Global solvability of a problem on two fluid motion without surface tension

Unsteady motion of viscous incompressible fluids is considered in a bounded domain. The liquids are separated by an unknown interface on which the surface tension is neglected. This motion is governed by an interface problem for the Navier–Stokes system. First, a local existence theorem is established for the problem in Hölder classes of functions. The proof is based on the solvability of a model problem for the Stokes system with a plane interface which was obtained earlier. Next, for a small initial velocity vector field and small mass forces, we prove the existence of a unique smooth solution to the problem on the infinite time interval.