Geodesic
Solvability of a~problem on the motion of a~viscous incompressible fluid bounded by a~free surface
Izvestiya. Mathematics , Tome 11 (1977) no. 6, pp. 1323-1358.

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We prove the existence of a single-valued classical solution, local with respect to the time, of an initial-boundary value problem for the system of Navier–Stokes equations describing, in a specified force field, the motion of a finite mass of fluid with a free surface. In this problem, not only are the velocity and pressure of the fluid to be determined, but also the region which the fluid occupies at each instant of time. In studying this problem we employ Lagrangian coordinates. Bibliography: 10 titles. 
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V. A. Solonnikov. Solvability of a~problem on the motion of a~viscous incompressible fluid bounded by a~free surface. Izvestiya. Mathematics , Tome 11 (1977) no. 6, pp. 1323-1358. http://geodesic.mathdoc.fr/item/IM2_1977_11_6_a7/

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[2] Solonnikov V. A., “O razreshimosti vtoroi nachalno-kraevoi zadachi dlya lineinoi nestatsionarnoi sistemy uravnenii Nave–Stoksa”, Zap. nauchn. seminarov Leningr. otd. Matem. in-ta AN SSSR, 69, 1977, 200–218 | MR | Zbl

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[7] Pukhnachev V. V., Zadachi so svobodnoi granitsei dlya uravnenii Nave–Stoksa, Dissertatsiya na soiskanie uchenoi stepeni doktora fiz.-matem. nauk, Novosibirsk, 1974

[8] Solonnikov V. A., “Otsenki reshenii nestatsionarnoi sistemy Nave–Stoksa”, Zap. nauchn. seminarov Leningr. otd. Matem. in-ta AN SSSR, 38, 1973, 153–231 | MR | Zbl

[9] Bytev V. I., “Neustanovivshiesya dvizheniya vraschayuschegosya koltsa vyazkoi neszhimaemoi zhidkosti so svobodnoi granitsei”, PMTF, 1970, no. 3, 82–88 | MR

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