ABSTRACT
This paper is written on the basis of lectures given by the author at a mathematical school on the Navier-Stokes equations in Lisbon in September 1991, and it is concerned with questions of the solvability of stationary and evolution problems of a viscous incompressible flow in domains with noncompact boundaries. A special attention has been given to these problems in recent 15 years when it became evident that they should be formulated in a slightly different way, compared to problems in bounded or exterior domains: in addition to standard initial and boundary conditions there should be prescribed such quantities as fluxes of the velocity vector field through some apertures or differences of limiting values of the pressure at infinity. Another characteristic feature of these problems consists in the fact that very often their solutions have unbounded energy integral which means that a basic a priori estimate given by the energy inequality may become deficient. In recent years there was elaborated a special techniques of integral estimates (so called “techniques of the Saint-Venanťs principle”) that makes it possible to construct and to investigate such solutions. Finally, some problems with noncompact free boundaries were considered.
