Geodesic
Solvability of a~problem on the plane motion of a~heavy viscous incompressible capillary liquid partially filling a~container
Izvestiya. Mathematics , Tome 14 (1980) no. 1, pp. 193-221

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The solvability of a plane stationary boundary value problem for the Navier–Stokes system of equations describing the motion of a viscous incompressible capillary liquid with a free surface is proved. It is assumed that the free surface maintains contact with the wall of the container holding the liquid. Bibliography: 21 titles. 
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     title = {Solvability of a~problem on the plane motion of a~heavy viscous incompressible capillary liquid partially filling a~container},
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V. A. Solonnikov. Solvability of a~problem on the plane motion of a~heavy viscous incompressible capillary liquid partially filling a~container. Izvestiya. Mathematics , Tome 14 (1980) no. 1, pp. 193-221. http://geodesic.mathdoc.fr/item/IM2_1980_14_1_a8/