-
Previous Article
$W$-Sobolev spaces: Higher order and regularity
- CPAA Home
- This Issue
-
Next Article
On the growth of the energy of entire solutions to the vector Allen-Cahn equation
Note on global regularity of 3D generalized magnetohydrodynamic-$\alpha$ model with zero diffusivity
1. | School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China |
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. |
[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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2022, 42 (6) : 2927-2943. 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 and 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 and 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 and Related Models, 2019, 12 (1) : 37-58. doi: 10.3934/krm.2019002 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]