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

Mark Steinhauer ()
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Mark Steinhauer: Universität Bonn, Mathematisches Seminar

A chapter in Geometric Analysis and Nonlinear Partial Differential Equations, 2003, pp 543-557 from Springer

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.

Keywords: Banach Space; Weak Solution; Hardy Space; Sobolev Inequality; Energy Inequality (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-55627-2_28

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DOI: 10.1007/978-3-642-55627-2_28

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