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

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## 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 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

*through the bounded components Γ*

**α**^{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 }.

## Keywords

Weak Solution Sobolev Inequality Unbounded Domain Lipschitzian Domain Stokes Problem
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