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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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