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Large time behavior for the full compressible magnetohydrodynamic flows
A new regularity criterion for the 3D MHD equations in $R^3$
1. | Department of Mathematics, University of Mostaganem, Box 227, Mostaganem 27000 |
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). |
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. |
[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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