### Abstract

Consider a smooth bounded domain Ω ⊆ ℝ ^{3} with boundary ∂Ω, a time interval [0, T), 0<T ≤ ∞, and the Navier-Stokes system in [0, T) × Ω, with initial value u _{0} ∈ L ^{2} _{σ}(Ω) and external force f = div F, F ∈ L ^{2}(0, T;L ^{2}(Ω)). Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u{pipe} _{∂Ω} = 0, div u = 0 to the more general class of Leray-Hopf type weak solutions u with general data u{pipe} _{∂Ω} = g, div u = k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u = υ + E with some vector field E in [0, T) × Ω satisfying the (linear) Stokes system with f = 0 and nonhomogeneous data. This reduces the general system to a perturbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system.

Original language | English |
---|---|

Pages (from-to) | 51-70 |

Number of pages | 20 |

Journal | Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova |

Volume | 125 |

Publication status | Published - 2011 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Geometry and Topology
- Mathematical Physics

### Cite this

**Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence.** / Farwig, R.; Kozono, Hideo; Sohr, H.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence

AU - Farwig, R.

AU - Kozono, Hideo

AU - Sohr, H.

PY - 2011

Y1 - 2011

N2 - Consider a smooth bounded domain Ω ⊆ ℝ 3 with boundary ∂Ω, a time interval [0, T), 0<T ≤ ∞, and the Navier-Stokes system in [0, T) × Ω, with initial value u 0 ∈ L 2 σ(Ω) and external force f = div F, F ∈ L 2(0, T;L 2(Ω)). Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u{pipe} ∂Ω = 0, div u = 0 to the more general class of Leray-Hopf type weak solutions u with general data u{pipe} ∂Ω = g, div u = k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u = υ + E with some vector field E in [0, T) × Ω satisfying the (linear) Stokes system with f = 0 and nonhomogeneous data. This reduces the general system to a perturbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system.

AB - Consider a smooth bounded domain Ω ⊆ ℝ 3 with boundary ∂Ω, a time interval [0, T), 0<T ≤ ∞, and the Navier-Stokes system in [0, T) × Ω, with initial value u 0 ∈ L 2 σ(Ω) and external force f = div F, F ∈ L 2(0, T;L 2(Ω)). Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u{pipe} ∂Ω = 0, div u = 0 to the more general class of Leray-Hopf type weak solutions u with general data u{pipe} ∂Ω = g, div u = k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u = υ + E with some vector field E in [0, T) × Ω satisfying the (linear) Stokes system with f = 0 and nonhomogeneous data. This reduces the general system to a perturbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system.

UR - http://www.scopus.com/inward/record.url?scp=84856151283&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856151283&partnerID=8YFLogxK

M3 - Article

VL - 125

SP - 51

EP - 70

JO - Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova

JF - Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova

SN - 0041-8994

ER -