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

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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

Abstract
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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 and 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.
[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.
[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.
[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]	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.
[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.
[7]	Q. Jiu and J. Zhao, Global regularity of 2D generalized MHD equations with magnetic diffusion, Z. Angew. Math. Phys., (2014), 1-11.
[8]	J. Fan, H. Malaikah, S. Monaquel, Nakamura and Y. Zhou, Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math., (2013), 1-5.
[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]	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.
[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.
[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.
[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.
[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.
[15]	A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2001.
[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.
[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.
[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.
[19]	J. Wu, The generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.
[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.
[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.
[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.
[23]	K. Yamazaki, A remark on the two-dimensional magnetohydrodynamics-alpha system, http://arxiv.org/abs/1401.6237v1.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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]	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.
[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.
[7]	Q. Jiu and J. Zhao, Global regularity of 2D generalized MHD equations with magnetic diffusion, Z. Angew. Math. Phys., (2014), 1-11.
[8]	J. Fan, H. Malaikah, S. Monaquel, Nakamura and Y. Zhou, Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math., (2013), 1-5.
[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]	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.
[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.
[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.
[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.
[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.
[15]	A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2001.
[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.
[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.
[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.
[19]	J. Wu, The generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.
[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.
[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.
[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.
[23]	K. Yamazaki, A remark on the two-dimensional magnetohydrodynamics-alpha system, http://arxiv.org/abs/1401.6237v1.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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