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 & 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.  Google Scholar

[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.  Google Scholar

[3]

D. Chae and M. schonbek, On the temporal decay for the Hall-magnetohydrodynamics equations, J. Diff. Equations 255 (2013), 3971-3982. Google Scholar

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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, Commun. Pure and Appl. Math., 36 (1983), 635-666. doi: 10.1002/cpa.3160360506.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

J. Chemin, Perfect Incompressible Fluids, Oxford Lecture Ser. Math. Appl., {14}, The Clarendon Press Oxford Univ. Press, New York, 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, Kine. Rela. Mode., 4 (2011), 901-918. doi: 10.3934/krm.2011.4.901.  Google Scholar

[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.  Google Scholar

[3]

D. Chae and M. schonbek, On the temporal decay for the Hall-magnetohydrodynamics equations, J. Diff. Equations 255 (2013), 3971-3982. 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, Commun. Pure and Appl. Math., 36 (1983), 635-666. doi: 10.1002/cpa.3160360506.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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, Acta Math. Scientia, 32 (2012), 1063-1072. doi: 10.1016/S0252-9602(12)60079-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

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