### 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 |
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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 |

DOIs | |

Publication status | Published - 2011 Jan 1 |

Externally published | Yes |

### ASJC Scopus subject areas

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

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## Cite this

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

*125*, 51-70. https://doi.org/10.4171/RSMUP/125-4