The problem of the evolution of two viscous immiscible liquids with an unknown interface belongs to the currently intensively studied class of problems with free boundaries, since in it, along with the vector velocity field and the pressure function of both liquids, their interface must be determined. The theory of these problems for the Navier - Stokes equations dates back only a little more than four decades in its development, although their formulation dates back to the classical works of the 19th century.