connected with the presence of symmetry groups defined by the initial manifold of perturbations of the velocity field. In particular, we demonstrate that fluid motions are cellular-like and synchronous appearance or disappearance of vortex structures hold in every cell.

Bibliographie

[1] Aleksandryan R. A., “Ob odnoi zadache Soboleva dlya spetsialnykh uravnenii s chastnymi proizvodnymi”, Dokl. AN SSSR, 73:5 (1950), 631–634 | MR

[2] Aleksandryan R. A., “Spektralnye svoistva operatorov, porozhdennykh sistemami differentsialnykh uravnenii tipa S. L. Soboleva”, Tr. Mosk. mat. o-va, 9, 1960, 455–505 | Zbl

[3] Arnold V. I., “Zamechaniya o povedenii techenii trekhmernoi idealnoi zhidkosti pri malom vozmuschenii nachalnogo polya skorostei”, Prikl. matematika i mekhanika, 36:2 (1972), 255–262

[4] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[5] Arnold V. I., Dopolnitelnye glavy teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1978 | MR

[6] Rabov S. A., Sveshnikov A. G., Zadachi dinamiki stratifitsirovannykh zhidkostei, Nauka, M., 1986 | MR

[7] Grigorev Yu. N., Organizovannye struktury v razvitoi pristennoi turbulentnosti, In-t vychisl. tekhnologii SO RAN, Novosibirsk, 1993

[8] Zaslavskii G. M., Sagdeev R. Z., Usikov D. A., Chernikov A. A., Slabyi khaos i kvaziregulyarnye struktury, Nauka, M., 1990

[9] Zelenik T. I., “Ob asimptotike reshenii odnoi smeshannoi zadachi”, Differents. uravneniya, 2:1 (1966), 47–64

[10] Zelenik T. I., “Ob obobschennykh sobstvennykh funktsiyakh operatora, svyazannogo s odnoi zadachei S. L. Soboleva”, Sib. mat. zhurn., 9:5 (1968), 1075–1092

[11] Zelenik T. I., Izbrannye voprosy kachestvennoi teorii uravnenii s chastnymi proizvodnymi, Izd-vo Novosibirskogo unta, Novosibirsk, 1970

[12] Zelenik T. I., “O nekotorykh svoistvakh reshenii uravnenii malykh kolebanii vraschayuscheisya zhidkosti”, Dinamika sploshnoi sredy, 107, In-t gidrodinamiki SO RAN, Novosibirsk, 1993, 58–64

[13] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[14] Kopachevskii N. D., Krein S. G., Ngo Zui Kan, Operatornye metody v lineinoi gidrodinamike, Nauka, M., 1989 | MR

[15] Lamb G., Gidrodinamika, Gostekhizdat, M.; L., 1947

[16] Lyuk Yu., Spetsialnye matematicheskie funktsii i ikh approksimatsii, Mir, M., 1980

[17] Maslov V. P., “Kogerentnye struktury, rezonansy i asimptoticheskaya needinstvennost dlya uravnenii Nave — Stoksa pri bolshikh chislakh Reinoldsa”, Uspekhi mat. nauk, 41:6 (1986), 19–36 | MR

[18] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR

[19] Palis Zh., Di Melu V., Geometricheskaya teoriya dinamicheskikh sistem, Mir, M., 1986 | MR

[20] Pukhnachev V. V., “Mikrokonvektsiya v vertikalnom sloe”, Izv. RAN. MZhG, 1994, no. 5, 76–84 | MR | Zbl

[21] Sobolev S. L., “Ob odnoi novoi zadache matematicheskoi fiziki”, Izv. AN SSSR. Ser. mat., 18:1 (1954), 3–50 | MR | Zbl

[22] Faddeev L. D., “K teorii ustoichivosti statsionarnykh plosko-parallelnykh techenii idealnoi zhidkosti”, Kraevye zadachi matematicheskoi fiziki, Zap. nauch. seminarov LOMI, 21, Mat. in-t im. V. A. Steklova. Leningr. otd-nie, 1971, 164–172 | MR | Zbl

[23] Fokin M. V., “O razreshimosti zadachi Dirikhle dlya uravneniya kolebaniya struny”, Dokl. AN SSSR, 272:4 (1983), 801–805 | MR | Zbl

[24] Fokin M. V., “Suschestvovanie singulyarnogo spektra i asimptotika reshenii zadachi Soboleva”, Vychislitelnye metody i modeli prikladnoi matematiki, Tr. In-ta matematiki SO RAN, 26, Izd-vo In-ta matematiki, Novosibirsk, 1994, 107–195 | MR

[25] Mezić I., Wiggins S., “On the integrability and perturbation of threedimensional fluid flows with symmetry”, J. Nonlinear Sci., 4:2 (1994), 157–194 | DOI | MR | Zbl

[26] Poincaré H., “Sur l'equilibre d'une masse fluide animee d'un mouvement de rotation”, Acta Math., 7 (1885), 259–380 | DOI | MR

[27] Robinson S. K., “A review of vortex structures and associated coherent motion in turbulent boundary layers”, Proc. 2nd IUTAM Symp. on Structure of Turbulence and Drag Reduct, Springer, Berlin; Heidelberg, 1990, 23–50