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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
##### Keywords:
Navier-Stokes system; Leray-Hopf weak solution