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

We consider the question of the possibility of approximation by solenoidal vectors from $C_0^\infty(\Omega)$ of solenoidal vectors with finite Dirichlet integral, defined in a domain $\Omega$, $\Omega\subset\mathbf R^3$, with some “exits” to infinity in the form of rotation bodies and vanishing on $\partial\Omega$. 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 $\partial\Omega$ and at infinity, the prescription of the flows of the velocity vector across certain “exits”.