### Abstract

We investigate the nonstationary Navier-Stokes equations for an exterior domain Ω ⊂ R^{3} in a solution class L^{s}(0,T; L ^{q}(Ω)) of very low regularity in space and time, satisfying Serrin's condition 2/s + 3/q = 1 but not necessarily any differentiability property. The weakest possible boundary conditions, beyond the usual trace theorems, are given by u|_{∂Ω} = g ε L^{s}(0,T; W^{-1/q,q}(∂Ω)), and will be made precise in this paper. Moreover, we suppose the weakest possible divergence condition k = div u ε L^{s} (0,T; L^{r}(Ω)), where 1/3 + 1/q = 1/r.

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

Number of pages | 24 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 59 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 Jan 1 |

### Keywords

- Nonhomogeneous data
- Serrin's class
- Stokes and navier-stokes equations
- Very weak solutions

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Farwig, R., Kozono, H., & Sohr, H. (2007). Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data.

*Journal of the Mathematical Society of Japan*,*59*(1), 127-150. https://doi.org/10.2969/jmsj/1180135504