American Institute of Mathematical Sciences

Advanced
March  2015, 14(2): 585-595. doi: 10.3934/cpaa.2015.14.585

Note on global regularity of 3D generalized magnetohydrodynamic-$\alpha$ model with zero diffusivity

Xiaojing Xu 1, and  Zhuan Ye 1,

1. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received  June 2014 Revised  August 2014 Published  December 2014

In this paper we consider the Cauchy problem of a three-dimensional incompressible magnetohydrodynamic-$\alpha$ (MHD-$\alpha$) model with fractional velocity and zero magnetic diffusivity. We prove the global existence results of classical solutions for the model in the endpoint case with arbitrarily large initial data in Sobolev spaces.
Keywords: Incompressible 3D MHD-$\alpha$ model, fractional Laplacian, uniqueness., global regularity.
Mathematics Subject Classification: 35Q35, 76D03,.
Citation: Xiaojing Xu, Zhuan Ye. Note on global regularity of 3D generalized magnetohydrodynamic-$\alpha$ model with zero diffusivity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 585-595. doi: 10.3934/cpaa.2015.14.585
References:
[1]

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

[2]

D. Catania, Global existence for a regularized magnetohydrodynamic-$\alpha$ model, Ann. Univ. Ferrara, 56 (2010), 1-20. doi: 10.1007/s11565-009-0069-1.  Google Scholar

[3]

D. Catania, Finite dimensional global attractor for 3D MHD-$\alpha$ models: A comparison, J. Math. Fluid Mech, 14 (2012), 95-115. doi: 10.1007/s00021-010-0041-y.  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]

C. Cao, J. Wu and B, Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46 (2014), 588-602. doi: 10.1137/130937718.  Google Scholar

[6]

Q. Jiu and J. Zhao, A remark on global regularity of 2D generalized magnetohydrodynamic equations, J. Math. Anal. Appl., 412 (2014), 478-484. doi: 10.1016/j.jmaa.2013.10.074.  Google Scholar

[7]

Q. Jiu and J. Zhao, Global regularity of 2D generalized MHD equations with magnetic diffusion, Z. Angew. Math. Phys., (2014), 1-11. Google Scholar

[8]

J. Fan, H. Malaikah, S. Monaquel, Nakamura and Y. Zhou, Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math., (2013), 1-5. 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]

D. Holm, Average Lagrangians and the mean effects of fluctuations in ideal fluid dynamics, Physica D, 170 (2002), 253-286. doi: 10.1016/S0167-2789(02)00552-3.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

J. S. Linshiz and E. S. Titi, On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to the Euler equations, J. Stat. Phys., 138 (2010), 305-332. doi: 10.1007/s10955-009-9916-9.  Google Scholar

[15]

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2001.  Google Scholar

[16]

C. V. Tran, X. Yu and Z. Zhai, Note on solution regularity of the generalized magnetohydrodynamic equations with partial dissipation, Nonlinear Anal., 85 (2013), 43-51. doi: 10.1016/j.na.2013.02.019.  Google Scholar

[17]

C. V. Tran, X. Yu and Z. Zhai, On global regularity of 2D generalized magnetodydrodynamics equations, J. Differential. Equations, 254 (2013), 4194-4216. doi: 10.1016/j.jde.2013.02.016.  Google Scholar

[18]

G. Wu, Regularity criteria for the 3D generalized MHD equations in terms of vorticity, Nonlinear Anal., 71 (2009), 4251-4258. doi: 10.1016/j.na.2009.02.115.  Google Scholar

[19]

J. Wu, The generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.  Google Scholar

[20]

J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.  Google Scholar

[21]

K. Yamazaki, On the global regularity of generalized Leray-alpha type models, Nonlinear Anal., 75 (2012), 503-515. doi: 10.1016/j.na.2011.08.051.  Google Scholar

[22]

K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity, Appl. Math. Lett., 29 (2014), 46-51. doi: 10.1016/j.aml.2013.10.014.  Google Scholar

[23]

K. Yamazaki, A remark on the two-dimensional magnetohydrodynamics-alpha system,, http://arxiv.org/abs/1401.6237v1., ().   Google Scholar

[24]

Z. Ye and X. Xu, Global regularity of the two-dimensional incompressible generalized magnetohydrodynamics system, Nonlinear Anal., 100 (2014), 86-96. doi: 10.1016/j.na.2014.01.012.  Google Scholar

[25]

Z. Ye and X. Xu, Global regularity of 3D generalized incompressible magnetohydrodynamic-$\alpha$ model, Appl. Math. Lett., 35 (2014), 1-6. doi: 10.1016/j.aml.2014.03.018.  Google Scholar

