A steady weak solution of the equations of motion of a viscous incompressible fluid through porous media in a domain with a non-compact boundary

We assume that Ω is a domain in ℝ ² or in ℝ ³ with a non-compact boundary, representing a generally inhomogeneous and anisotropic porous medium. We prove the weak solvability of the boundary-value problem, describing the steady motion of a viscous incompressible fluid in Ω. We impose no restriction on sizes of the velocity fluxes through unbounded components of the boundary of Ω. The proof is based on the construction of appropriate Galerkin approximations and study of their convergence. In Sect. 4, we provide several examples of concrete forms of Ω and prescribed velocity profiles on ∂Ω, when our main theorem can be applied.