Geodesic
Certain spaces of solenoidal vectors and the solvability of the boundary problem for the Navier--Stokes system of equations in domains with noncompact boundaries
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 136-151

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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”. 
@article{ZNSL_1977_73_a9,
     author = {V. A. Solonnikov and K. I. Pileckas},
     title = {Certain spaces of solenoidal vectors and the solvability of the boundary problem for the {Navier--Stokes} system of equations in domains with noncompact boundaries},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {136--151},
     publisher = {mathdoc},
     volume = {73},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1977_73_a9/}
}
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V. A. Solonnikov; K. I. Pileckas. Certain spaces of solenoidal vectors and the solvability of the boundary problem for the Navier--Stokes system of equations in domains with noncompact boundaries. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 136-151. http://geodesic.mathdoc.fr/item/ZNSL_1977_73_a9/