Spectral properties of operators in the problem on normal oscillations of~a~mixture of viscous compressible fluids

D. A. Zakora

Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 1, pp. 73-97.

Résumé

In this paper, we study a problem of normal oscillations of a homogeneous mixture of several viscous compressible fluids filling a bounded domain of three-dimensional space with an infinitely smooth boundary. Two boundary conditions are considered: the no-slip condition and the slip condition without shear stresses. It is proved that the essential spectrum of the problem in both cases is a finite set of segments located on the real axis. The discrete spectrum lies on the real axis, except perhaps for a finite number of complex conjugate eigenvalues. The spectrum of the problem contains a subsequence of eigenvalues with a limit point at infinity and a power-law asymptotic distribution.

Détail

Keywords: mixture of fluids, spectral problem, essential spectrum, discrete spectrum.
Mots-clés : compressible viscous fluid

Bibliographie

[1] Azizov T. Ya., Iokhvidov I. S., Osnovy teorii lineinykh operatorov v prostranstvakh s indefinitnoi metrikoi, Nauka, M., 1986 | MR

[2] Azizov T. Ya., Kopachevskii N. D., Orlova L. D., “Evolyutsionnaya i spektralnaya zadachi, porozhdennye problemoi malykh dvizhenii vyazkouprugoi zhidkosti”, Tr. SPb. Mat. ob-va, 6 (1988), 5–33

[3] Birman M. Sh., Solomyak M. Z., “Asimptotika spektra differentsialnykh uravnenii”, Itogi nauki i tekhn. Ser. Mat. analyu, 14, no. 11, 1977, 5–58 | Zbl

[4] Volevich L. R., “Razreshimost kraevykh zadach dlya obschikh ellipticheskikh sistem”, Mat. sb., 68:3 (1965), 373–416 | Zbl

[5] Dorovskii V. N., Perepechko Yu. V., “Teoriya chastichnogo plavleniya”, Geolog. i geofiz., 1989, no. 9, 56–64

[6] Imomnazarov Kh. Kh., Imomnazarov Sh. Kh., Mamatkulov M. M., Chernykh E. G., “Fundamentalnoe reshenie dlya statsionarnogo uravneniya dvukhskorostnoi gidrodinamiki s odnim davleniem”, Sib. zh. industr. mat., 17:4 (2014), 60–66 | MR | Zbl

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

[8] Kozhevnikov A. N., Funktsionalnye metody matematicheskoi fiziki, Uchebnoe posobie, MAI, M., 1991

[9] Krein S. G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967

[10] Lankaster P., Teoriya matrits, Nauka, M., 1973

[11] Mamontov A. E., Prokudin D. A., “Razreshimost nachalno-kraevoi zadachi dlya uravnenii politropnogo dvizheniya smesei vyazkikh szhimaemykh zhidkostei”, Sib. elektron. mat. izv., 13 (2016), 541–583 | Zbl

[12] Mamontov A. E., Prokudin D. A., “Razreshimost nestatsionarnykh uravnenii mnogokomponentnykh vyazkikh szhimaemykh zhidkostei”, Izv. RAN. Ser. mat., 82:1 (2018), 151–197 | DOI | MR | Zbl

[13] Markus A. S., Matsaev V. I., “Teorema o sravnenii spektrov i spektralnaya asimptotika dlya puchka M. V. Keldysha”, Mat. sb., 123:3 (1984), 391–406 | MR | Zbl

[14] Nigmatulin R. I., Dinamika mnogofaznykh sred, v. 1, Nauka, M., 1987

[15] Oleinik O. A., Iosifyan G. A., Shamaev A. S., Matematicheskie zadachi teorii silno neodnorodnykh uprugikh sred, MGU, M., 1990 | MR

[16] Pal P. K., Maslennikova V. N., “Spektralnye svoistva operatorov v zadache o kolebanii szhimaemoi zhidkosti vo vraschayuschikhsya sosudakh”, Dokl. AN SSSR, 281:3 (1985), 529–534 | MR | Zbl

[17] Rektoris K., Variatsionnye metody v matematicheskoi fizike i tekhnike, Mir, M., 1985 | MR

[18] Solonnikov V. A., “Ob obschikh kraevykh zadachakh dlya sistem, ellipticheskikh v smysle A. Daglisa—L. Nirenberga. II”, Tr. MIAN, 92, 1966, 233–297 | Zbl

[19] Atkinson F. V., Langer H., Mennicken R., Shkalikov A. A., “The essential spectrum of some matrix operators”, Math. Nachr., 167 (1994), 5–20 | DOI | MR | Zbl

[20] Faierman M., Fries R. J., Mennicken R., Möller M., “On the essential spectrum of the linearized Navier—Stokes operator”, Integr. Equ. Oper. Theory, 38:1 (2000), 9–27 | DOI | MR | Zbl

[21] Frehse J., Goj S., Málek J., “A Stokes-like system for mixtures”, Nonlinear Problems in Mathematical Physics and Related Topics, v. II, Kluwer, Dordrecht—Norwell—New York—London, 2002, 119–136 | DOI | MR | Zbl

[22] Frehse J., Goj S., Málek J., “On power and non-power asymptotic behavior of positive solutions to Emden—Fowler type higher-order equations”, SIAM J. Math. Anal., 36:4 (2005), 1259–1281 | DOI | MR | Zbl

[23] Frehse J., Goj S., Málek J., “A uniqueness result for a model for mixtures in the absence of external forces and interaction momentum”, Appl. Math., 50 (2005), 527–541 | DOI | MR | Zbl

[24] Gohberg I., Goldberg S., Kaashoek M. A., Classes of linear operators, v. 1, Birkhäuser, Basel—Boston—Berlin, 1990 | Zbl

[25] Grubb G., Geymonat G., “The essential spectrum of elliptic systems of mixed order”, Math. Ann., 227 (1977), 247–276 | DOI | MR | Zbl

[26] Kozhevnikov A., Skubachevskaya T., “Some applications of pseudo-differential operators to elasticity”, Hokkaido Math. J., 26:2 (1997), 297–322 | DOI | MR | Zbl

[27] Mamontov A. E., Prokudin D. A., “Viscous compressible multi-fluids: modeling and multi-D existence”, Methods Appl. Anal., 20:2 (2013), 179–195 | DOI | MR

[28] Mennicken R., Shkalikov A. A., “Spectral decomposition of symmetric operator matrices”, Math. Nachr., 179 (1996), 259–273 | DOI | MR | Zbl

[29] Rajagopal K. L., Tao L., Mechanics of mixtures, World Sci. Publ., River Edge, 1995 | MR | Zbl