Weak solutions of the stationary Navier-Stokes equations for a viscous incompressible fluid past an obstacle

Horst Heck, Hyunseok Kim, Hideo Kozono

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Abstract

Consider the stationary Navier-Stokes equations in an exterior domain Ω ⊂ ℝ3 with smooth boundary. For every prescribed constant vector u ≠ 0 and every external force f ∈ Ḣ2-1(Ω), Leray (J. Math. Pures. Appl., 9:1-82, 1933) constructed a weak solution u with ∇u ∈ L2(Ω) and u-u ∈ L6(Ω). Here Ḣ2-1(Ω) denotes the dual space of the homogeneous Sobolev space Ḣ12(Ω). We prove that the weak solution u fulfills the additional regularity property u-u ∈ L4(Ω) and u · ∇u Ḣ2-(Ω) without any restriction on f except for f ∈ Ḣ2-1(Ω). As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that {double pipe}f{double pipe}Ḣ-12(Ω) and {pipe} u{pipe} are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1-82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as u→0 in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case u≠0.

Original languageEnglish
Pages (from-to)653-681
Number of pages29
JournalMathematische Annalen
Volume356
Issue number2
DOIs
Publication statusPublished - 2013 Jun 1

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ASJC Scopus subject areas

  • Mathematics(all)

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