### 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 |
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Pages (from-to) | 231-247 |

Number of pages | 17 |

Journal | Funkcialaj Ekvacioj |

Volume | 53 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 Aug 1 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology

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## Cite this

*Funkcialaj Ekvacioj*,

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