American Institute of Mathematical Sciences

May  2012, 11(3): 973-980. doi: 10.3934/cpaa.2012.11.973

A new regularity criterion for the 3D MHD equations in $R^3$

 1 Department of Mathematics, University of Mostaganem, Box 227, Mostaganem 27000

Received  October 2010 Revised  April 2011 Published  December 2011

In this paper, we establish some improved regularity conditions for the 3D incompressible magnetohydrodynamic equations via only two components of the velocity and magnetic fields. This is an improvement of the result given by Ji and Lee [8].
Citation: Sadek Gala. A new regularity criterion for the 3D MHD equations in $R^3$. Communications on Pure and Applied Analysis, 2012, 11 (3) : 973-980. doi: 10.3934/cpaa.2012.11.973
References:
 [1] H.-O. Bae and H.-J. Choe, A regularity criterion for the Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 1173-1187. doi: 10.1080/03605300701257500. [2] R. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455. doi: 10.1007/s002200050067. [3] C. Cao and E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2662. doi: 10.1512/iumj.2008.57.3719. [4] Q. Chen and C. Miao, Existence theorem and blow-up criterion of the strong solutions to the two-fluid MHD equations in $R^3$, J. Differential Equations, 239 (2007), 251-271. doi: 10.1016/j.jde.2007.03.029. [5] Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919-930. doi: 10.1007/s00220-008-0545-y. [6] C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020. [7] C. He and Z. Xin, On the regularity of solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002. [8] E. Ji and J. Lee, Some regularity criteria for the 3D incompressible magnetohydrodynamics, J. Math. Anal. Appl., 369 (2010), 317-322. doi: 10.1016/j.jmaa.2010.03.015. [9] S. Gala and P. G. Lemarié-Rieusset, Multipliers between Sobolev spaces and fractional differentiation, J. Math. Anal. Appl., 322 (2006), 1030-1054. doi: 10.1016/j.jmaa.2005.07.043. [10] I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012. [11] J. Neustupa, A. Novotný and P. Penel, An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity, Topics in Mathematical Fluid Mechanics, Quaderni di Matematica Vol. 10 Seconda Universita di Napoli, Caserta, 2002, 163-183. [12] P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493. doi: 10.1023/B:APOM.0000048124.64244.7e. [13] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506. [14] J. Wu, Viscous and inviscid magnetohydrodynamics equations, J. Anal. Math., 73 (1997), 251-265. doi: 10.1007/BF02788146. [15] Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886. doi: 10.3934/dcds.2005.12.881. [16] Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199. doi: 10.1007/s00033-009-0023-1. [17] Y. Zhou and S. Gala, On the existence of global solutions for the magneto-hydrodynamic equations, Preprint, (2010).

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References:
 [1] H.-O. Bae and H.-J. Choe, A regularity criterion for the Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 1173-1187. doi: 10.1080/03605300701257500. [2] R. Caflisch, I. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455. doi: 10.1007/s002200050067. [3] C. Cao and E. S. Titi, Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57 (2008), 2643-2662. doi: 10.1512/iumj.2008.57.3719. [4] Q. Chen and C. Miao, Existence theorem and blow-up criterion of the strong solutions to the two-fluid MHD equations in $R^3$, J. Differential Equations, 239 (2007), 251-271. doi: 10.1016/j.jde.2007.03.029. [5] Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919-930. doi: 10.1007/s00220-008-0545-y. [6] C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274. doi: 10.1016/j.jde.2009.09.020. [7] C. He and Z. Xin, On the regularity of solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. doi: 10.1016/j.jde.2004.07.002. [8] E. Ji and J. Lee, Some regularity criteria for the 3D incompressible magnetohydrodynamics, J. Math. Anal. Appl., 369 (2010), 317-322. doi: 10.1016/j.jmaa.2010.03.015. [9] S. Gala and P. G. Lemarié-Rieusset, Multipliers between Sobolev spaces and fractional differentiation, J. Math. Anal. Appl., 322 (2006), 1030-1054. doi: 10.1016/j.jmaa.2005.07.043. [10] I. Kukavica and M. Ziane, One component regularity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 453-469. doi: 10.1088/0951-7715/19/2/012. [11] J. Neustupa, A. Novotný and P. Penel, An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity, Topics in Mathematical Fluid Mechanics, Quaderni di Matematica Vol. 10 Seconda Universita di Napoli, Caserta, 2002, 163-183. [12] P. Penel and M. Pokorný, Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., 49 (2004), 483-493. doi: 10.1023/B:APOM.0000048124.64244.7e. [13] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. doi: 10.1002/cpa.3160360506. [14] J. Wu, Viscous and inviscid magnetohydrodynamics equations, J. Anal. Math., 73 (1997), 251-265. doi: 10.1007/BF02788146. [15] Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886. doi: 10.3934/dcds.2005.12.881. [16] Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space, Z. Angew. Math. Phys., 61 (2010), 193-199. doi: 10.1007/s00033-009-0023-1. [17] Y. Zhou and S. Gala, On the existence of global solutions for the magneto-hydrodynamic equations, Preprint, (2010).
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