On the Regularity Criterion of Weak Solution for the 3D Viscous Magneto-Hydrodynamics Equations

Beirão da Veiga H.: A new regularity class for the Navier-Stokes equations in \({\mathbb{R}^n}\) . Chinese Ann. of Math. 16B, 407–412 (1995) Bony J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. 14, 209–246 (1981) Chen Q., Zhang Z.: Space-time estimates in the Besov spaces and the Navier-Stokes equations. Meth. Appl. Anal. 13, 107–122 (2006) Chen Q., Miao C., Zhang Z.: The Beale-Kato-Majda criterion to the 3D Magneto-hydrodynamics equations. Commun. Math. Phys. 275, 861–872 (2007) Chemin, J.-Y.: Perfect Incompressible Fluids. New York: Oxford University Press, 1998 Cheskidov, A., Shvydkoy, R.: On the regularity of weak solutions of the 3D Navier-Stokes equations in \({B^{-1}_{\infty,\infty}}\) . http://arXiv.org/list/0708.3067v2, 2007 Constantin P., Fefferman C.: Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42, 775–788 (1993) Duvaut G., Lions J.-L.: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ratl. Mech. Anal. 46, 241–279 (1972) Escauriaza L., Seregin G., S̆verák V.: L 3,∞-solutions to the Navier-Stokes equations and backward uniqueness. Russ. Math. Surv. 58, 211–250 (2003) Fujita H., Kato T.: On the Navier-Stokes initial value problem I. Archiv. Rat. Mech. Anal. 16, 269–315 (1964) Giga Y.: Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J. Differ. Eq. 62, 186–212 (1986) He C., Xin Z.: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Diff. Eqs. 213, 235–254 (2005) Kato T.: Nonstationary flows of viscous and ideal fluids in \({\mathbb{R}^3}\) . J. Funct. Anal. 9, 296–305 (1972) Kozono H., Ogawa T., Taniuchi Y.: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math. Z. 242, 251–278 (2002) Politano H., Pouquet A., Sulem P.L.: Current and vorticity dynamics in three-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 2, 2931–2939 (1995) Serrin J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 9, 187–195 (1962) Sermange M., Teman R.: Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36, 635–664 (1983) Triebel, H.: Theory of Function Spaces. Monograph in Mathematics, Vol. 78, Basel: Birkhauser Verlag, 1983 Von Wahl W.: Regularity of weak solutions of the Navier-Stokes equations. Proc. Symp. Pure Appl. Math. 45, 497–503 (1986) Wu J.: Bounds and new approaches for the 3D MHD equations. J. Nonlinear Sci. 12, 395–413 (2002) Wu J.: Regularity results for weak solutions of the 3D MHD equations. Discrete. Contin. Dynam. Syst. 10, 543–556 (2004) Wu J.: Generalized MHD equations. J. Differ. Eqs. 195, 284–312 (2003) Wu J.: Regularity criteria for the generalized MHD equations. Comm. PDE 33(2), 285–306 (2008) Zhou Y.: Remarks on regularities for the 3D MHD equations. Discrete. Contin. Dynam. Syst. 12, 881–886 (2005)