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Regularity Criteria for Navier-Stokes Solutions

In: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Author

Listed:
  • Gregory Seregin

    (University of Oxford, Mathematical Institute)

  • Vladimir Šverák

    (University of Minnesota, School of Mathematics)

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.

Suggested Citation

  • Gregory Seregin & Vladimir Šverák, 2018. "Regularity Criteria for Navier-Stokes Solutions," Springer Books, in: Yoshikazu Giga & Antonín Novotný (ed.), Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, chapter 16, pages 829-867, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-13344-7_16
    DOI: 10.1007/978-3-319-13344-7_16
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