# zbMATH — the first resource for mathematics

Non blow-up criterion for the 3-D magneto-hydrodynamics equations in the limiting case. (English) Zbl 1372.35050
Summary: In this paper, we prove that suitable weak solution $$(u,b)$$ of the 3-D MHD equations can be extended beyond $$T$$ if $$u \in L^{\infty}(0,T;L^3(\mathbb{R}^3))$$ and the horizontal components $$b_h$$ of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition, which improves the corresponding regularity criterion by Mahalov-Nicolaenko-Shilkin.
##### MSC:
 35B44 Blow-up in context of PDEs 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 76W05 Magnetohydrodynamics and electrohydrodynamics 35D30 Weak solutions to PDEs
Full Text:
##### References:
 [1] Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35, 771-831, (1982) · Zbl 0509.35067 [2] Cao, C.; Wu, J., Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248, 2263-2274, (2010) · Zbl 1190.35046 [3] Chen, Q.; Miao, C.; Zhang, Z., The beale-Kato-Majda criterion to the 3D magneto-hydrodynamics equations, Comm. Math. Phys., 275, 861-872, (2007) · Zbl 1138.76066 [4] Chen, Q.; Miao, C.; Zhang, Z., On the regularity criterion of weak solution for the 3D viscous magnetohydrodynamics equations, Comm. Math. Phys., 284, 919-930, (2008) · Zbl 1168.35035 [5] Dong, H.; Du, D., The Navier-Stokes equations in the critical Lebesgue space, Comm. Math. Phys., 292, 811-827, (2009) · Zbl 1216.35085 [6] Duvaut, G.; Lions, J. L., Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational. Mech. Anal., 46, 241-279, (1972) · Zbl 0264.73027 [7] Escauriaza, L.; Seregin, G. A.; Šverák, V., $$L$$\^{3,∞} solutions to the Navier-Stokes equations and backward uniqueness, Russian Math. Surveys, 58, 211-250, (2003) · Zbl 1064.35134 [8] He, C.; Xin, Z., On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213, 235-254, (2005) · Zbl 1072.35154 [9] He, C.; Xin, Z., Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227, 113-152, (2005) · Zbl 1083.35110 [10] Kang, K.; Lee, J., Interior regularity criteria for suitable weak solutions of the magnetohydrodynamics equations, J. Differentional Equations, 247, 2310-2330, (2009) · Zbl 1397.35212 [11] Kang, K.; Kim, J., Boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations, J. Funct. Anal., 266, 99-120, (2014) · Zbl 1299.35236 [12] Lin, F. H., A new proof of the caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51, 241-257, (1998) · Zbl 0958.35102 [13] Mahalov, A.; Nicolaenko, B.; Shilkin, T., $$L$$\^{3,∞}-solutions to the MHD equations, Zap. Naučn. Sem. S. Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 336, 112-132, (2006) · Zbl 1127.35047 [14] Politano, H.; Pouquet, A.; Sulem, P. L., Current and vorticity dynamics in three-dimensional magnetohydrodynamic turbulence, Phys. Plasmas, 2, 2931-2939, (1995) · Zbl 0875.76699 [15] Seregin, G. A., Estimate of suitable solutions to the Navier-Stokes equations in critical Morrey spaces, J. Math. Sci., 143, 2961-2968, (2007) [16] Seregin, G. A., On the local regularity of suitable weak solutions of the Navier-Stokes equations, Russian Math. Surveys, 62, 595-614, (2007) · Zbl 1139.76018 [17] Sermange, M.; Teman, R., Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36, 635-664, (1983) · Zbl 0524.76099 [18] Wang, W.; Zhang, Z., Limiting case for the regularity criterion to the 3-D magneto-hydrodynamics equations, J. Differential Equations, 252, 5751-5762, (2012) · Zbl 1332.35302 [19] Wang, W.; Zhang, Z., On the interior regularity criteria for suitable weak solutions of the magnetohydrodynamics equations, SIAM J. Math. Anal., 45, 2666-677, (2013) · Zbl 1284.35104 [20] Wu, J., Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12, 395-413, (2002) · Zbl 1029.76062 [21] Wu, J., Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10, 543-556, (2004) · Zbl 1055.76062 [22] Zhou, Y., Remarks on regularities for the 3D MHD equations, Discrete. Contin. Dyn. Syst., 12, 881-886, (2005) · Zbl 1068.35117
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.