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Stability of the two dimensional magnetohydrodynamic flows in $\mathbb{R}^3$
1. | Department of Mathematics, Shanghai Finance University, Shanghai 201209 |
2. | Department of Mathematical Sciences, South China Normal University, Guangzhou, 510631 |
References:
[1] |
O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, "Integral Representations of Functions, and Imbedding Theorems," Izdat. "Nauka," Moskow, 1975. |
[2] |
L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.
doi: 10.1002/cpa.3160350604. |
[3] |
Chongsheng Cao and Jiahong Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Advances in Mathematics, 226 (2011), 1803-1822. |
[4] |
Cheng He and Zhouping Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152.
doi: 10.1016/j.jfa.2005.06.009. |
[5] |
Cheng He and Zhouping Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[6] |
G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, (French) Arch. Rational Mech. Anal., 46 (1972), 241-279. |
[7] |
Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.
doi: 10.1016/0022-1236(91)90136-S. |
[8] |
H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194.
doi: 10.1007/s002090000130. |
[9] |
Zhen Lei, Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chinese Ann. Math. Ser. B, 27 (2006), 565-580.
doi: 10.1007/s11401-005-0041-z. |
[10] |
Zhen Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37.
doi: 10.1007/s00205-010-0346-2. |
[11] |
F.-H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.
doi: 10.1002/cpa.20074. |
[12] |
Zhen Lei, Chun Liu and Yi Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398. |
[13] |
Zhen Lei, Chun Liu and Yi Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Comm. Math. Sci., 5 (2007), 595-616. |
[14] |
Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete and Continuous Dynamical Systems, 25 (2009), 575-583.
doi: 10.3934/dcds.2009.25.575. |
[15] |
Z. Lei and Y. Zhou, Global existence of classical solutions for two-dimensional Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814.
doi: 10.1137/040618813. |
[16] |
Fanghua Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257.
doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A. |
[17] |
P. B. Mucha, Stability of 2D incompressible flows in $\mathbb{R}^3$, J. Differential Equations, 245 (2008), 2355-2367.
doi: 10.1016/j.jde.2008.07.033. |
[18] |
Keyan Wang, On global regularity of incompressible Navier-Stokes equations in $\mathbb{R}^3$, Comm. Pure Appl. Anal., 8 (2009), 1067-1072.
doi: 10.3934/cpaa.2009.8.1067. |
[19] |
Jiahong Wu, Regularity results for weak solutions of the 3D MHD equations. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 543-556.
doi: 10.3934/dcds.2004.10.543. |
[20] |
M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.
doi: 10.1002/cpa.3160360506. |
[21] |
Fan Wang and Keyan Wang, Global regularity for the 3D MHD equations with mixed partial dissipation with small initial data, preprint. |
[22] |
Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. |
[23] |
Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 41 (2006), 1174-1180.
doi: 10.1016/j.ijnonlinmec.2006.12.001. |
[24] |
Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886.
doi: 10.3934/dcds.2005.12.881. |
show all references
References:
[1] |
O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, "Integral Representations of Functions, and Imbedding Theorems," Izdat. "Nauka," Moskow, 1975. |
[2] |
L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.
doi: 10.1002/cpa.3160350604. |
[3] |
Chongsheng Cao and Jiahong Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Advances in Mathematics, 226 (2011), 1803-1822. |
[4] |
Cheng He and Zhouping Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152.
doi: 10.1016/j.jfa.2005.06.009. |
[5] |
Cheng He and Zhouping Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[6] |
G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, (French) Arch. Rational Mech. Anal., 46 (1972), 241-279. |
[7] |
Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.
doi: 10.1016/0022-1236(91)90136-S. |
[8] |
H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z., 235 (2000), 173-194.
doi: 10.1007/s002090000130. |
[9] |
Zhen Lei, Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chinese Ann. Math. Ser. B, 27 (2006), 565-580.
doi: 10.1007/s11401-005-0041-z. |
[10] |
Zhen Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37.
doi: 10.1007/s00205-010-0346-2. |
[11] |
F.-H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.
doi: 10.1002/cpa.20074. |
[12] |
Zhen Lei, Chun Liu and Yi Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398. |
[13] |
Zhen Lei, Chun Liu and Yi Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Comm. Math. Sci., 5 (2007), 595-616. |
[14] |
Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete and Continuous Dynamical Systems, 25 (2009), 575-583.
doi: 10.3934/dcds.2009.25.575. |
[15] |
Z. Lei and Y. Zhou, Global existence of classical solutions for two-dimensional Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814.
doi: 10.1137/040618813. |
[16] |
Fanghua Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257.
doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A. |
[17] |
P. B. Mucha, Stability of 2D incompressible flows in $\mathbb{R}^3$, J. Differential Equations, 245 (2008), 2355-2367.
doi: 10.1016/j.jde.2008.07.033. |
[18] |
Keyan Wang, On global regularity of incompressible Navier-Stokes equations in $\mathbb{R}^3$, Comm. Pure Appl. Anal., 8 (2009), 1067-1072.
doi: 10.3934/cpaa.2009.8.1067. |
[19] |
Jiahong Wu, Regularity results for weak solutions of the 3D MHD equations. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 543-556.
doi: 10.3934/dcds.2004.10.543. |
[20] |
M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.
doi: 10.1002/cpa.3160360506. |
[21] |
Fan Wang and Keyan Wang, Global regularity for the 3D MHD equations with mixed partial dissipation with small initial data, preprint. |
[22] |
Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505. |
[23] |
Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 41 (2006), 1174-1180.
doi: 10.1016/j.ijnonlinmec.2006.12.001. |
[24] |
Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886.
doi: 10.3934/dcds.2005.12.881. |
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