Evolution free boundary problem for the Navier–Stokes equations

The subject of this paper is a two-dimensional evolution free boundary problem for a viscous incompressible fluid that partially fills a container. The purpose is to prove the time-local unique solvability of the problem for the Navier–Stokes system in the Sobolev–Slobodetsky spaces and to estimate the obtained solution in these spaces. This result is achieved for small wetting angles.