Regularity criteria for navier-stokes solutions

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.