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On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity
1. | School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011 |
2. | School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China |
References:
[1] |
M. Acheritogaray, P. Degond, A. Frouvelle and J. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system,, \emph{Kine. Rela. Mode.}, 4 (2011), 901.
doi: 10.3934/krm.2011.4.901. |
[2] |
D. Chae, P. Degond and J. Liu, Well-posedness for Hall-magnetohydrodynamics,, \emph{Ann. de l'Institut Henri Poincare (C) Non Linear Anal.}, ().
doi: 10.1016/j.anihpc.2013.04.006. |
[3] |
D. Chae and M. schonbek, On the temporal decay for the Hall-magnetohydrodynamics equations,, \emph{J. Diff. Equations} \textbf{255} (2013), 255 (2013), 3971. Google Scholar |
[4] |
D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics,, arXiv:1305.4681v1., (). Google Scholar |
[5] |
M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, \emph{Commun. Pure and Appl. Math.}, 36 (1983), 635.
doi: 10.1002/cpa.3160360506. |
[6] |
J. Wu, Regularity results for weak solutions of the 3D MHD equations,, \emph{Discrete Contin. Dyn. Syst.}, 10 (2004), 543.
doi: 10.3934/dcds.2004.10.543. |
[7] |
C. He and Z. P. Xin, Partial regularity of suitable weak sokutions to the incompressible magnetohydrodynamics equations,, \emph{J. Funct. Anal.}, 227 (2005), 113.
doi: 10.1016/j.jfa.2005.06.009. |
[8] |
Gala Sadek, A note on the blow-up criterion of smooth solutions to the 3D incompressible MHD equations,, \emph{Acta Math. Appl. Sin. Engl. Ser.}, 28 (2012), 639.
doi: 10.1007/s10255-012-0175-1. |
[9] |
Gala Sadek, A new regularity criterion for the 3D MHD equations in $\mathbbR^3$,, \emph{Commun. Pure Appl. Anal.}, 11 (2012), 1353.
doi: 10.3934/cpaa.2012.11.973. |
[10] |
Y. Zhou, Remarks on regularities for the 3D MHD equations,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 881.
doi: 10.3934/dcds.2005.12.881. |
[11] |
Y. Zhou, Regularity criteria for the generalized viscous MHD equations,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 24 (2007), 491.
doi: 10.1016/j.anihpc.2006.03.014. |
[12] |
Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space,, \emph{Z. Angew. Math. Phys.}, 61 (2010), 193.
doi: 10.1007/s00033-009-0023-1. |
[13] |
Y. Zhou and S. Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field,, \emph{Nonlinear Anal}, 72 (2010), 3643.
doi: 10.1016/j.na.2009.12.045. |
[14] |
Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations,, \emph{Forum Math.}, 24 (2012), 691.
doi: 10.1515/form.2011.079. |
[15] |
X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations involving partial components,, \emph{Nonlinear Anal. Real World Appl.}, 13 (2012), 410.
doi: 10.1016/j.nonrwa.2011.07.055. |
[16] |
C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations,, \emph{J. Diff. Equations}, 248 (2010), 2263.
doi: 10.1016/j.jde.2009.09.020. |
[17] |
Z. Lei and Y. Zhou, BKM criterion and global weak solutions for magnetohydrodynamics with zero viscosity,, \emph{Discrete Contin. Dyn. Syst.}, 25 (2009), 575.
doi: 10.3934/dcds.2009.25.575. |
[18] |
Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions,, arXiv:1212.5968v1., (). Google Scholar |
[19] |
Y.-Z. Wang, S. Wang and Y.-X. Wang, Regularity criteria for weak solution to the 3D magnetohydrodynamic equations,, \emph{Acta Math. Scientia}, 32 (2012), 1063.
doi: 10.1016/S0252-9602(12)60079-4. |
[20] |
Y.-Z. Wang, H. J. Zhao and Y.-X. Wang, A logarithmally improved blow up criterion of smooth solutions for the three-dimensional MHD equations,, \emph{International Journal of Mathematics}, 23 (2012).
doi: 10.1142/S0129167X12500279. |
[21] |
J. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations,, \emph{Comm. Math. Phys.}, 94 (1984), 61.
doi: 10.1007/BF01212349. |
[22] |
H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,, \emph{Math. Z.}, 242 (2002), 251.
doi: 10.1007/s002090100332. |
[23] |
Y.-Z. Wang, L. Hu and Y.-X. Wang, A Beale-Kato-Madja Criterion for Magneto-Micropolar Fluid Equations with Partial Viscosity,, \emph{Boundary Value Problems}, (2011).
doi: 10.1155/2011/128614. |
[24] |
Y.-Z. Wang, Y. Li and Y.-X. Wang, Blow-up criterion of smooth solutions for magneto-micropolar fluid equations with partial viscosity,, Boundary Value Problems, (2011).
doi: 10.1186/1687-2770-2011-11. |
[25] |
Y.-Z. Wang and Y.-X. Wang, Blow-up criterion for two-dimensional magneto-micropolar fluid equations with partial viscosity,, \emph{Mathematical Methods in the Applied Sciences}, 34 (2011), 2125.
doi: 10.1002/mma.1510. |
[26] |
H. Triebel, Theory of Function Spaces,, Monograph in Mathematics, (1983). Google Scholar |
[27] |
J. Chemin, Perfect Incompressible Fluids,, Oxford Lecture Ser. Math. Appl., (1998). Google Scholar |
[28] |
A. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press: Cambridge, (2002). Google Scholar |
show all references
References:
[1] |
M. Acheritogaray, P. Degond, A. Frouvelle and J. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system,, \emph{Kine. Rela. Mode.}, 4 (2011), 901.
