Certain spaces of solenoidal vectors and the solvability of the boundary problem for the Navier-Stokes system of equations in domains with noncompact boundaries

Abstract

We consider the question of the possibility of approximation by solenoidal vectors from Co∞(Ω) of solenoidal vectors with finite Dirichlet integral, defined in a domain Ω, Ω⊂ℝ3, with some “exits” to infinity in the form of rotation bodies and vanishing on ∂Ω. A large class of domains is found for which such an approximation is impossible. It is shown that in these domains the formulation of the boundary problem for a stationary Navier-Stokes system of equations must include, besides the ordinary boundary conditions on ∂Ω and at infinity, the prescription of the flows of the velocity vector across certain “exits.”