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March  2014, 7(1): 45-56. doi: 10.3934/krm.2014.7.45

Regularity criteria for the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion

Jishan Fan 1, and  Tohru Ozawa 2,

1. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037
2. 

Department of Applied Physics, Waseda University, Tokyo, 169-8555

Received  July 2012 Revised  July 2013 Published  December 2013

This paper proves some regularity criteria for the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion. We also prove the global existence of strong solutions of its regularized MHD-$\alpha$ system.
Keywords: MHD, regularity criterion, BMO space..
Mathematics Subject Classification: Primary: 35Q30, 76D03; Secondary: 76D0.
Citation: Jishan Fan, Tohru Ozawa. Regularity criteria for the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion. Kinetic & Related Models, 2014, 7 (1) : 45-56. doi: 10.3934/krm.2014.7.45
References:
[1]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1.  Google Scholar

[2]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embedding and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.  Google Scholar

[3]

C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differential Equations, 254 (2013), 2661-2681. doi: 10.1016/j.jde.2013.01.002.  Google Scholar

[4]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.  Google Scholar

[5]

E. Casella, P. Secchi and P. Trebeschi, Global classical solutions for MHD system, J. Math. Fluid Mech., 5 (2003), 70-91. doi: 10.1007/s000210300003.  Google Scholar

[6]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar

[7]

H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541-544.  Google Scholar

[8]

J. Fan and T. Ozawa, Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model, Kinet. Relat. Models, 2 (2009), 293-305. doi: 10.3934/krm.2009.2.293.  Google Scholar

[9]

J. Fan and T. Ozawa, Global Cauchy problem for the 2-D magnetohydrodynamic-$\alpha$ models with partial viscous terms, J. Math. Fluid Mech., 12 (2010), 306-319. doi: 10.1007/s00021-008-0289-7.  Google Scholar

[10]

J. Fan and T. Ozawa, Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model, Discrete and Continuous Dynamical Systems. Suppl., I (2011), 400-409.  Google Scholar

[11]

C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. doi: 10.1007/BF02392215.  Google Scholar

[12]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[13]

C. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de-Vries equations, J. Amer. Math. Soc., 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[14]

H. Kozono, Weak and classical solutions of the 2-D MHD equations, Tohoku Math. J., 41 (1989), 471-488. doi: 10.2748/tmj/1178227774.  Google Scholar

[15]

H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations, Math. Z., 242 (2002), 251-278. doi: 10.1007/s002090100332.  Google Scholar

[16]

Z. Lei, N. Masmoudi and Y. Zhou, Remarks on the blow-up criteria for Oldroyd models, J. Differential Equations, 248 (2010), 328-341. doi: 10.1016/j.jde.2009.07.011.  Google Scholar

[17]

Z. Lei and Y. Zhou, BKM's 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]

J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models, J. Math. Phys., 48 (2007), 065504 (28 pages). doi: 10.1063/1.2360145.  Google Scholar

[19]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012.  Google Scholar

[20]

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

[21]

Y. Zhou and J. Fan, A regularity criterion for the 2D MHD system with zero magnetic diffusivity, J. Math. Anal. Appl., 378 (2011), 169-172. doi: 10.1016/j.jmaa.2011.01.014.  Google Scholar

show all references

References:
[1]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1.  Google Scholar

[2]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embedding and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154.  Google Scholar

[3]

C. Cao, D. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differential Equations, 254 (2013), 2661-2681. doi: 10.1016/j.jde.2013.01.002.  Google Scholar

[4]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822. doi: 10.1016/j.aim.2010.08.017.  Google Scholar

[5]

E. Casella, P. Secchi and P. Trebeschi, Global classical solutions for MHD system, J. Math. Fluid Mech., 5 (2003), 70-91. doi: 10.1007/s000210300003.  Google Scholar

[6]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar

[7]

H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541-544.  Google Scholar

[8]

J. Fan and T. Ozawa, Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model, Kinet. Relat. Models, 2 (2009), 293-305. doi: 10.3934/krm.2009.2.293.  Google Scholar

[9]

J. Fan and T. Ozawa, Global Cauchy problem for the 2-D magnetohydrodynamic-$\alpha$ models with partial viscous terms, J. Math. Fluid Mech., 12 (2010), 306-319. doi: 10.1007/s00021-008-0289-7.  Google Scholar

[10]

J. Fan and T. Ozawa, Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model, Discrete and Continuous Dynamical Systems. Suppl., I (2011), 400-409.  Google Scholar

[11]

C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. doi: 10.1007/BF02392215.  Google Scholar

[12]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[13]

C. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de-Vries equations, J. Amer. Math. Soc., 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[14]

H. Kozono, Weak and classical solutions of the 2-D MHD equations, Tohoku Math. J., 41 (1989), 471-488. doi: 10.2748/tmj/1178227774.  Google Scholar

[15]

H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations, Math. Z., 242 (2002), 251-278. doi: 10.1007/s002090100332.  Google Scholar

[16]

Z. Lei, N. Masmoudi and Y. Zhou, Remarks on the blow-up criteria for Oldroyd models, J. Differential Equations, 248 (2010), 328-341. doi: 10.1016/j.jde.2009.07.011.  Google Scholar

[17]

Z. Lei and Y. Zhou, BKM's 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]

J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models, J. Math. Phys., 48 (2007), 065504 (28 pages). doi: 10.1063/1.2360145.  Google Scholar

[19]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012.  Google Scholar

[20]

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

[21]

Y. Zhou and J. Fan, A regularity criterion for the 2D MHD system with zero magnetic diffusivity, J. Math. Anal. Appl., 378 (2011), 169-172. doi: 10.1016/j.jmaa.2011.01.014.  Google Scholar

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