Stability of the two dimensional magnetohydrodynamic flows in \mathbb{R}^3

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

Stability of the two dimensional magnetohydrodynamic flows in $\mathbb{R}^3$

Keyan Wang ^1, and Yi Du ^2,

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

Abstract
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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.

Keywords: large data, Magnetohydrodynamic equations, global existence., stability.

Mathematics Subject Classification: Primary: 35Q30, 35B35; Secondary: 76D05,76E0.

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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