On the Nonhomogeneous Boundary Value Problem for the Navier–Stokes System in a Class of Unbounded Domains

The stationary Navier–Stokes system with nonhomogeneous boundary conditions is studied in a class of domains Ω having “paraboloidal” outlets to infinity. The boundary is multiply connected and consists of M infinite connected components S m , which form the outer boundary, and I compact connected components Γ i forming the inner boundary Γ. The boundary value a is assumed to have a compact support and it is supposed that the fluxes of a over the components Γ i of the inner boundary are sufficiently small. We do not pose any restrictions on fluxes of a over the infinite components S m . The existence of at least one weak solution to the Navier–Stokes problem is proved. The solution may have finite or infinite Dirichlet integral depending on geometrical properties of outlets to infinity.