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We prove the existence of a unique Poiseuille-type solution to the time almost-periodic (in Stepanov sense) Stokes problem in an infinite spatially periodic pipe.

In this paper, we study the nonhomogeneous boundary value problem for the stationary Navier–Stokes equations in two-dimensional symmetric domains with finitely many outlets to infinity. The domains may have no self-symmetric outlet (-type domain), one self-symmetric outlet (-type domain) or two self-symmetric outlets (-type domain). We construct a symmetric solenoidal extension of the boundary value satisfying the Leray–Hopf inequality. After having such an extension, the nonhomogeneous boundary value problem is reduced to homogeneous one and the existence of at least one weak solution follows. Notice that we do not impose any restrictions on the size of the fluxes over the inner and outer boundaries. Moreover, the Dirichlet integral of the solution can be either finite or infinite depending on the geometry of the domains.

The time periodic linearized Navier–Stokes system (Stokes system) with nonhomogeneous boundary conditions is studied in a domain Ω which has a “paraboloidal” outlet to infinity. The time periodic boundary value a ( x , t ) is assumed to have a compact support and it is supposed that the flux of a over ∂Ω is nonzero. The existence and uniqueness of a weak solution is proved. The solution can have either finite or infinite Dirichlet integral depending on geometrical properties of the outlet to infinity.

The nonhomogeneous boundary value problem for the stationary Navier–Stokes equations in a two-dimensional domain with a cusp point on the boundary is studied. The case when the flux of the boundary value $$\mathbf{a }$$ is nonzero, i.e., when there is a source or sink in the cusp point, is considered. The existence of at least one weak solution having infinite Dirichlet integral is proved without any restrictions on the size of the flux $$F=\int \limits _{\partial \Omega }\mathbf{a }\cdot \mathbf{n }dS$$ .

We study the stationary nonhomogeneous Navier–Stokes problem in a two-dimensional symmetric domain with a semi-infinite outlet (for instance, either paraboloidal or channel-like). Under the symmetry assumptions on the domain, boundary value and external force, we prove the existence of at least one weak symmetric solution without any restriction on the size of the fluxes, i.e. the fluxes of the boundary value [Formula: see text] over the inner and the outer boundaries may be arbitrarily large. Only the necessary compatibility condition (the total flux is equal to zero) has to be satisfied. Moreover, the Dirichlet integral of the solution can be finite or infinite depending on the geometry of the domain.

The stationary Navier–Stokes system with nonhomogeneous boundary conditions is studied in a class of domains Ω having “paraboloidal” outlets to infinity. The boundary $${\partial\Omega}$$ 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.

In this paper the stationary Navier–Stokes system with non-homogeneous boundary condition is studied in domain which consists of two connected layers. The extension of the boundary value, which reduces the non-homogeneous boundary problem to the homogeneous one, is constructed in this paper.

In this paper the stationary Navier–Stokes system with non-homogeneous boundary condition is studied in the unbounded domain. The extension of the boundary value satisfying Leray’s inequality is constructed. Therefore the non-homogeneous boundary problem could be reduced to the homogeneous one which was already investigated before.