We study the problem on small motions of two non mixing ideal stratified fluids with a free surface, covered with crumbling ice. Using method of orthogonal projecting the boundary conditions on the moving surface and the introduction of auxiliary problems of the original initial-boundary value problem is reduced to the equivalent Cauchy problem for a differential equation of second order in some Hilbert space. We find sufficient existence conditions for a strong (with respect to the time variable) solution of the initial-boundary value problem describing the evolution of the specified hydrodynamics system.

Denis O. Tsvetkov, Cand. Sci. (Phys.–Math.), Taurida Academy, Crimean Federal University, 4, Vernadskii pr., Simferopol, 295007, Russian Federation, e-mail: tsvetdo@gmail.com

Tsvetkov D.O. Small Movements of a System of Ideal Stratified Fluids Completely Covered with Crumbled Ice. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 26, pp. 105-120. (in Russian) https://doi.org/10.26516/1997-7670.2018.26.105

1. Agranovich M.S. Spectral problems for second-order strongly elliptic systems in smooth and non-smooth domains. Russian Math. Surveys, 2002, vol. 57, no. 5, pp. 847–920. https://doi.org/10.4213/rm552
2. Gabov S.A., Sveshnikov A.G. Problems in the dynamics of flotation liquids. J. Math. Sci., 1991, vol. 54, no.4, pp. 979-1041. https://doi.org/10.1007/BF01138947
3. Daletsky Yu.L., Krein M.G. Stability of solutions of differential equations in a Banach space. Moscow, 1970, 534 p. (in Russian)
4. Kopachevskii N.D. Abstract green’s formula and its applications. Simferopol, 2016, 280 p. (in Russian)
5. Kopachevskii N.D., Krein S.G., Ngo Zuy Can. Operator methods are in linear hydrodynamics: evolution and spectral problems. Moscow, Nauka Publ., 1989, 416 p. (in Russian)
6. Kopachevskii N.D., Tsvetkov D.O. Oscillations of stratified fluids. J. Math. Sci., 2010, vol. 164, no. 4, pp. 574-602. https://doi.org/10.1007/s10958-010-9764-9
7. Kopachevskii N.D., Tsvetkov D.O. Cauchy problem generated by oscillations of stratified fluid partially closed by ice. Tavricheskij vestnik informatiki i matematiki, 2018, vol. 1, no. 38, pp. 31-39 (in Russian)
8. Kopachevskii N.D., Tsvetkov D.O. Small motions of ideal stratified liquid with a free surface totally covered by a crumbled ice. Ufa Mathematical Journal, 2018, vol. 10, no. 3, pp. 44–59. https://doi.org/10.13108/2018-10-3-43
9. Temchenko T.P. Spectral and evolutionary the problems of the oscillations of a stratified fluids. Cand. sci. diss. Moscow, 1989, 147 p. https://search.rsl.ru/ru/record/01007932205
10. Petters A.S. The effect of a floating mat on the water waves. Communs Pure and Appl. Math., 1950, vol. 3, no. 4, pp. 319-354.