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On the singular set and the uniqueness of weak solutions of the Navier- Stokes equations. (English) Zbl 0567.35069
This paper establishes three results for weak solutions to the Navier- Stokes equations. Firstly it proves a uniqueness theorem by showing that if u, v are weak solutions with the same data, if $$u\in L^{\infty}((0,T),L^ n(\Omega))$$ with $$\Omega$$ bounded and if v satisfies the energy inequality everywhere then $$u=v$$. Secondly, let u be any weak solution in the class $$L^{\infty}((0,T),L^ n(\Omega)).$$ Then there are at most countably many points in [0,T] where $$\| u(t)\|_{L^ n(\Omega)}$$ is discontinuous from the left while u is continuous from the right at each t in [0,T], in the $$L^ n(\Omega)$$- topology. Finally it is shown that for $$n=4$$ any weak solution for which the energy inequality holds in regular with the exception of a bounded set of measure 0.
Reviewer: B.Straughan

MSC:
 35Q30 Navier-Stokes equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000)
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References:
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