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

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

Sadek Gala 1,

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].
Keywords: weak solution, Magneto-hydrodynamic equations, regularity criterion., multiplier space.
Mathematics Subject Classification: 35Q35, 35B65, 76D0.
Citation: Sadek Gala. A new regularity criterion for the 3D MHD equations in $R^3$. Communications on Pure & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Wu, Viscous and inviscid magnetohydrodynamics equations, J. Anal. Math., 73 (1997), 251-265. doi: 10.1007/BF02788146.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Y. Zhou and S. Gala, On the existence of global solutions for the magneto-hydrodynamic equations, Preprint, (2010). Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. Wu, Viscous and inviscid magnetohydrodynamics equations, J. Anal. Math., 73 (1997), 251-265. doi: 10.1007/BF02788146.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Y. Zhou and S. Gala, On the existence of global solutions for the magneto-hydrodynamic equations, Preprint, (2010). Google Scholar

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