[26]

B. Yuan and L. Bai, Remarks on global regularity of 2D generalized MHD equations, J. Math. Anal. Appl., 413 (2014), 633-640. doi: 10.1016/j.jmaa.2013.12.024.  Google Scholar

[27]

J. Zhao and M. Zhu, Global regularity for the incompressible MHD-$\alpha$ system with fractional diffusion, Appl. Math. Lett., 29 (2014), 26-29. doi: 10.1016/j.aml.2013.10.009.  Google Scholar

[28]

Y. Zhou and J. Fan, Global well-posedness for two modified-Leray-$\alpha$-MHD models with partial viscous terms, Math. Meth. Appl. Sci., 33 (2010), 856-862. doi: 10.1002/mma.1198.  Google Scholar

[29]

Y. Zhou and J. Fan, On the Cauchy problem for a Leray-$\alpha$-MHD model, Nonlinear Anal., 12 (2011), 648-657. doi: 10.1016/j.nonrwa.2010.07.007.  Google Scholar

[30]

Y. Zhou and J. Fan, Regularity criteria for a magnetohydrodynamical-$\alpha$ model, Commun. Pure Appl. Anal., 10 (2011), 309-326. doi: 10.3934/cpaa.2011.10.309.  Google Scholar

show all references

References:
[1]

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

[2]

D. Catania, Global existence for a regularized magnetohydrodynamic-$\alpha$ model, Ann. Univ. Ferrara, 56 (2010), 1-20. doi: 10.1007/s11565-009-0069-1.  Google Scholar

[3]

D. Catania, Finite dimensional global attractor for 3D MHD-$\alpha$ models: A comparison, J. Math. Fluid Mech, 14 (2012), 95-115. doi: 10.1007/s00021-010-0041-y.  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]

C. Cao, J. Wu and B, Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46 (2014), 588-602. doi: 10.1137/130937718.  Google Scholar

[6]

Q. Jiu and J. Zhao, A remark on global regularity of 2D generalized magnetohydrodynamic equations, J. Math. Anal. Appl., 412 (2014), 478-484. doi: 10.1016/j.jmaa.2013.10.074.  Google Scholar

[7]

Q. Jiu and J. Zhao, Global regularity of 2D generalized MHD equations with magnetic diffusion, Z. Angew. Math. Phys., (2014), 1-11. Google Scholar

[8]

J. Fan, H. Malaikah, S. Monaquel, Nakamura and Y. Zhou, Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math., (2013), 1-5. 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]

D. Holm, Average Lagrangians and the mean effects of fluctuations in ideal fluid dynamics, Physica D, 170 (2002), 253-286. doi: 10.1016/S0167-2789(02)00552-3.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

J. S. Linshiz and E. S. Titi, On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to the Euler equations, J. Stat. Phys., 138 (2010), 305-332. doi: 10.1007/s10955-009-9916-9.  Google Scholar

[15]

A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2001.  Google Scholar

[16]

C. V. Tran, X. Yu and Z. Zhai, Note on solution regularity of the generalized magnetohydrodynamic equations with partial dissipation, Nonlinear Anal., 85 (2013), 43-51. doi: 10.1016/j.na.2013.02.019.  Google Scholar

[17]

C. V. Tran, X. Yu and Z. Zhai, On global regularity of 2D generalized magnetodydrodynamics equations, J. Differential. Equations, 254 (2013), 4194-4216. doi: 10.1016/j.jde.2013.02.016.  Google Scholar

[18]

G. Wu, Regularity criteria for the 3D generalized MHD equations in terms of vorticity, Nonlinear Anal., 71 (2009), 4251-4258. doi: 10.1016/j.na.2009.02.115.  Google Scholar

[19]

J. Wu, The generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.  Google Scholar

[20]

J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.  Google Scholar

[21]

K. Yamazaki, On the global regularity of generalized Leray-alpha type models, Nonlinear Anal., 75 (2012), 503-515. doi: 10.1016/j.na.2011.08.051.  Google Scholar

[22]

K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity, Appl. Math. Lett., 29 (2014), 46-51. doi: 10.1016/j.aml.2013.10.014.  Google Scholar

[23]

K. Yamazaki, A remark on the two-dimensional magnetohydrodynamics-alpha system,, http://arxiv.org/abs/1401.6237v1., ().   Google Scholar

[24]

Z. Ye and X. Xu, Global regularity of the two-dimensional incompressible generalized magnetohydrodynamics system, Nonlinear Anal., 100 (2014), 86-96. doi: 10.1016/j.na.2014.01.012.  Google Scholar