doi: 10.3934/krm.2011.4.901. |
[2] |
D. Chae, P. Degond and J. Liu, Well-posedness for Hall-magnetohydrodynamics,, \emph{Ann. de l'Institut Henri Poincare (C) Non Linear Anal.}, ().
doi: 10.1016/j.anihpc.2013.04.006. |
[3] |
D. Chae and M. schonbek, On the temporal decay for the Hall-magnetohydrodynamics equations,, \emph{J. Diff. Equations} \textbf{255} (2013), 255 (2013), 3971. Google Scholar |
[4] |
D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics,, arXiv:1305.4681v1., (). Google Scholar |
[5] |
M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, \emph{Commun. Pure and Appl. Math.}, 36 (1983), 635.
doi: 10.1002/cpa.3160360506. |
[6] |
J. Wu, Regularity results for weak solutions of the 3D MHD equations,, \emph{Discrete Contin. Dyn. Syst.}, 10 (2004), 543.
doi: 10.3934/dcds.2004.10.543. |
[7] |
C. He and Z. P. Xin, Partial regularity of suitable weak sokutions to the incompressible magnetohydrodynamics equations,, \emph{J. Funct. Anal.}, 227 (2005), 113.
doi: 10.1016/j.jfa.2005.06.009. |
[8] |
Gala Sadek, A note on the blow-up criterion of smooth solutions to the 3D incompressible MHD equations,, \emph{Acta Math. Appl. Sin. Engl. Ser.}, 28 (2012), 639.
doi: 10.1007/s10255-012-0175-1. |
[9] |
Gala Sadek, A new regularity criterion for the 3D MHD equations in $\mathbbR^3$,, \emph{Commun. Pure Appl. Anal.}, 11 (2012), 1353.
doi: 10.3934/cpaa.2012.11.973. |
[10] |
Y. Zhou, Remarks on regularities for the 3D MHD equations,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 881.
doi: 10.3934/dcds.2005.12.881. |
[11] |
Y. Zhou, Regularity criteria for the generalized viscous MHD equations,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 24 (2007), 491.
doi: 10.1016/j.anihpc.2006.03.014. |
[12] |
Y. Zhou and S. Gala, Regularity criteria for the solutions to the 3D MHD equations in the multiplier space,, \emph{Z. Angew. Math. Phys.}, 61 (2010), 193.
doi: 10.1007/s00033-009-0023-1. |
[13] |
Y. Zhou and S. Gala, A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field,, \emph{Nonlinear Anal}, 72 (2010), 3643.
doi: 10.1016/j.na.2009.12.045. |
[14] |
Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations,, \emph{Forum Math.}, 24 (2012), 691.
doi: 10.1515/form.2011.079. |
[15] |
X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations involving partial components,, \emph{Nonlinear Anal. Real World Appl.}, 13 (2012), 410.
doi: 10.1016/j.nonrwa.2011.07.055. |
[16] |
C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations,, \emph{J. Diff. Equations}, 248 (2010), 2263.
doi: 10.1016/j.jde.2009.09.020. |
[17] |
Z. Lei and Y. Zhou, BKM criterion and global weak solutions for magnetohydrodynamics with zero viscosity,, \emph{Discrete Contin. Dyn. Syst.}, 25 (2009), 575.
doi: 10.3934/dcds.2009.25.575. |
[18] |
Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions,, arXiv:1212.5968v1., (). Google Scholar |
[19] |
Y.-Z. Wang, S. Wang and Y.-X. Wang, Regularity criteria for weak solution to the 3D magnetohydrodynamic equations,, \emph{Acta Math. Scientia}, 32 (2012), 1063.
doi: 10.1016/S0252-9602(12)60079-4. |
[20] |
Y.-Z. Wang, H. J. Zhao and Y.-X. Wang, A logarithmally improved blow up criterion of smooth solutions for the three-dimensional MHD equations,, \emph{International Journal of Mathematics}, 23 (2012).
doi: 10.1142/S0129167X12500279. |
[21] |
J. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations,, \emph{Comm. Math. Phys.}, 94 (1984), 61.
doi: 10.1007/BF01212349. |
[22] |
H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,, \emph{Math. Z.}, 242 (2002), 251.
doi: 10.1007/s002090100332. |
[23] |
Y.-Z. Wang, L. Hu and Y.-X. Wang, A Beale-Kato-Madja Criterion for Magneto-Micropolar Fluid Equations with Partial Viscosity,, \emph{Boundary Value Problems}, (2011).
doi: 10.1155/2011/128614. |
[24] |
Y.-Z. Wang, Y. Li and Y.-X. Wang, Blow-up criterion of smooth solutions for magneto-micropolar fluid equations with partial viscosity,, Boundary Value Problems, (2011).
doi: 10.1186/1687-2770-2011-11. |
[25] |
Y.-Z. Wang and Y.-X. Wang, Blow-up criterion for two-dimensional magneto-micropolar fluid equations with partial viscosity,, \emph{Mathematical Methods in the Applied Sciences}, 34 (2011), 2125.
doi: 10.1002/mma.1510. |
[26] |
H. Triebel, Theory of Function Spaces,, Monograph in Mathematics, (1983). Google Scholar |
[27] |
J. Chemin, Perfect Incompressible Fluids,, Oxford Lecture Ser. Math. Appl., (1998). Google Scholar |
[28] |
A. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press: Cambridge, (2002). Google Scholar |
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