Leray's inequality in general multi-connected domains in ℝ n

Reinhard Farwig, Hideo Kozono, Taku Yanagisawa

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Abstract

Consider the stationary Navier-Stokes equations in a bounded domain Ω ⊃ ℝ n whose boundary ∂Ω consists of L + 1 smooth (n - 1)-dimensional closed hypersurfaces Γ 0, Γ 1, . . ., Γ L, where Γ 1, . . ., Γ L lie inside of Γ 0 and outside of one another. The Leray inequality of the given boundary data β on ∂Ω plays an important role for the existence of solutions. It is known that if the flux γ i ≡ ∫ Γi β · νd S = 0 on Γ i(ν: the unit outer normal to Γ i) is zero for each i = 0, 1, . . ., L, then the Leray inequality holds. We prove that if there exists a sphere S in Ω separating ∂Ω in such a way that Γ 1, . . ., Γ k (1 ≦ k ≦ L) are contained inside of S and that the others Γ k+1, . . ., Γ L are outside of S, then the Leray inequality necessarily implies that γ 1 + · · · + γ k = 0. In particular, suppose that there are L spheres S 1, . . ., S L in Ω lying outside of one another such that Γ i lies inside of S i for all i = 1, . . ., L. Then the Leray inequality holds if and only if γ 0 = γ 1 = · · · = γ L = 0.

Original languageEnglish
Pages (from-to)137-145
Number of pages9
JournalMathematische Annalen
Volume354
Issue number1
DOIs
Publication statusPublished - 2012 Sep 1

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ASJC Scopus subject areas

  • Mathematics(all)

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