On a steady motion of a drop in an infinite liquid medium

There is given a rigorous proof of the solvability of the free boundary problem for the Navier–Stokes equations governing a steady fall (or uprising) of a drop in an infinite liquid medium. It is assumed that the densities of both liquids are close to each other and the solution is obtained as a perturbation of the rest state. However, in comparison with other problems of this type, the proof requires much more delicate arguments since the Frechet derivative of the corresponding operator computed at the rest state is not invertible. Bibl. 8 titles.