# American Institute of Mathematical Sciences

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 & 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., (1975).   Google Scholar [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.  doi: 10.1002/cpa.3160350604.  Google Scholar [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.   Google Scholar [4] Cheng He and Zhouping Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations,, J. Funct. Anal., 227 (2005), 113.  doi: 10.1016/j.jfa.2005.06.009.  Google Scholar [5] Cheng He and Zhouping Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, J. Differential Equations, 213 (2005), 235.  doi: 10.1016/j.jde.2004.07.002.  Google Scholar [6] G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, (French) Arch. Rational Mech. Anal., 46 (1972), 241.   Google Scholar [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.  doi: 10.1016/0022-1236(91)90136-S.  Google Scholar [8] H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.  doi: 10.1007/s002090000130.  Google Scholar [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.  doi: 10.1007/s11401-005-0041-z.  Google Scholar [10] Zhen Lei, On 2D viscoelasticity with small strain,, Arch. Ration. Mech. Anal., 198 (2010), 13.  doi: 10.1007/s00205-010-0346-2.  Google Scholar [11] F.-H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids,, Comm. Pure Appl. Math., 58 (2005), 1437.  doi: 10.1002/cpa.20074.  Google Scholar [12] Zhen Lei, Chun Liu and Yi Zhou, Global solutions for incompressible viscoelastic fluids,, Arch. Ration. Mech. Anal., 188 (2008), 371.   Google Scholar [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.   Google Scholar [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.  doi: 10.3934/dcds.2009.25.575.  Google Scholar [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.  doi: 10.1137/040618813.  Google Scholar [16] Fanghua Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem,, Comm. Pure Appl. Math., 51 (1998), 241.  doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.  Google Scholar [17] P. B. Mucha, Stability of 2D incompressible flows in $\mathbbR^3$,, J. Differential Equations, 245 (2008), 2355.  doi: 10.1016/j.jde.2008.07.033.  Google Scholar [18] Keyan Wang, On global regularity of incompressible Navier-Stokes equations in $\mathbbR^3$,, Comm. Pure Appl. Anal., 8 (2009), 1067.  doi: 10.3934/cpaa.2009.8.1067.  Google Scholar [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.  doi: 10.3934/dcds.2004.10.543.  Google Scholar [20] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, Comm. Pure Appl. Math., 36 (1983), 635.  doi: 10.1002/cpa.3160360506.  Google Scholar [21] Fan Wang and Keyan Wang, Global regularity for the 3D MHD equations with mixed partial dissipation with small initial data,, preprint., ().   Google Scholar [22] Y. Zhou, Regularity criteria for the generalized viscous MHD equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491.   Google Scholar [23] Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure,, Internat. J. Non-Linear Mech., 41 (2006), 1174.  doi: 10.1016/j.ijnonlinmec.2006.12.001.  Google Scholar [24] Y. Zhou, Remarks on regularities for the 3D MHD equations,, Discrete Contin. Dyn. Syst., 12 (2005), 881.  doi: 10.3934/dcds.2005.12.881.  Google Scholar

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##### References:
 [1] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, "Integral Representations of Functions, and Imbedding Theorems,", Izdat., (1975).   Google Scholar [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.  doi: 10.1002/cpa.3160350604.  Google Scholar [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.   Google Scholar [4] Cheng He and Zhouping Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations,, J. Funct. Anal., 227 (2005), 113.  doi: 10.1016/j.jfa.2005.06.009.  Google Scholar [5] Cheng He and Zhouping Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, J. Differential Equations, 213 (2005), 235.  doi: 10.1016/j.jde.2004.07.002.  Google Scholar [6] G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, (French) Arch. Rational Mech. Anal., 46 (1972), 241.   Google Scholar [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.  doi: 10.1016/0022-1236(91)90136-S.  Google Scholar [8] H. Kozono and Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations,, Math. Z., 235 (2000), 173.  doi: 10.1007/s002090000130.  Google Scholar [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.  doi: 10.1007/s11401-005-0041-z.  Google Scholar [10] Zhen Lei, On 2D viscoelasticity with small strain,, Arch. Ration. Mech. Anal., 198 (2010), 13.  doi: 10.1007/s00205-010-0346-2.  Google Scholar [11] F.-H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids,, Comm. Pure Appl. Math., 58 (2005), 1437.  doi: 10.1002/cpa.20074.  Google Scholar [12] Zhen Lei, Chun Liu and Yi Zhou, Global solutions for incompressible viscoelastic fluids,, Arch. Ration. Mech. Anal., 188 (2008), 371.   Google Scholar [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.   Google Scholar [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.  doi: 10.3934/dcds.2009.25.575.  Google Scholar [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.  doi: 10.1137/040618813.  Google Scholar [16] Fanghua Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem,, Comm. Pure Appl. Math., 51 (1998), 241.  doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.  Google Scholar [17] P. B. Mucha, Stability of 2D incompressible flows in $\mathbbR^3$,, J. Differential Equations, 245 (2008), 2355.  doi: 10.1016/j.jde.2008.07.033.  Google Scholar [18] Keyan Wang, On global regularity of incompressible Navier-Stokes equations in $\mathbbR^3$,, Comm. Pure Appl. Anal., 8 (2009), 1067.  doi: 10.3934/cpaa.2009.8.1067.  Google Scholar [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.  doi: 10.3934/dcds.2004.10.543.  Google Scholar [20] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, Comm. Pure Appl. Math., 36 (1983), 635.  doi: 10.1002/cpa.3160360506.  Google Scholar [21] Fan Wang and Keyan Wang, Global regularity for the 3D MHD equations with mixed partial dissipation with small initial data,, preprint., ().   Google Scholar [22] Y. Zhou, Regularity criteria for the generalized viscous MHD equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491.   Google Scholar [23] Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure,, Internat. J. Non-Linear Mech., 41 (2006), 1174.  doi: 10.1016/j.ijnonlinmec.2006.12.001.  Google Scholar [24] Y. Zhou, Remarks on regularities for the 3D MHD equations,, Discrete Contin. Dyn. Syst., 12 (2005), 881.  doi: 10.3934/dcds.2005.12.881.  Google Scholar
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