Regularity criteria for the generalized viscous MHD equations
Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 3, p. 491-505
@article{AIHPC_2007__24_3_491_0,
author = {Zhou, Yong},
title = {Regularity criteria for the generalized viscous MHD equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {24},
number = {3},
year = {2007},
pages = {491-505},
doi = {10.1016/j.anihpc.2006.03.014},
zbl = {1130.35110},
mrnumber = {2321203},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2007__24_3_491_0}
}

Zhou, Yong. Regularity criteria for the generalized viscous MHD equations. Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 3, pp. 491-505. doi : 10.1016/j.anihpc.2006.03.014. http://www.numdam.org/item/AIHPC_2007__24_3_491_0/

[1] Beirão Da Veiga H., A new regularity class for the Navier-Stokes equations in ${R}^{n}$, Chinese Ann. Math. 16 (1995) 407-412. | Zbl 0837.35111

[2] Beirão Da Veiga H., Vorticity and smoothness in viscous flows, in: Nonlinear Problems in Mathematical Physics and Related Topics, II, Int. Math. Ser. (N.Y.), vol. 2, Kluwer/Plenum, New York, 2002, pp. 61-67. | MR 1971989 | Zbl pre01985313

[3] Beirão Da Veiga H., Berselli L.C., On the regularizing effect of the vorticity direction in incompressible viscous flows, Differential Integral Equations 15 (3) (2002) 345-356. | MR 1870646 | Zbl 1014.35072

[4] Caffarelli L., Kohn R., Nirenberg L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982) 771-831. | Zbl 0509.35067

[5] Caflisch R., Klapper I., Steele G., Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys. 184 (2) (1997) 443-455. | MR 1462753 | Zbl 0874.76092

[6] Chemin J.Y., Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 14, The Clarendon Press, Oxford University Press, New York, 1998. | MR 1688875 | Zbl 0927.76002

[7] Constantin P., Fefferman C., Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J. 42 (1993) 775-789. | Zbl 0837.35113

[8] Duoandikoetxea J., Fourier Analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, 2001, Translated and revised from the 1995 Spanish original by David Cruz-Uribe. | MR 1800316 | Zbl 0969.42001

[9] He C., On partial regularity for weak solutions to the Navier-Stokes equations, J. Funct. Anal. 211 (1) (2004) 153-162. | Zbl 1062.35065

[10] He C., Xin Z., On the regularity of solutions to the magnetohydrodynamic equations, J. Differential Equations 213 (2) (2005) 235-254. | MR 2142366 | Zbl 1072.35154

[11] Sermange M., Temam R., Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math. 36 (5) (1983) 635-664. | MR 716200 | Zbl 0524.76099

[12] Serrin J., On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962) 187-195. | Zbl 0106.18302

[13] Stein E.M., Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, NJ, 1970. | MR 290095 | Zbl 0207.13501

[14] Tian G., Xin Z., Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom. 7 (1999) 221-257. | Zbl 0939.35139

[15] Wu J., Generalized MHD equations, J. Differential Equations 195 (2003) 284-312. | MR 2016814 | Zbl 1057.35040

[16] Wu J., Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci. 12 (4) (2002) 395-413. | MR 1915942 | Zbl 1029.76062

[17] Zhou Y., A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component, Method Appl. Anal. 9 (4) (2002) 563-578.

[18] Zhou Y., Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain, Math. Ann. 328 (1-2) (2004) 173-192. | Zbl 1054.35062

[19] Zhou Y., A new regularity criterion for the Navier-Stokes equations in terms of the direction of vorticity, Monatsh. Math. 144 (2005) 251-257. | Zbl 1072.35148

[20] Zhou Y., A new regularity criterion for weak solutions to the Navier-Stokes equations, J. Math. Pures Appl. (9) 84 (11) (2005) 1496-1514. | Zbl 1092.35081

[21] Zhou Y., Remarks on regularities for the 3D MHD equations, Discrete Contin. Dynam. Syst. 12 (5) (2005) 881-886. | MR 2128731 | Zbl 1068.35117

[22] Zhou Y., On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in ${R}^{N}$, Z. Angew. Math. Phys. 57 (2006) 384-392. | Zbl 1099.35091