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 $C_0^\infty(\Omega)$ 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”.