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Regularity criteria for weak solutions of the Navier-Stokes system. (English) Zbl 0598.35094
Nonlinear functional analysis and its applications, Proc. Summer. Res. Inst., Berkeley/Calif. 1983, Proc. Symp. Pure Math. 45, Pt. 1, 449-453 (1986).
[For the entire collection see Zbl 0583.00018.]
The initial boundary value problem for the Navier-Stokes system is considered \[ \partial u/\partial t-\Delta u+(u,\text{grad})u+\text{grad} p=0, \] div u\(=0\) in \(D\times (0,\infty)\), \(u=0\) on \(\partial D\times (0,\infty)\), \(u(x,D)=a(x)\) in D. The author gives some sufficient conditions for regularity of a global in time Leray-Hopf weak solution. The basic results are: 1. Let \(u\in L^ p(0,T;L^ q(D))\) with \(k\equiv 2-q+nq/p<0\) and \(p>n\), then \(u\in C^{\infty}(\bar D\times (0,T))\); 2. Let \(u\in L^ p(0,T;L^ q(D))\), \(k>0,p>n\), then there is a closed subset of (0,T) with vanishing k/2-dimensional Hausdorff measure such that \(u\in C^{\infty}(\bar D\times ((0,T)\setminus E))\); 3. Let \(u\in L^ n(0,T;L^ q(D))\), then there is a closed subset E of (0,T) with zero Lebesgue measure such that \(u\in C^{\infty}(\bar D\times ((0,T)\setminus E))\).
Reviewer: T.Shaposhnikova

MSC:
35Q30 Navier-Stokes equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
76D05 Navier-Stokes equations for incompressible viscous fluids