[25]

Z. Ye and X. Xu, Global regularity of 3D generalized incompressible magnetohydrodynamic-$\alpha$ model, Appl. Math. Lett., 35 (2014), 1-6. doi: 10.1016/j.aml.2014.03.018.  Google Scholar

[26]

B. Yuan and L. Bai, Remarks on global regularity of 2D generalized MHD equations, J. Math. Anal. Appl., 413 (2014), 633-640. doi: 10.1016/j.jmaa.2013.12.024.  Google Scholar

[27]

J. Zhao and M. Zhu, Global regularity for the incompressible MHD-$\alpha$ system with fractional diffusion, Appl. Math. Lett., 29 (2014), 26-29. doi: 10.1016/j.aml.2013.10.009.  Google Scholar

[28]

Y. Zhou and J. Fan, Global well-posedness for two modified-Leray-$\alpha$-MHD models with partial viscous terms, Math. Meth. Appl. Sci., 33 (2010), 856-862. doi: 10.1002/mma.1198.  Google Scholar

[29]

Y. Zhou and J. Fan, On the Cauchy problem for a Leray-$\alpha$-MHD model, Nonlinear Anal., 12 (2011), 648-657. doi: 10.1016/j.nonrwa.2010.07.007.  Google Scholar

[30]

Y. Zhou and J. Fan, Regularity criteria for a magnetohydrodynamical-$\alpha$ model, Commun. Pure Appl. Anal., 10 (2011), 309-326. doi: 10.3934/cpaa.2011.10.309.  Google Scholar

[1]

Jishan Fan, Tohru Ozawa. Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model. Conference Publications, 2011, 2011 (Special) : 400-409. doi: 10.3934/proc.2011.2011.400
[2]

Fei Chen, Yongsheng Li, Huan Xu. Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 2945-2967. doi: 10.3934/dcds.2016.36.2945
[3]

Sadek Gala. A new regularity criterion for the 3D MHD equations in $R^3$. Communications on Pure & Applied Analysis, 2012, 11 (3) : 973-980. doi: 10.3934/cpaa.2012.11.973
[4]

Quansen Jiu, Jitao Liu. Global regularity for the 3D axisymmetric MHD Equations with horizontal dissipation and vertical magnetic diffusion. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 301-322. doi: 10.3934/dcds.2015.35.301
[5]

Jiahong Wu. Regularity results for weak solutions of the 3D MHD equations. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 543-556. doi: 10.3934/dcds.2004.10.543
[6]

Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. Kinetic & Related Models, 2012, 5 (3) : 505-516. doi: 10.3934/krm.2012.5.505
[7]

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
[8]

Feng Cheng, Chao-Jiang Xu. On the Gevrey regularity of solutions to the 3D ideal MHD equations. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6485-6506. doi: 10.3934/dcds.2019281
[9]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[10]

Edriss S. Titi, Saber Trabelsi. Global well-posedness of a 3D MHD model in porous media. Journal of Geometric Mechanics, 2019, 11 (4) : 621-637. doi: 10.3934/jgm.2019031
[11]

Junxiong Jia, Jigen Peng, Kexue Li. On the decay and stability of global solutions to the 3D inhomogeneous MHD system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 745-780. doi: 10.3934/cpaa.2017036
[12]

Tomás Caraballo, Antonio M. Márquez-Durán, José Real. Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 559-578. doi: 10.3934/dcds.2006.15.559
[13]

Jitao Liu. On the initial boundary value problem for certain 2D MHD-$\alpha$ equations without velocity viscosity. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1179-1191. doi: 10.3934/cpaa.2016.15.1179
[14]

T. Tachim Medjo. A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor. Communications on Pure & Applied Analysis, 2011, 10 (2) : 415-433. doi: 10.3934/cpaa.2011.10.415
[15]

Anne Bronzi, Ricardo Rosa. On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 19-49. doi: 10.3934/dcds.2014.34.19
[16]

Anthony Suen. Global regularity for the 3D compressible magnetohydrodynamics with general pressure. Discrete & Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022004
[17]

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
[18]

Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. II. Kinetic & Related Models, 2014, 7 (2) : 291-304. doi: 10.3934/krm.2014.7.291
[19]

Yong Zhou. Remarks on regularities for the 3D MHD equations. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 881-886. doi: 10.3934/dcds.2005.12.881
[20]

Fei Chen, Boling Guo, Xiaoping Zhai. Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density. Kinetic & Related Models, 2019, 12 (1) : 37-58. doi: 10.3934/krm.2019002

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy