Geodesic
Cauchy problem in a~scale of Banach spaces
Informatics and Automation, Modern problems of mechanics, Tome 281 (2013), pp. 7-15.

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

The concept of quasidifferential operator in a scale of Banach spaces is formulated. A theorem of existence and uniqueness of a solution to the Cauchy problem for the equation with a nonlinear quasidifferential operator is proved. As an example of application of the theorem, the correctness of the nonlinear nonlocal problem of plane-parallel unsteady potential motion of a liquid with free boundary is proved. 
@article{TRSPY_2013_281_a1,
     author = {L. V. Ovsyannikov},
     title = {Cauchy problem in a~scale of {Banach} spaces},
     journal = {Informatics and Automation},
     pages = {7--15},
     publisher = {mathdoc},
     volume = {281},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2013_281_a1/}
}
TY  - JOUR
AU  - L. V. Ovsyannikov
TI  - Cauchy problem in a~scale of Banach spaces
JO  - Informatics and Automation
PY  - 2013
SP  - 7
EP  - 15
VL  - 281
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2013_281_a1/
LA  - ru
ID  - TRSPY_2013_281_a1
ER  - 
%0 Journal Article
%A L. V. Ovsyannikov
%T Cauchy problem in a~scale of Banach spaces
%J Informatics and Automation
%D 2013
%P 7-15
%V 281
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2013_281_a1/
%G ru
%F TRSPY_2013_281_a1
L. V. Ovsyannikov. Cauchy problem in a~scale of Banach spaces. Informatics and Automation, Modern problems of mechanics, Tome 281 (2013), pp. 7-15. http://geodesic.mathdoc.fr/item/TRSPY_2013_281_a1/

[1] Stoker J.J., Water waves: The mathematical theory with applications, Interscience, New York, 1957 | MR | Zbl

[2] Ovsyannikov L.V., “About motion of a finite mass of liquid”, Fluid dynamics transactions, V. 3, PWN, Warszawa, 1967, 75–81

[3] Ovsyannikov L.V., “Singulyarnyi operator v shkale banakhovykh prostranstv”, DAN SSSR, 163:4 (1965), 819–822 | MR | Zbl

[4] Ovsyannikov L.V., “Obschie uravneniya i primery”, Zadacha o neustanovivshemsya dvizhenii zhidkosti so svobodnoi granitsei, Nauka, Novosibirsk, 1967, 3–75

[5] Trèves F., Ovcyannikov theorem and hyperdifferential operators, Notas Mat., 46, Inst. Mat. Pura Apl., Rio de Janeiro, 1968 | MR

[6] Krein S.G., Petunin Yu.I., “Shkaly banakhovykh prostranstv”, UMN, 21:2 (1966), 89–168 | MR | Zbl

[7] Ovsyannikov L.V., “Nelineinaya zadacha Koshi v shkale banakhovykh prostranstv”, DAN SSSR, 200:4 (1971), 789–792 | Zbl

[8] Nalimov V.I., “Apriornye otsenki reshenii ellipticheskikh uravnenii v klasse analiticheskikh funktsii i ikh prilozheniya k zadache Koshi–Puassona”, DAN SSSR, 189:1 (1969), 45–48 | MR | Zbl

[9] Demidov G.V., “Nekotorye prilozheniya obobschennoi teoremy Kovalevskoi”, Chislennye metody mekhaniki sploshnoi sredy, T. 1, No2, Izd-vo In-ta teor. i prikl. mekh. SO AN SSSR, Novosibirsk, 1970, 10–32 | MR

[10] Ovsyannikov L.V., Makarenko N.I., Nalimov V.I. i dr., Nelineinye problemy teorii poverkhnostnykh i vnutrennikh voln, Nauka, Novosibirsk, 1985

[11] Nishida T., “A note on a theorem of Nirenberg”, J. Diff. Geom., 12:4 (1977), 629–633 | MR | Zbl

[12] Nirenberg L., Topics in nonlinear functional analysis, Courant Inst. Math. Sci., New York, 1974 | MR | MR | Zbl | Zbl

[13] Ovsyannikov L.V., “Obosnovanie teorii melkoi vody”, Tr. Vsesoyuz. konf. po uravneniyam s chastnymi proizvodnymi, posv. 75-letiyu I.G. Petrovskogo, M., 1976, MGU, M., 1978, 185–188