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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 |
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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