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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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