### Abstract

Consider the Navier-Stokes equations in a smooth bounded domain Ω ⊂ R^{3} and a time interval [0,T), 0 < T ≤ ∞. It is well-known that there exists at least one global weak solution u with vanishing boundary values u_{∂Ω} = 0 for any given initial value u_{0} ∈ L^{2}
_{σ}(Ω) external force f = div F, F ∈ L^{2} (0,T;L^{2}(Ω)), and satisfying the strong energy inequality. Our aim is to extend this existence result to a much larger class of global in time "Leray-Hopf type" weak solutions u with nonzero boundary values u_{∂Ω} = g ∈ W ^{1/2,2}(∂Ω). As for usual weak solutions we do not need any smallness condition on g; indeed, our generalized weak solutions u exist globally in time. The solutions will satisfy an energy estimate with exponentially increasing terms in time, but for simply connected domains the energy increases at most linearly in time.

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

Pages (from-to) | 231-247 |

Number of pages | 17 |

Journal | Funkcialaj Ekvacioj |

Volume | 53 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 Aug |

Externally published | Yes |

### Fingerprint

### Keywords

- Navier-stokes equations
- Nonhomogeneous boundary values
- Strong energy inequality
- Weak solution

### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Geometry and Topology

### Cite this

*Funkcialaj Ekvacioj*,

*53*(2), 231-247. https://doi.org/10.1619/fesi.53.231

**Global weak solutions of the Navier-Stokes system with nonzero boundary conditions.** / Farwig, R.; Kozono, Hideo; Sohr, H.

Research output: Contribution to journal › Article

*Funkcialaj Ekvacioj*, vol. 53, no. 2, pp. 231-247. https://doi.org/10.1619/fesi.53.231

}

TY - JOUR

T1 - Global weak solutions of the Navier-Stokes system with nonzero boundary conditions

AU - Farwig, R.

AU - Kozono, Hideo

AU - Sohr, H.

PY - 2010/8

Y1 - 2010/8

N2 - Consider the Navier-Stokes equations in a smooth bounded domain Ω ⊂ R3 and a time interval [0,T), 0 < T ≤ ∞. It is well-known that there exists at least one global weak solution u with vanishing boundary values u∂Ω = 0 for any given initial value u0 ∈ L2 σ(Ω) external force f = div F, F ∈ L2 (0,T;L2(Ω)), and satisfying the strong energy inequality. Our aim is to extend this existence result to a much larger class of global in time "Leray-Hopf type" weak solutions u with nonzero boundary values u∂Ω = g ∈ W 1/2,2(∂Ω). As for usual weak solutions we do not need any smallness condition on g; indeed, our generalized weak solutions u exist globally in time. The solutions will satisfy an energy estimate with exponentially increasing terms in time, but for simply connected domains the energy increases at most linearly in time.

AB - Consider the Navier-Stokes equations in a smooth bounded domain Ω ⊂ R3 and a time interval [0,T), 0 < T ≤ ∞. It is well-known that there exists at least one global weak solution u with vanishing boundary values u∂Ω = 0 for any given initial value u0 ∈ L2 σ(Ω) external force f = div F, F ∈ L2 (0,T;L2(Ω)), and satisfying the strong energy inequality. Our aim is to extend this existence result to a much larger class of global in time "Leray-Hopf type" weak solutions u with nonzero boundary values u∂Ω = g ∈ W 1/2,2(∂Ω). As for usual weak solutions we do not need any smallness condition on g; indeed, our generalized weak solutions u exist globally in time. The solutions will satisfy an energy estimate with exponentially increasing terms in time, but for simply connected domains the energy increases at most linearly in time.

KW - Navier-stokes equations

KW - Nonhomogeneous boundary values

KW - Strong energy inequality

KW - Weak solution

UR - http://www.scopus.com/inward/record.url?scp=79251547052&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79251547052&partnerID=8YFLogxK

U2 - 10.1619/fesi.53.231

DO - 10.1619/fesi.53.231

M3 - Article

AN - SCOPUS:79251547052

VL - 53

SP - 231

EP - 247

JO - Funkcialaj Ekvacioj

JF - Funkcialaj Ekvacioj

SN - 0532-8721

IS - 2

ER -