Regularity Criteria for Navier-Stokes Solutions

Gregory Seregin () and Vladimir Šverák ()
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Gregory Seregin: University of Oxford, Mathematical Institute
Vladimir Šverák: University of Minnesota, School of Mathematics

Chapter 16 in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 2018, pp 829-867 from Springer

Abstract: 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.

Keywords: Liouville Type Theorem; Scaling Symmetry; Suitable Weak Solutions; Ancient Solutions; Scale Invariant Quantity (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/978-3-319-13344-7_16

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