$L_2$-theory for two incompressible fluids separated by a free interface

Abstract

The paper is devoted to the problem of non-stationary motion of two viscous incompressible fluids separated by a free surface and contained in a bounded vessel. It is assumed that the fluids are subject to mass forces and capillary forces at the interface. We prove the stability of a rest state under the assumption that initial velocities are small, a free interface is close to a sphere at an initial instant of time, and mass forces decay as $t\to\infty$.