Abstract
We consider the question of the possibility of approximation by solenoidal vectors from C ∞o (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.”
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Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 136–151, 1977.
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Solonnikov, V.A., Piletskas, K.I. Certain spaces of solenoidal vectors and the solvability of the boundary problem for the Navier-Stokes system of equations in domains with noncompact boundaries. J Math Sci 34, 2101–2111 (1986). https://doi.org/10.1007/BF01741584
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DOI: https://doi.org/10.1007/BF01741584