## Abstract

Let u be a weak solution of the Navier-Stokes equations in a smooth domain Ω ⊆ ℝ and a time interval [0, T), 0 < T < ∞, with initial value u_{0}, and vanishing external force. As is well known, global regularity of u for general u_{0} is an unsolved problem unless we pose additional assumptions on u_{0} or on the solution u itself such as Serrin's condition ||u||L^{s}(0,T;L^{q}(Ω)) < ∞ where 2/s + 3/q = 1. In the present paper we prove several new local and global regularity properties by using assumptions beyond Serrin's condition e.g. as follows: If Ω is bounded and the norm ||u||L^{1}(0, T;L^{q}(Ω)), with Serrin's number 2/1 + 3/q strictly larger than 1, is sufficiently small, then u is regular in (0, T). Further local regularity conditions for general smooth domains are based on energy quantities such as ||u||L^{∞}(T_{0},T_{1}L^{2}(Ω)) |w||i»(r0,Ti;Z.2(i))) and || ▽ u|| ||u||L^{2}(T _{0},T_{1}L^{2}(Ω)). Indiana University Mathematics Journal

Original language | English |
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Pages (from-to) | 2111-2131 |

Number of pages | 21 |

Journal | Indiana University Mathematics Journal |

Volume | 56 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2007 |

## Keywords

- Instationary Navier-Stokes equations
- Local in time regularity
- Serrin's condition