### Abstract

We consider the stationary Navier-Stokes equations on a multiply connected bounded domain Ω in R{double-struck}^{n} for n = 2; 3 under nonhomogeneous boundary conditions. We present a new sufficient condition for the existence of weak solutions. This condition is a variational estimate described in terms of the harmonic part of solenoidal extensions of the given boundary data; we prove it by using the Helmholtz-Weyl decomposition of vector fields over Ω satisfying adequate boundary conditions. We also study the validity of Leray's inequality for various assumptions about the symmetry of Ω.

Original language | English |
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Pages (from-to) | 127-150 |

Number of pages | 24 |

Journal | Pacific Journal of Mathematics |

Volume | 243 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Nov |

Externally published | Yes |

### Fingerprint

### Keywords

- Helmholtz-Weyl decomposition
- Nonhomogeneous Boundary value problems
- Stationary Navier-Stokes equations

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Nonhomogeneous boundary value problems for stationary navier-stokes equations in a multiply connected bounded domain.** / Kozono, Hideo; Yanagisawa, Taku.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 243, no. 1, pp. 127-150. https://doi.org/10.2140/pjm.2009.243.127

}

TY - JOUR

T1 - Nonhomogeneous boundary value problems for stationary navier-stokes equations in a multiply connected bounded domain

AU - Kozono, Hideo

AU - Yanagisawa, Taku

PY - 2009/11

Y1 - 2009/11

N2 - We consider the stationary Navier-Stokes equations on a multiply connected bounded domain Ω in R{double-struck}n for n = 2; 3 under nonhomogeneous boundary conditions. We present a new sufficient condition for the existence of weak solutions. This condition is a variational estimate described in terms of the harmonic part of solenoidal extensions of the given boundary data; we prove it by using the Helmholtz-Weyl decomposition of vector fields over Ω satisfying adequate boundary conditions. We also study the validity of Leray's inequality for various assumptions about the symmetry of Ω.

AB - We consider the stationary Navier-Stokes equations on a multiply connected bounded domain Ω in R{double-struck}n for n = 2; 3 under nonhomogeneous boundary conditions. We present a new sufficient condition for the existence of weak solutions. This condition is a variational estimate described in terms of the harmonic part of solenoidal extensions of the given boundary data; we prove it by using the Helmholtz-Weyl decomposition of vector fields over Ω satisfying adequate boundary conditions. We also study the validity of Leray's inequality for various assumptions about the symmetry of Ω.

KW - Helmholtz-Weyl decomposition

KW - Nonhomogeneous Boundary value problems

KW - Stationary Navier-Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=77950281365&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950281365&partnerID=8YFLogxK

U2 - 10.2140/pjm.2009.243.127

DO - 10.2140/pjm.2009.243.127

M3 - Article

VL - 243

SP - 127

EP - 150

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -