Geodesic
On blowup of gravity-gyroscopic waves with nonlinear sources and sinks on the boundary
Matematičeskie trudy, Tome 14 (2011) no. 2, pp. 83-126.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the article, we consider the equation of internal gravity-gyroscopic waves in an exponentially stratified fluid whose model is the great oceans. We study the case in which a fluid occupies a bounded domain and a boundary condition of the third kind is imposed on the boundary, All sinks and sources localized on the boundary are taken into account. The local solvability of the problem in a weak generalized sense is established and sufficient conditions of finite time blowup of solutions are exposed. 
@article{MT_2011_14_2_a4,
     author = {M. O. Korpusov},
     title = {On blowup of gravity-gyroscopic waves with nonlinear sources and sinks on the boundary},
     journal = {Matemati\v{c}eskie trudy},
     pages = {83--126},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2011_14_2_a4/}
}
TY  - JOUR
AU  - M. O. Korpusov
TI  - On blowup of gravity-gyroscopic waves with nonlinear sources and sinks on the boundary
JO  - Matematičeskie trudy
PY  - 2011
SP  - 83
EP  - 126
VL  - 14
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2011_14_2_a4/
LA  - ru
ID  - MT_2011_14_2_a4
ER  - 
%0 Journal Article
%A M. O. Korpusov
%T On blowup of gravity-gyroscopic waves with nonlinear sources and sinks on the boundary
%J Matematičeskie trudy
%D 2011
%P 83-126
%V 14
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2011_14_2_a4/
%G ru
%F MT_2011_14_2_a4
M. O. Korpusov. On blowup of gravity-gyroscopic waves with nonlinear sources and sinks on the boundary. Matematičeskie trudy, Tome 14 (2011) no. 2, pp. 83-126. http://geodesic.mathdoc.fr/item/MT_2011_14_2_a4/

[1] Baiokki K., Kapelo A., Variatsionnye i kvazivariatsionnye neravenstva, Nauka, M., 1988 | MR

[2] Borovikov V. A., “Asimptoticheskoe razlozhenie funktsii Grina uravneniya vnutrennikh gravitatsionnykh voln pri $t\to+\infty$”, Prikl. mekhanika i tekhnich. fizika, 37:4 (1996), 173–182 | MR | Zbl

[3] Bulatov V. V., Vladimirov Yu. V., Vnutrennie gravitatsionnye volny v neodnorodnykh sredakh, Nauka, M., 2005 | MR | Zbl

[4] Gabov S. A., Sveshnikov A. G., Lineinye zadachi teorii nestatsionarnykh vnutrennikh voln, Nauka, M., 1990 | MR

[5] Gaevskii Kh., Greger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978 | MR

[6] Galaktionov V. A., Pokhozhaev S. I., “Uravneniya nelineinoi dispersii tretego poryadka: udarnye volny, volny razrezheniya i razrusheniya”, Zhurn. vychisl. matem. i matem. fiz., 48:10 (2008), 1819–1846 | MR | Zbl

[7] Demidenko G. V., Uspenskii S. V., Uravneniya i sistemy, ne razreshennye otnositelno starshei proizvodnoi, Nauchnaya kniga, Novosibirsk, 1998 | MR | Zbl

[8] Kalantarov V. K., Ladyzhenskaya O. A., “Formirovanie kollapsov v kvazilineinykh uravneniyakh parabolicheskogo i giperbolicheskogo tipov”, Zapiski LOMI, 69, 1977, 77–102 | MR | Zbl

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

[10] Korpusov M. O., Sveshnikov A. G., “O razrushenii resheniya sistemy uravnenii Oskolkova”, Matem. sb., 200:4 (2009), 83–108 | MR | Zbl

[11] Krasnoselskii M. A., Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, GITTL, M., 1956 | MR

[12] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR | Zbl

[13] Mitidieri E. L., Pokhozhaev S. I., Apriornye otsenki i otsutstvie reshenii differentsialnykh neravenstv v chastnykh proizvodnykh, Tr. Mat. in-ta im. V. A. Steklova, 234, 2001 | MR | Zbl

[14] Ovsyannikov L. V., Monakhov V. N., Nelineinye problemy teorii poverkhnostnykh i vnutrennikh voln, Nauka, Novosibirsk, 1985 | Zbl

[15] Pletner Yu. D., “Fundamentalnye resheniya operatorov tipa Soboleva i nekotorye nachalno-kraevye zadachi”, Zhurn. vychisl. matem. i matem. fiz., 32:12 (1992), 1885–1899 | MR | Zbl

[16] Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Nauka, M., 1987 | MR

[17] Sveshnikov A. G., Alshin A. B., Korpusov M. O., Pletner Yu. D., Lineinye i nelineinye uravneniya sobolevskogo tipa, Fizmatlit, M., 2007 | Zbl

[18] Sveshnikov A. G., Alshin A. B., Korpusov M. O., Nelineinyi funktsionalnyi analiz i ego prilozheniya k uravneniyam v chastnykh proizvodnykh, Nauchnyi Mir, M., 2008

[19] Temam R., Uravneniya Nave–Stoksa. Teoriya i chislennyi analiz, Mir, M., 1981 | MR | Zbl

[20] Helfrich K. R., Melville W. K., “Long nonlinear internal waves”, Annu. Rev. Fluid Mech., 38 (2006), 395–425 | DOI | MR | Zbl

[21] Gasinski L., Papageorgiou N. S., Nonlinear Analysis, Series in Mathematical Analysis and Applications, 9, Chapman and Hall, Boca Raton, FL, 2005 | MR

[22] Levine H. A., “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+\mathscr F(u)$”, Arch. Rational. Mech. Anal., 51 (1973), 371–386 | DOI | MR | Zbl

[23] Levine H. A., “Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+\mathscr F(u)$”, Trans. Amer. Math. Soc., 192 (1974), 1–21 | MR | Zbl