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On the result of He concerning the smoothness of solutions to the Navier-Stokes equationsKeywords: Navier-Stokes equations , regularity of systems of PDE's , anisotropic regularity criteria. Abstract: We improve the regularity criterion for the Navier-Stokes equations proved by He [4]. We show that for the Cauchy problem the Leray-Hopf weak solution is smooth provided $ abla u_3 in L^t(0,T;L^s)$, $frac 2t + frac 3s leq frac 32$.
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