### Abstract

We present several new regularity criteria for weak solutions u of the instationary Navier-Stokes system which additionally satisfy the strong energy inequality. (i) If the kinetic energy 1/2|u(t)\|_{2}
^{2} is Hölder continuous as a function of time t with Hölder exponent α in (1/2, 1), then u is regular. (ii) If for some α in (1/2, 1) the dissipation energy satisfies the left-side condition lim rm inf δrightarrow 01δα int-δ^{t}∇ u _{2}
^{2}dτ <∞ for all t of the given time interval, then u is regular. The proofs use local regularity results which are based on the theory of very weak solutions, see [1], [4], and on uniqueness arguments for weak solutions. Finally, in the last section we mention a local space-time regularity condition.

Original language | English |
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Pages (from-to) | 428-442 |

Number of pages | 15 |

Journal | Journal of Mathematical Fluid Mechanics |

Volume | 11 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2009 Oct |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Mathematical Physics
- Computational Mathematics
- Condensed Matter Physics

### Cite this

*Journal of Mathematical Fluid Mechanics*,

*11*(3), 428-442. https://doi.org/10.1007/s00021-008-0267-0