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

Previous Article
Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation
KRM Home
This Issue
Next Article
A mathematical model for value estimation with public information and herding

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

Abstract
Related Papers

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 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.

[1]	Sadek Gala. A new regularity criterion for the 3D MHD equations in $R^3$. Communications on Pure and Applied Analysis, 2012, 11 (3) : 973-980. doi: 10.3934/cpaa.2012.11.973
[2]	Ahmad Mohammad Alghamdi, Sadek Gala, Chenyin Qian, Maria Alessandra Ragusa. The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations. Electronic Research Archive, 2020, 28 (1) : 183-193. doi: 10.3934/era.2020012
[3]	Jishan Fan, Tohru Ozawa. A regularity criterion for 3D density-dependent MHD system with zero viscosity. Conference Publications, 2015, 2015 (special) : 395-399. doi: 10.3934/proc.2015.0395
[4]	Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics and Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005
[5]	Zhang Chao, Minghua Yang. BMO type space associated with Neumann operator and application to a class of parabolic equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1629-1645. doi: 10.3934/dcdsb.2021104
[6]	Ming Lu, Yi Du, Zheng-An Yao, Zujin Zhang. A blow-up criterion for the 3D compressible MHD equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1167-1183. doi: 10.3934/cpaa.2012.11.1167
[7]	Sun-Sig Byun, Lihe Wang. $W^{1,p}$ regularity for the conormal derivative problem with parabolic BMO nonlinearity in reifenberg domains. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 617-637. doi: 10.3934/dcds.2008.20.617
[8]	Sibei Yang, Dachun Yang, Wenxian Ma. Global regularity estimates for Neumann problems of elliptic operators with coefficients having a BMO anti-symmetric part in NTA domains. Communications on Pure and Applied Analysis, 2022, 21 (3) : 959-998. doi: 10.3934/cpaa.2022006
[9]	Jiahong Wu. Regularity results for weak solutions of the 3D MHD equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 543-556. doi: 10.3934/dcds.2004.10.543
[10]	Tomoyuki Suzuki. Regularity criteria in weak spaces in terms of the pressure to the MHD equations. Conference Publications, 2011, 2011 (Special) : 1335-1343. doi: 10.3934/proc.2011.2011.1335
[11]	Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. Kinetic and Related Models, 2012, 5 (3) : 505-516. doi: 10.3934/krm.2012.5.505
[12]	Feng Cheng, Chao-Jiang Xu. On the Gevrey regularity of solutions to the 3D ideal MHD equations. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6485-6506. doi: 10.3934/dcds.2019281
[13]	Zijin Li, Xinghong Pan. One component regularity criteria for the axially symmetric MHD-Boussinesq system. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2333-2353. doi: 10.3934/dcds.2021192
[14]	Zhengguang Guo, Sadek Gala. Regularity criterion of the Newton-Boussinesq equations in $R^3$. Communications on Pure and Applied Analysis, 2012, 11 (2) : 443-451. doi: 10.3934/cpaa.2012.11.443
[15]	Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299
[16]	Mimi Dai, Han Liu. Low modes regularity criterion for a chemotaxis-Navier-Stokes system. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2713-2735. doi: 10.3934/cpaa.2020118
[17]	Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. II. Kinetic and Related Models, 2014, 7 (2) : 291-304. doi: 10.3934/krm.2014.7.291
[18]	Jishan Fan, Tohru Ozawa. Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model. Kinetic and Related Models, 2009, 2 (2) : 293-305. doi: 10.3934/krm.2009.2.293
[19]	Quansen Jiu, Jitao Liu. Global regularity for the 3D axisymmetric MHD Equations with horizontal dissipation and vertical magnetic diffusion. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 301-322. doi: 10.3934/dcds.2015.35.301
[20]	Cheng-Jie Liu, Feng Xie, Tong Yang. Uniform regularity and vanishing viscosity limit for the incompressible non-resistive MHD system with TMF. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2725-2750. doi: 10.3934/cpaa.2021073

2020 Impact Factor: 1.432

Tools

Metrics

Other articles
by authors

on AIMS
Jishan Fan Tohru Ozawa
on Google Scholar
Jishan Fan Tohru Ozawa

[Back to Top]