Geodesic
Small motions of an ideal stratified fluid in a basin covered with ice
Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 64 (2018) no. 3, pp. 573-590 
N. D. Kopachevsky; D. O. Tsvetkov. Small motions of an ideal stratified fluid in a basin covered with ice. Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 64 (2018) no. 3, pp. 573-590. http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a4/
@article{CMFD_2018_64_3_a4,
     author = {N. D. Kopachevsky and D. O. Tsvetkov},
     title = {Small motions of an ideal stratified fluid in a~basin covered with ice},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {573--590},
     year = {2018},
     volume = {64},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a4/}
}
TY  - JOUR
AU  - N. D. Kopachevsky
AU  - D. O. Tsvetkov
TI  - Small motions of an ideal stratified fluid in a basin covered with ice
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2018
SP  - 573
EP  - 590
VL  - 64
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a4/
LA  - ru
ID  - CMFD_2018_64_3_a4
ER  - 
%0 Journal Article
%A N. D. Kopachevsky
%A D. O. Tsvetkov
%T Small motions of an ideal stratified fluid in a basin covered with ice
%J Contemporary Mathematics. Fundamental Directions
%D 2018
%P 573-590
%V 64
%N 3
%U http://geodesic.mathdoc.fr/item/CMFD_2018_64_3_a4/
%G ru
%F CMFD_2018_64_3_a4

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We study the problem on small motions of an ideal stratified fluid with a free surface partially covered with crushed ice. The crushed ice is supposed to be ponderable particles of some matter floating on the free surface. These particles do not interact with each other during oscillations of the free boundary (or this interaction is neglible) and stay on the surface during these oscillations. Using the method of orthogonal projecting of boundary-value conditions on the free surface and introducing auxiliary problems, we reduce the original initial-boundary value problem to the equivalent Cauchy problem for a second-order differential equation in some Hilbert space. We obtain conditions under which there exists a strong with respect to time solution of the initial-boundary value problem describing the evolution of this hydraulic system.

[1] S. A. Gabov, “On one problem of hydrodynamics of an ideal fluid related to flotation”, Differ. Equ., 24:1 (1986), 16–21 (in Russian) | MR

[2] S. A. Gabov, A. G. Sveshnikov, “Mathematical problems of hydrodynamics of floating fluid”, Totals Sci. Tech. Ser. Math. Anal., 28, 1990, 3–86 (in Russian) | MR | Zbl

[3] J. Goldstein, Semigroups of Linear Operators and Applications, Vyshcha shkola, Kiev, 1989, (Russian translation)

[4] I. V. Ivanov, I. V. Mel'nikov, A. I. Filinkov, Differential-Operator Equations and Ill-Posed Problems, Fizmatlit, M., 1995 (in Russian)

[5] N. D. Kopachevsky, Volterra Integrodifferential Equations in a Hilbert Space, A Special Course, FLP O. A. Bondarenko, Simferopol', 2012 (in Russian)

[6] N. D. Kopachevsky, S. G. Kreyn, Ngo Zuy Kan, Operator Methods in Linear Hydrodynamics: Evolution and Spectral Problems, Nauka, Moscow, 1989 (in Russian)

[7] N. D. Kopachevsky, D. O. Tsvetkov, “Oscillations of stratified fluids”, Contemp. Math. Fundam. Directions, 29, 2008, 103–130 (in Russian) | MR

[8] S. G. Kreyn, Linear Differential Equations in Banach Space, Nauka, Moscow, 1967 (in Russian)

[9] M. A. Soldatov, “Oscillations of a fluid in a basin partially covered with ice”, Sci. Notes SGU, 12:2 (2000), 80–83 (in Russian) | MR

[10] Sowa M., “Cosine operator functions”, Rozpr. Math., 49 (1966), 1–47 | MR