### Abstract

Consider a weak solution u of the instationary Navier-Stokes system in a bounded domain of R^{3} satisfying the strong energy inequality. Extending previous results by Farwig et al., J. Math. Fluid Mech. 11, 1-14 (2008), we prove among other things that u is regular if either the kinetic energy 1/2 ∥u(t) ∥^{2}
_{2} or the dissipation energy ∫ ^{t} _{0} ∥∇u(τ ) ∥^{2} _{2} dτ is (left-side) Hölder continuous as a function of time t with Hölder exponent 1/2 and with sufficiently small Hölder seminorm. The proofs use local regularity results which are based on the theory of very weak solutions and on uniqueness arguments for weak solutions.

Original language | English |
---|---|

Title of host publication | Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday |

Pages | 215-227 |

Number of pages | 13 |

DOIs | |

Publication status | Published - 2010 |

Externally published | Yes |

Event | 2007 International Conference on Mathematical Fluid Mechanics - Estoril Duration: 2007 May 21 → 2007 May 25 |

### Other

Other | 2007 International Conference on Mathematical Fluid Mechanics |
---|---|

City | Estoril |

Period | 07/5/21 → 07/5/25 |

### Fingerprint

### Keywords

- Energy criteria
- Hölder continuity
- Navier-Stokes equations
- Regularity criteria
- Weak solutions

### ASJC Scopus subject areas

- Fluid Flow and Transfer Processes

### Cite this

*Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday*(pp. 215-227) https://doi.org/10.1007/978-3-642-04068-9-13

**Regularity of weak solutions for the Navier-Stokes equations via energy criteria.** / Farwig, Reinhard; Kozono, Hideo; Sohr, Hermann.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday.*pp. 215-227, 2007 International Conference on Mathematical Fluid Mechanics, Estoril, 07/5/21. https://doi.org/10.1007/978-3-642-04068-9-13

}

TY - GEN

T1 - Regularity of weak solutions for the Navier-Stokes equations via energy criteria

AU - Farwig, Reinhard

AU - Kozono, Hideo

AU - Sohr, Hermann

PY - 2010

Y1 - 2010

N2 - Consider a weak solution u of the instationary Navier-Stokes system in a bounded domain of R3 satisfying the strong energy inequality. Extending previous results by Farwig et al., J. Math. Fluid Mech. 11, 1-14 (2008), we prove among other things that u is regular if either the kinetic energy 1/2 ∥u(t) ∥2 2 or the dissipation energy ∫ t 0 ∥∇u(τ ) ∥2 2 dτ is (left-side) Hölder continuous as a function of time t with Hölder exponent 1/2 and with sufficiently small Hölder seminorm. The proofs use local regularity results which are based on the theory of very weak solutions and on uniqueness arguments for weak solutions.

AB - Consider a weak solution u of the instationary Navier-Stokes system in a bounded domain of R3 satisfying the strong energy inequality. Extending previous results by Farwig et al., J. Math. Fluid Mech. 11, 1-14 (2008), we prove among other things that u is regular if either the kinetic energy 1/2 ∥u(t) ∥2 2 or the dissipation energy ∫ t 0 ∥∇u(τ ) ∥2 2 dτ is (left-side) Hölder continuous as a function of time t with Hölder exponent 1/2 and with sufficiently small Hölder seminorm. The proofs use local regularity results which are based on the theory of very weak solutions and on uniqueness arguments for weak solutions.

KW - Energy criteria

KW - Hölder continuity

KW - Navier-Stokes equations

KW - Regularity criteria

KW - Weak solutions

UR - http://www.scopus.com/inward/record.url?scp=84896762645&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84896762645&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-04068-9-13

DO - 10.1007/978-3-642-04068-9-13

M3 - Conference contribution

SN - 9783642040672

SP - 215

EP - 227

BT - Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday

ER -