May  2012, 17(3): 1061-1073. doi: 10.3934/dcdsb.2012.17.1061

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

Received  March 2011 Revised  September 2011 Published  January 2012

We prove that two dimensional incompressible magnetohydrodynamic flows are stable in $\mathbb{R}^3$. As a corollary, we show the global existence of classical solutions to the three dimensional incompressible magnetohydrodynamic equations with small initial data. Furthermore, our smallness assumption of the perturbed initial data $(u_0, B_0)$ from that of the two dimensional case is only imposed on the scaling invariant quantity $\|u_0\|_{L^2}\|(\xi\cdot\nabla)u_0\|_{L^2} + \|B_0\|_{L^2}\|(\xi\cdot\nabla)B_0\|_{L^2}$ for one direction $\xi$, while $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} + \|B_0\|_{L^2}\|\nabla B_0\|_{L^2}$ may be arbitrarily large.
Citation: Keyan Wang, Yi Du. Stability of the two dimensional magnetohydrodynamic flows in $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 1061-1073. doi: 10.3934/dcdsb.2012.17.1061
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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