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On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component. (English) Zbl 1373.35226

Summary: We improve the regularity criterion for the incompressible Navier-Stokes equations in the full three-dimensional space involving the gradient of one velocity component. The method is based on recent results of C.-S. Cao and E. S. Titi [Indiana Univ. Math. J. 57, No. 6, 2643–2661 (2008; Zbl 1159.35053)] and I. Kukavica and M. Ziane [J. Math. Phys. 48, No. 6, 065203, 10 p. (2007; Zbl 1144.81373)]. In particular, for \(s\in [2,3]\), we get that the solution is regular if \(\nabla u_3\in L^t(0,T;L^s(\mathbb R^3))\), \(2/t+3/s\leq \frac{23}{12}\).
©2009 American Institute of Physics

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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