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 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 ΓM+1, . . . , ΓN of ∂Ω are “sufficiently small”, but we impose no restriction on the size of fluxes through the unbounded components Γ1, . . . , ΓM.
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Neustupa, J. A New Approach to the Existence of Weak Solutions of the Steady Navier–Stokes System with Inhomogeneous Boundary Data in Domains with Noncompact Boundaries. Arch Rational Mech Anal 198, 331–348 (2010). https://doi.org/10.1007/s00205-010-0297-7
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DOI: https://doi.org/10.1007/s00205-010-0297-7