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,, \emph{Comm. Partial Differential Equations}, 5 (1980), 773.  doi: 10.1080/03605308008820154.  Google Scholar

[2]

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

[3]

D. Catania, Finite dimensional global attractor for 3D MHD-$\alpha$ models: A comparison,, \emph{J. Math. Fluid Mech}, 14 (2012), 95.  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,, \emph{Adv. Math.}, 226 (2011), 1803.  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,, \emph{SIAM J. Math. Anal.}, 46 (2014), 588.  doi: 10.1137/130937718.  Google Scholar

[6]

Q. Jiu and J. Zhao, A remark on global regularity of 2D generalized magnetohydrodynamic equations,, \emph{J. Math. Anal. Appl.}, 412 (2014), 478.  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,, \emph{Z. Angew. Math. Phys.}, (2014), 1.   Google Scholar

[8]

J. Fan, H. Malaikah, S. Monaquel, Nakamura and Y. Zhou, Global Cauchy problem of 2D generalized MHD equations,, \emph{Monatsh. Math.}, (2013), 1.   Google Scholar

[9]

J. Fan and T. Ozawa, Global Cauchy problem for the 2-D magnetohydrodynamic-$\alpha$ models with partial viscous terms,, \emph{J. Math. Fluid Mech.}, 12 (2010), 306.  doi: 10.1007/s00021-008-0289-7.  Google Scholar

[10]

D. Holm, Average Lagrangians and the mean effects of fluctuations in ideal fluid dynamics,, \emph{Physica D}, 170 (2002), 253.  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,, \emph{Comm. Pure Appl. Math.}, 41 (1988), 891.  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,, \emph{Math. Z.}, 242 (2002), 251.  doi: 10.1007/s002090100332.  Google Scholar

[13]

J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models,, \emph{J. Math. Phys.}, 48 (2007).  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,, \emph{J. Stat. Phys.}, 138 (2010), 305.  doi: 10.1007/s10955-009-9916-9.  Google Scholar

[15]

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

[16]

C. V. Tran, X. Yu and Z. Zhai, Note on solution regularity of the generalized magnetohydrodynamic equations with partial dissipation,, \emph{Nonlinear Anal.}, 85 (2013), 43.  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,, \emph{J. Differential. Equations}, 254 (2013), 4194.  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,, \emph{Nonlinear Anal.}, 71 (2009), 4251.  doi: 10.1016/j.na.2009.02.115.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity,, \emph{Appl. Math. Lett.}, 29 (2014), 46.  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,, \emph{Nonlinear Anal.}, 100 (2014), 86.  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,, \emph{Appl. Math. Lett.}, 35 (2014), 1.  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,, \emph{J. Math. Anal. Appl.}, 413 (2014), 633.  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,, \emph{Appl. Math. Lett.}, 29 (2014), 26.  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,, \emph{Math. Meth. Appl. Sci.}, 33 (2010), 856.  doi: 10.1002/mma.1198.  Google Scholar

[29]

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

[30]

Y. Zhou and J. Fan, Regularity criteria for a magnetohydrodynamical-$\alpha$ model,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 309.  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,, \emph{Comm. Partial Differential Equations}, 5 (1980), 773.  doi: 10.1080/03605308008820154.  Google Scholar

[2]

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

[3]

D. Catania, Finite dimensional global attractor for 3D MHD-$\alpha$ models: A comparison,, \emph{J. Math. Fluid Mech}, 14 (2012), 95.  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,, \emph{Adv. Math.}, 226 (2011), 1803.  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,, \emph{SIAM J. Math. Anal.}, 46 (2014), 588.  doi: 10.1137/130937718.  Google Scholar

[6]

Q. Jiu and J. Zhao, A remark on global regularity of 2D generalized magnetohydrodynamic equations,, \emph{J. Math. Anal. Appl.}, 412 (2014), 478.  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,, \emph{Z. Angew. Math. Phys.}, (2014), 1.   Google Scholar

[8]

J. Fan, H. Malaikah, S. Monaquel, Nakamura and Y. Zhou, Global Cauchy problem of 2D generalized MHD equations,, \emph{Monatsh. Math.}, (2013), 1.   Google Scholar

[9]

J. Fan and T. Ozawa, Global Cauchy problem for the 2-D magnetohydrodynamic-$\alpha$ models with partial viscous terms,, \emph{J. Math. Fluid Mech.}, 12 (2010), 306.  doi: 10.1007/s00021-008-0289-7.  Google Scholar

[10]

D. Holm, Average Lagrangians and the mean effects of fluctuations in ideal fluid dynamics,, \emph{Physica D}, 170 (2002), 253.  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,, \emph{Comm. Pure Appl. Math.}, 41 (1988), 891.  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,, \emph{Math. Z.}, 242 (2002), 251.  doi: 10.1007/s002090100332.  Google Scholar

[13]

J. S. Linshiz and E. S. Titi, Analytical study of certain magnetohydrodynamic-$\alpha$ models,, \emph{J. Math. Phys.}, 48 (2007).  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,, \emph{J. Stat. Phys.}, 138 (2010), 305.  doi: 10.1007/s10955-009-9916-9.  Google Scholar

[15]

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

[16]

C. V. Tran, X. Yu and Z. Zhai, Note on solution regularity of the generalized magnetohydrodynamic equations with partial dissipation,, \emph{Nonlinear Anal.}, 85 (2013), 43.  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,, \emph{J. Differential. Equations}, 254 (2013), 4194.  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,, \emph{Nonlinear Anal.}, 71 (2009), 4251.  doi: 10.1016/j.na.2009.02.115.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity,, \emph{Appl. Math. Lett.}, 29 (2014), 46.  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,, \emph{Nonlinear Anal.}, 100 (2014), 86.  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,, \emph{Appl. Math. Lett.}, 35 (2014), 1.  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,, \emph{J. Math. Anal. Appl.}, 413 (2014), 633.  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,, \emph{Appl. Math. Lett.}, 29 (2014), 26.  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,, \emph{Math. Meth. Appl. Sci.}, 33 (2010), 856.  doi: 10.1002/mma.1198.  Google Scholar

[29]

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

[30]

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

[1]

Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143
[2]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[3]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[4]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[5]

Balázs Kósa, Karol Mikula, Markjoe Olunna Uba, Antonia Weberling, Neophytos Christodoulou, Magdalena Zernicka-Goetz. 3D image segmentation supported by a point cloud. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 971-985. doi: 10.3934/dcdss.2020351
[6]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
[7]

Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039
[8]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[9]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310
[10]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034
[11]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268
[12]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299
[13]

Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020289
[14]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075
[15]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267
[16]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326
[17]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156
[18]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445
[19]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293
[20]

Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021007

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences