A New Approach to the Existence of Weak Solutions of the Steady Navier-Stokes System with Inhomogeneous Boundary Data in Domains with Noncompact Boundaries

We prove the existence of a weak solution to the steady Navier-Stokes problem in a three dimensional domain Omega, whose boundary 80 consists of M unbounded components Gamma(1),..., Gamma(M) and N - M bounded components Gamma(M+1),, Gamma(N). We use the inhomogeneous Dirichlet boundary condition on partial derivative Omega. The prescribed velocity profile a on a partial derivative Omega is assumed to have an L(3)-extension to Omega with the gradient in L(2) (Omega)(3x3). We assume that the fluxes of a through the bounded components Gamma(M+1),..., Gamma(N) of a Omega are "sufficiently small", but we impose no restriction on the size of fluxes through the unbounded components Gamma(1),..., Gamma(M).