American Institute of Mathematical Sciences

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May  2014, 13(3): 1327-1336. doi: 10.3934/cpaa.2014.13.1327

On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity

Yu-Zhu Wang 1, and  Weibing Zuo 2,

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

Received  September 2013 Revised  October 2013 Published  December 2013

In this paper we study the initial problem for the Hall-magnetohydrodynamics system with partial viscosity in $ R^n(n=2, 3)$. We obtain a Beale-Kato-Majda type blow up criterion of smooth solutions.
Keywords: blow-up criterion., smooth solutions, Hall-magnetohydrodynamics system.
Mathematics Subject Classification: Primary: 76B03; Secondary: 76W0.
Citation: Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327
References:
[1]

M. Acheritogaray, P. Degond, A. Frouvelle and J. Liu, Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system, Kine. Rela. Mode., 4 (2011), 901-918. 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, J. Diff. Equations 255 (2013), 3971-3982.

[4]

D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics,, arXiv:1305.4681v1., (). 

[5]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Commun. Pure and Appl. Math., 36 (1983), 635-666. doi: 10.1002/cpa.3160360506.

[6]

J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10 (2004), 543-556. 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, J. Funct. Anal., 227 (2005), 113-152. 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, Acta Math. Appl. Sin. Engl. Ser., 28 (2012), 639-642. doi: 10.1007/s10255-012-0175-1.

[9]

Gala Sadek, A new regularity criterion for the 3D MHD equations in $\mathbbR^3$, Commun. Pure Appl. Anal., 11 (2012), 1353-1360. doi: 10.3934/cpaa.2012.11.973.

[10]

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.

[11]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. 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, Z. Angew. Math. Phys., 61 (2010),193-199. 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, Nonlinear Anal, 72 (2010), 3643-3648. doi: 10.1016/j.na.2009.12.045.

[14]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708. doi: 10.1515/form.2011.079.

[15]

X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations involving partial components, Nonlinear Anal. Real World Appl., 13 (2012), 410-418. doi: 10.1016/j.nonrwa.2011.07.055.

[16]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Diff. Equations, 248 (2010), 2263-2274. 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, Discrete Contin. Dyn. Syst., 25 (2009), 575-583. doi: 10.3934/dcds.2009.25.575.

[18]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions,, arXiv:1212.5968v1., (). 

[19]

Y.-Z. Wang, S. Wang and Y.-X. Wang, Regularity criteria for weak solution to the 3D magnetohydrodynamic equations, Acta Math. Scientia, 32 (2012), 1063-1072. 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, International Journal of Mathematics, 23 (2012), 1250027 (12 pages). 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, Comm. Math. Phys., 94 (1984), 61-66. 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, Math. Z., 242 (2002), 251-278. 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, Boundary Value Problems, Volume 2011, Article ID 128614, 14 pages. 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, Volume 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, Mathematical Methods in the Applied Sciences, 34 (2011), 2125-2135. doi: 10.1002/mma.1510.

[26]

H. Triebel, Theory of Function Spaces, Monograph in Mathematics, vol. 78. Birkhauser: Basel, 1983.

[27]

J. Chemin, Perfect Incompressible Fluids, Oxford Lecture Ser. Math. Appl., {14}, The Clarendon Press Oxford Univ. Press, New York, 1998.

[28]

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press: Cambridge, 2002.

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, Kine. Rela. Mode., 4 (2011), 901-918. 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, J. Diff. Equations 255 (2013), 3971-3982.

[4]

D. Chae and J. Lee, On the blow-up criterion and small data global existence for the Hall-magnetohydrodynamics,, arXiv:1305.4681v1., (). 

[5]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Commun. Pure and Appl. Math., 36 (1983), 635-666. doi: 10.1002/cpa.3160360506.

[6]

J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10 (2004), 543-556. 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, J. Funct. Anal., 227 (2005), 113-152. 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, Acta Math. Appl. Sin. Engl. Ser., 28 (2012), 639-642. doi: 10.1007/s10255-012-0175-1.

[9]

Gala Sadek, A new regularity criterion for the 3D MHD equations in $\mathbbR^3$, Commun. Pure Appl. Anal., 11 (2012), 1353-1360. doi: 10.3934/cpaa.2012.11.973.

[10]

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.

[11]

Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. 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, Z. Angew. Math. Phys., 61 (2010),193-199. 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, Nonlinear Anal, 72 (2010), 3643-3648. doi: 10.1016/j.na.2009.12.045.

[14]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708. doi: 10.1515/form.2011.079.

[15]

X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations involving partial components, Nonlinear Anal. Real World Appl., 13 (2012), 410-418. doi: 10.1016/j.nonrwa.2011.07.055.

[16]

C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Diff. Equations, 248 (2010), 2263-2274. 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, Discrete Contin. Dyn. Syst., 25 (2009), 575-583. doi: 10.3934/dcds.2009.25.575.

[18]

Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions,, arXiv:1212.5968v1., (). 

[19]

Y.-Z. Wang, S. Wang and Y.-X. Wang, Regularity criteria for weak solution to the 3D magnetohydrodynamic equations, Acta Math. Scientia, 32 (2012), 1063-1072. 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, International Journal of Mathematics, 23 (2012), 1250027 (12 pages). 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, Comm. Math. Phys., 94 (1984), 61-66. 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, Math. Z., 242 (2002), 251-278. 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, Boundary Value Problems, Volume 2011, Article ID 128614, 14 pages. 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, Volume 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, Mathematical Methods in the Applied Sciences, 34 (2011), 2125-2135. doi: 10.1002/mma.1510.

[26]

H. Triebel, Theory of Function Spaces, Monograph in Mathematics, vol. 78. Birkhauser: Basel, 1983.

[27]

J. Chemin, Perfect Incompressible Fluids, Oxford Lecture Ser. Math. Appl., {14}, The Clarendon Press Oxford Univ. Press, New York, 1998.

[28]

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press: Cambridge, 2002.

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