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

摘要

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 Γ~1, . .., Γ~M and N - M bounded components Γ~(M+1), . .., Γ~N. We use the inhomogeneous Dirichlet boundary condition on ?Ω. The prescribed velocity profile α on ?Ω is assumed to have an L~3-extension to Ω with the gradient in L~2(Ω)~(3×3). We assume that the fluxes of α through the bounded components Γ~(M+1), . .., Γ~N of ?Ω are "sufficiently small", but we impose no restriction on the size of fluxes through the unbounded components Γ~1, . .., Γ~M.