## Abstract

In this chapter some of the known regularity criteria for the weak solution of the incompressible 3D Navier-Stokes equations are discussed. At present the problem of regularity of general solutions starting from smooth data is open, and all the criteria involve an assumption on a suitable quantity which is invariant under the scaling symmetry of the equations. Both interior regularity and boundary regularity are addressed. The methods developed by Scheffer and Caffarelli-Kohn-Nirenberg play an important role. Simple but important considerations based on dimensional analysis and the scaling symmetry are recalled, together with some heuristics. Connections between the Liouville-type theorems and Type I singularities are also discussed. Proofs of some statements which are not easily accessible in the literature are presented.

Original language | English (US) |
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Title of host publication | Handbook of Mathematical Analysis in Mechanics of Viscous Fluids |

Publisher | Springer International Publishing |

Pages | 829-867 |

Number of pages | 39 |

ISBN (Electronic) | 9783319133447 |

ISBN (Print) | 9783319133430 |

DOIs | |

State | Published - Apr 19 2018 |