Weak solutions of Navier-Stokes equations. (English) Zbl 0568.35077

This paper considers the boundary-initial value problem for weak solutions to the Navier-Stokes equations on a domain (possibly unbounded) in \({\mathbb{R}}^ n\) with homogeneous boundary data. The first result proved is existence of a weak solution in a class of functions introduced by J. L. Lions, this class having somewhat stronger properties than that of the Hopf weak solution. A further result gives a uniqueness theorem in the spirit of the papers of Foias and of Serrin, but without the restriction imposed on the space dimension by Serrin. Finally, it is shown that if there is a weak solution in \(L^{n,\infty}\) which is right continuous in t as an \(L^ n\)-valued function, then it is the only weak solution.
Reviewer: B.Straughan


35Q30 Navier-Stokes equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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