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

Abstract

We prove the existence of a weak solution to the steady Navier–Stokes problem in a three dimensional domain Ω, whose boundary ∂Ω consists of M unbounded components Γ1, . . . , ΓM and N − M bounded components ΓM+1, . . . , ΓN. We use the inhomogeneous Dirichlet boundary condition on ∂Ω. The prescribed velocity profile α on ∂Ω is assumed to have an L3-extension to Ω with the gradient in L2(Ω)3×3. We assume that the fluxes of α through the bounded components ΓM+1, . . . , ΓN of ∂Ω are “sufficiently small”, but we impose no restriction on the size of fluxes through the unbounded components Γ1, . . . , ΓM.