Abstract
We assume that Omega is a domain in a"e(2) or in a"e(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 Omega. We impose no restriction on sizes of the velocity fluxes through unbounded components of the boundary of Omega. 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 Omega and prescribed velocity profiles on a,Omega, when our main theorem can be applied.