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On Uniqueness- and Regularity Criteria for the Navier-Stokes Equations

In: Geometric Analysis and Nonlinear Partial Differential Equations

Author

Listed:
  • Mark Steinhauer

    (Universität Bonn, Mathematisches Seminar)

Abstract

Summary We survey and improve some results concerning uniqueness and regularity of solutions to the instationary Navier-Stokes equations in three (and higher) dimensions. In particular we show that the class of weak solutions which additionally belong to the space L 2(0,T; BMO) guarantees uniqueness as well as regularity. The method of proof which we present is elementary and depends deeply on the “div-curl” structure of the nonlinear convective term u · ∇u of the Navier-Stokes equations together with div u = 0 and according to Coifman, Lions, Meyer & Semmes it belongs to the Hardy space H 1. This also shows that it is applicable to other equations in hydrodynamics as for example the Boussinesq equations, the equations of Magneto-Hydrodynamics and the equations of higher grade type fluids.

Suggested Citation

  • Mark Steinhauer, 2003. "On Uniqueness- and Regularity Criteria for the Navier-Stokes Equations," Springer Books, in: Stefan Hildebrandt & Hermann Karcher (ed.), Geometric Analysis and Nonlinear Partial Differential Equations, pages 543-557, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55627-2_28
    DOI: 10.1007/978-3-642-55627-2_28
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