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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,, \emph{Comm. Partial Differential Equations}, 5 (1980), 773.
doi: 10.1080/03605308008820154. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
A. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2001).
|
[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. |
[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. |
[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. |
[19] |
J. Wu, The generalized MHD equations,, \emph{J. Differential Equations}, 195 (2003), 284.
doi: 10.1016/j.jde.2003.07.007. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
A. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2001).
|
[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. |
[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. |
[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. |
[19] |
J. Wu, The generalized MHD equations,, \emph{J. Differential Equations}, 195 (2003), 284.
doi: 10.1016/j.jde.2003.07.007. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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