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Improvement of some anisotropic regularity criteria for the Navier--Stokes equations

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  • We consider the incompressible Navier--Stokes equations in the entire three-dimensional space. Assuming additional regularity on the components of the vector field $\partial_3$u we show intermediate anisotropic regularity results between the results by Kukavica and Ziane [5] and by Cao and Titi [3]and improve the result from the paper by Penel and Pokorný [9].
    Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D05.

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    C. Cao, Sufficient conditions for the regularity to the 3D Navier-Stokes equations, Discrete Contin. Dyn. Syst., 26 (2010), 1141-1151.doi: 10.3934/dcds.2010.26.1141.

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    C. Cao and E. S. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., 202 (2011), 919-932.doi: 10.1007/s00205-011-0439-6.

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    P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493.doi: 10.1023/B:APOM.0000048124.64244.7e.

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    P. Penel and M. Pokorný, On anisotropic regularity criteria for the solutions to 3D Navier-Stokes equations, J. Math. Fluid Mech., 13 (2011), 341-353.doi: 10.1007/s00021-010-0038-6.

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    Y. Zhou and M. Pokorný, On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component, J. Math. Phys., 50 (2009), 123514, 11 pp.doi: 10.1063/1.3268589.

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    Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107.doi: 10.1088/0951-7715/23/5/004.

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