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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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[1] Beirão da Veiga, H.: Vorticity and smoothness in viscous flows. In: Birman, M.S., Hildebrandt, S., Solonnikov, V.A., Uraltseva, N.N. (eds.) Nonlinear Problems in Mathematical Physics and Related Topics, II, Volume 2 of Int. Math. Ser. N. Y., pp. 61-67. Kluwer/Plenum, New York (2002)
[2] Beirão da Veiga, H.; Berselli, LC, On the regularizing effect of the vorticity direction in incompressible viscous flows, Differ. Integral Equ., 15, 345-356, (2002) · Zbl 1014.35072
[3] Beirão da Veiga, H., Giga, Y., Grujić, Z.: Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations. In: Giga, Y., Novotny, A. (eds.) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, pp. 901-932. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-10151-4_18-1
[4] Calderón, AP; Zygmund, A., On singular integrals, Am. J. Math., 78, 289-309, (1956) · Zbl 0072.11501
[5] Chae, D., On the regularity conditions for the Navier-Stokes and related equations, Rev. Mat. Iberoam., 23, 371-384, (2007) · Zbl 1130.35100
[6] Chorin, AJ, The evolution of a turbulent vortex, Commun. Math. Phys., 83, 517-535, (1982) · Zbl 0494.76024
[7] Constantin, P., Geometric statistics in turbulence, SIAM Rev., 36, 73-98, (1994) · Zbl 0803.35106
[8] Constantin, P.; Fefferman, C., Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42, 775-789, (1993) · Zbl 0837.35113
[9] Fan, J.; Ozawa, T., On the regularity criteria for the generalized Navier-Stokes equations and Lagrangian averaged Euler equations, Differ. Integral Equ., 21, 443-457, (2008) · Zbl 1224.35338
[10] Frisch, U.; Sulem, PL; Nelkin, M., A simple dynamical model of intermittent fully developed turbulence, J. Fluid Mech., 87, 719-736, (1978) · Zbl 0395.76051
[11] Ju, N., The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Commun. Math. Phys., 255, 161-181, (2005) · Zbl 1088.37049
[12] Katz, NH; Pavlović, N., A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12, 355-379, (2002) · Zbl 0999.35069
[13] Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod; Gauthier-Villars, Paris (1969) · Zbl 0189.40603
[14] Serrin, J.: The initial value problem for the Navier-Stokes equations. In: Nonlinear Problems (Proceedings of a Symposium, Madison, Wisconsin, 1962), pp. 69-98. University of Wisconsin Press, Madison, Wisconsin (1963)
[15] Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)
[16] Tanahashi, M., Miyauchi, T., Ikeda, J.: Scaling law of coherent fine scale structure in homogeneous isotropic turbulence. In: Proceedings of 11th Symposium on Turbulence Shear Flows (1997)
[17] Triebel, H.: The Structure of Functions, Monographs in Mathematics, vol. 97. Birkhäuser Verlag, Basel (2001)
[18] Wu, J., Generalized MHD equations, J. Differ. Equ., 195, 284-312, (2003) · Zbl 1057.35040
[19] Zhou, Y., Direction of vorticity and a new regularity criterion for the Navier-Stokes equations, ANZIAM J., 46, 309-316, (2005) · Zbl 1072.35565
[20] Zhou, Y., A new regularity criterion for the Navier-Stokes equations in terms of the direction of vorticity, Monatsh. Math., 144, 251-257, (2005) · Zbl 1072.35148
[21] Zhou, Y., Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24, 491-505, (2007) · Zbl 1130.35110
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