### Abstract

Consider the stationary Navier-Stokes equations in an exterior domain Ω ⊂ ℝ^{3} with smooth boundary. For every prescribed constant vector u_{∞} ≠ 0 and every external force f ∈ Ḣ_{2}^{-1}(Ω), Leray (J. Math. Pures. Appl., 9:1-82, 1933) constructed a weak solution u with ∇u ∈ L_{2}(Ω) and u-u_{∞} ∈ L_{6}(Ω). Here Ḣ_{2}^{-1}(Ω) denotes the dual space of the homogeneous Sobolev space Ḣ^{1}_{2}(Ω). We prove that the weak solution u fulfills the additional regularity property u-u_{∞} ∈ L_{4}(Ω) and u_{∞} · ∇u Ḣ_{2}^{-}(Ω) without any restriction on f except for f ∈ Ḣ_{2}^{-1}(Ω). As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that {double pipe}f{double pipe}_{Ḣ-12(Ω)} and {pipe} u_{∞}{pipe} are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1-82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as u_{∞}→0 in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case u_{∞}≠0.

Original language | English |
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Pages (from-to) | 653-681 |

Number of pages | 29 |

Journal | Mathematische Annalen |

Volume | 356 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2013 Jun 1 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Annalen*,

*356*(2), 653-681. https://doi.org/10.1007/s00208-012-0861-6