### 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 language | English |
---|---|

Pages (from-to) | 137-145 |

Number of pages | 9 |

Journal | Mathematische Annalen |

Volume | 354 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2012 Sep |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

^{n}

*Mathematische Annalen*,

*354*(1), 137-145. https://doi.org/10.1007/s00208-011-0716-6

**Leray's inequality in general multi-connected domains in ℝ ^{n} .** / Farwig, Reinhard; Kozono, Hideo; Yanagisawa, Taku.

Research output: Contribution to journal › Article

^{n}',

*Mathematische Annalen*, vol. 354, no. 1, pp. 137-145. https://doi.org/10.1007/s00208-011-0716-6

^{n}Mathematische Annalen. 2012 Sep;354(1):137-145. https://doi.org/10.1007/s00208-011-0716-6

}

TY - JOUR

T1 - Leray's inequality in general multi-connected domains in ℝ n

AU - Farwig, Reinhard

AU - Kozono, Hideo

AU - Yanagisawa, Taku

PY - 2012/9

Y1 - 2012/9

N2 - 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.

AB - 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.

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U2 - 10.1007/s00208-011-0716-6

DO - 10.1007/s00208-011-0716-6

M3 - Article

VL - 354

SP - 137

EP - 145

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 1

ER -