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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

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.
Citation: Jishan Fan, Tohru Ozawa. Regularity criteria for the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion. Kinetic and 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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

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