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.
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Steinhauer, M. (2003). On Uniqueness- and Regularity Criteria for the Navier-Stokes Equations. In: Hildebrandt, S., Karcher, H. (eds) Geometric Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55627-2_28
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DOI: https://doi.org/10.1007/978-3-642-55627-2_28
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