Solvability of the problem of evolution of an isolated volume of viscous, incompressible capillary fluid

The initial boundary-value problem for the Navier–Stokes equation describing the flow of a viscous, incompressible capillary fluid bounded only by a free surface is considered. At the initial time the region occupied by the fluid and the velocity field of the fluid are given. A theorem is formulated regarding the unique solvability of the problem for a finite time interval, and a model linearized problem in a half space is obtained.