Vsevolod Solonnikov
Steklov Mathematics Institute - St.Petersburg

Abstract: The communication is concerned with the problem governing the non-stationary motion of two immiscible fluids (both incompressible or incompressible and compressible), contained in a bounded vessel and separated with a free interface. The motion is described by the system of two Navier-Stokes equations completed by initial and boundary conditions at the exterior boundary and at the free interface that is given at the initial instant . It is proved that the problem is uniquely solvable in the Sobolev spaces of functions locally in time or in the infinite time interval , provided that the initial data are close to the rest state: the velocity vector fields of both fluids vanish, the pressure and the density of the compressible fluid are constant, the free boundary is a sphere. As , the solution tends to the equilibrium state. The resuls are obtained in collaboration with I.V. Denisova.

This seminar is supported by National Funding from FCT - Fundação para a Ciência e a Tecnologia, under the project: UID/MAT/04561/2013.