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Direction of vorticity and a refined regularity criterion for the Navier-Stokes equations with fractional Laplacian. (English) Zbl 1416.35190
Summary: We give a refined regularity criterion for solutions of the three-dimensional Navier-Stokes equations with fractional dissipative term $$(-\Delta )^{\alpha /2}v$$. The criterion is composed by the direction field of the vorticity and its magnitude simultaneously. Our result is a generalized of previous results by H. Beirão da Veiga and L. C. Berselli [Differ. Integral Equ. 15, No. 3, 345–356 (2002; Zbl 1014.35072)], and Y. Zhou [ANZIAM J. 46, No. 3, 309–316 (2005; Zbl 1072.35565); Monatsh. Math. 144, No. 3, 251–257 (2005; Zbl 1072.35148)]. Moreover, our result mentioned about the relation between the solution of the Navier-Stokes equations and the Euler equations.
##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 35R11 Fractional partial differential equations 35D35 Strong solutions to PDEs 35B65 Smoothness and regularity of solutions to PDEs
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##### References:
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