Solvability of a problem on the motion of a viscous incompressible fluid bounded by a free surface

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.
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