### 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 |
---|---|

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 |

### Fingerprint

### 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

**Energy-based regularity criteria for the Navier-Stokes equations.** / Farwig, Reinhard; Kozono, Hideo; Sohr, Hermann.

Research output: Contribution to journal › Article

*Journal of Mathematical Fluid Mechanics*, vol. 11, no. 3, pp. 428-442. https://doi.org/10.1007/s00021-008-0267-0

}

TY - JOUR

T1 - Energy-based regularity criteria for the Navier-Stokes equations

AU - Farwig, Reinhard

AU - Kozono, Hideo

AU - Sohr, Hermann

PY - 2009/10

Y1 - 2009/10

N2 - 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-δt∇ u 2 2dτ <∞ 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.

AB - 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-δt∇ u 2 2dτ <∞ 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.

KW - Energy inequality

KW - Instationary Navier-Stokes equations

KW - Local in time regularity

KW - Serrin's condition

UR - http://www.scopus.com/inward/record.url?scp=70350336961&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350336961&partnerID=8YFLogxK

U2 - 10.1007/s00021-008-0267-0

DO - 10.1007/s00021-008-0267-0

M3 - Article

AN - SCOPUS:70350336961

VL - 11

SP - 428

EP - 442

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

SN - 1422-6928

IS - 3

ER -