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

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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

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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

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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

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., (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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