doi: 10.3934/dcds.2021141
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Global stability solution of the 2D MHD equations with mixed partial dissipation

1. 

Institutes of Physical Science and Information Technology, Anhui University, Hefei 230601, China

2. 

School of Mathematical Science, Anhui University, Hefei 230601, China

3. 

College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China

* Corresponding author: Bo-Qing Dong

Received  April 2021 Revised  July 2021 Early access September 2021

This paper is devoted to understanding the global stability of perturbations near a background magnetic field of the 2D magnetohydrodynamic (MHD) equations with partial dissipation. We establish the global stability for the solutions of the nonlinear MHD system by the bootstrap argument.

Citation: Yana Guo, Yan Jia, Bo-Qing Dong. Global stability solution of the 2D MHD equations with mixed partial dissipation. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021141
References:
[1]

H. Abidi and P. Zhang, On the global well-posedness of 3-D MHD system with initial data near the equilibrium state, Comm. Pure Appl. Math., 70 (2017), 1509-1561.  doi: 10.1002/cpa.21645.  Google Scholar

[2]

H. Alfvén, Existence of electromagnetic-hydrodynamic waves, Nature, 150 (1942), 405-406.   Google Scholar

[3] D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511599965.  Google Scholar
[4]

N. BoardmanH. Lin and J. Wu, Stabilization of a background magnetic field on a 2 dimensional magnetohydrodynamic flow, SIAM J. Math. Anal., 52 (2020), 5001-5035.  doi: 10.1137/20M1324776.  Google Scholar

[5]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Rational Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4.  Google Scholar

[6]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822.  doi: 10.1016/j.aim.2010.08.017.  Google Scholar

[7] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, England, 2001.  doi: 10.1017/CBO9780511626333.  Google Scholar
[8]

W. Deng and P. Zhang, Large time behavior of solutions to 3-D MHD system with initial data near equilibrium, Arch. Ration. Mech. Anal., 230 (2018), 1017-1102.  doi: 10.1007/s00205-018-1265-x.  Google Scholar

[9]

B.-Q. DongY. JiaJ. Li and J. Wu, Global regularity and time decay for the 2D magnetohydrodynamic equations with fractional dissipation and partial magnetic diffusion, J. Math. Fluid Mech., 20 (2018), 1541-1565.  doi: 10.1007/s00021-018-0376-3.  Google Scholar

[10]

B.-Q. DongJ. Li and J. Wu, Global regularity for the 2D MHD equation with partial hyper-resistivity, Int. Math. Res. Not., 14 (2019), 4261-4280.  doi: 10.1093/imrn/rnx240.  Google Scholar

[11]

L. Du and D. Zhou, Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion, SIAM J. Math. Anal., 47 (2015), 1562-1589.  doi: 10.1137/140959821.  Google Scholar

[12]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch, Ration. Mech. Anal., 46 (1972), 241-279.  doi: 10.1007/BF00250512.  Google Scholar

[13]

L.-B. He, L. Xu and P. Yu, On global dynamics of three dimensional magnetohydrodynamics: Nonlinear stability of Alfvén waves, Ann. PDE, 4 (2018), Paper No. 5,105 pp. doi: 10.1007/s40818-017-0041-9.  Google Scholar

[14]

R. Ji and J. Wu, The resistive magnetohydrodynamic equation near an equilibrium, J. Differential Equations, 268 (2020), 1854-1871.  doi: 10.1016/j.jde.2019.09.027.  Google Scholar

[15]

Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25 (2009), 575-583.  doi: 10.3934/dcds.2009.25.575.  Google Scholar

[16]

C. LiJ. Wu and X. Xu, Smoothing and stabilization effects of magnetic field on electrically conducting fluids, J. Differential Equations, 276 (2021), 368-403.  doi: 10.1016/j.jde.2020.12.012.  Google Scholar

[17]

F. LinL. Xu and P. Zhang, Global small solutions to 2-d incompressible mhd system, J. Differential Equations, 259 (2015), 5440-5485.  doi: 10.1016/j.jde.2015.06.034.  Google Scholar

[18]

F. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580.  doi: 10.1002/cpa.21506.  Google Scholar

[19]

H. Lin, R. Ji, J. Wu and L. Yan, Stability of perturbations near a background magnetic field of the 2D incompressible MHD equations with mixed partial dissipation, J. Funct. Anal., 279 (2020), 108519, 39 pp. doi: 10.1016/j.jfa.2020.108519.  Google Scholar

[20]

R. PanY. Zhou and Y. Zhu, Global classical solutions of three dimensional viscous mhd system without magnetic diffusion on periodic boxes, Arch. Ration. Mech. Anal., 227 (2018), 637-662.  doi: 10.1007/s00205-017-1170-8.  Google Scholar

[21]

X. RenJ. WuZ. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541.  doi: 10.1016/j.jfa.2014.04.020.  Google Scholar

[22]

M. Schonbek and T. Schonbek, Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows, Discrete Contin. Dyn. Syst., 13 (2005), 1277-1304.  doi: 10.3934/dcds.2005.13.1277.  Google Scholar

[23]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Commun. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.  Google Scholar

[24]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.  Google Scholar

[25]

D. Wei and Z. Zhang, Global well-posedness of the MHD equations in a homogeneous magnetic field, Anal. PDE, 10 (2017), 1361-1406.  doi: 10.2140/apde.2017.10.1361.  Google Scholar

[26]

J. Wu, The 2D magnetohydrodynamic equations with partial or fractional dissipation, Lectures on the Analysis of Nonlinear Partial Differential Equations, Part 5, MLM5, in Morningside Lectures on Mathematics, International Press, Somerville, MA, (2018), 283–332.  Google Scholar

[27]

J. WuY. Wu and X. Xu, Global small solution to the 2D mhd system with a velocity damping term, SIAM J. Math. Anal., 47 (2015), 2630-2656.  doi: 10.1137/140985445.  Google Scholar

[28]

J. Wu and Y. Zhu, Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium, Adv. Math., 377 (2021), 107466, 26 pp. doi: 10.1016/j.aim.2020.107466.  Google Scholar

[29]

T. Zhang, Global solutions to the 2D viscous, non-resistive MHD system with large background magnetic field, J. Differential Equations, 260 (2016), 5450-5480.  doi: 10.1016/j.jde.2015.12.005.  Google Scholar

show all references

References:
[1]

H. Abidi and P. Zhang, On the global well-posedness of 3-D MHD system with initial data near the equilibrium state, Comm. Pure Appl. Math., 70 (2017), 1509-1561.  doi: 10.1002/cpa.21645.  Google Scholar

[2]

H. Alfvén, Existence of electromagnetic-hydrodynamic waves, Nature, 150 (1942), 405-406.   Google Scholar

[3] D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511599965.  Google Scholar
[4]

N. BoardmanH. Lin and J. Wu, Stabilization of a background magnetic field on a 2 dimensional magnetohydrodynamic flow, SIAM J. Math. Anal., 52 (2020), 5001-5035.  doi: 10.1137/20M1324776.  Google Scholar

[5]

Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Rational Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4.  Google Scholar

[6]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822.  doi: 10.1016/j.aim.2010.08.017.  Google Scholar

[7] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, England, 2001.  doi: 10.1017/CBO9780511626333.  Google Scholar
[8]

W. Deng and P. Zhang, Large time behavior of solutions to 3-D MHD system with initial data near equilibrium, Arch. Ration. Mech. Anal., 230 (2018), 1017-1102.  doi: 10.1007/s00205-018-1265-x.  Google Scholar

[9]

B.-Q. DongY. JiaJ. Li and J. Wu, Global regularity and time decay for the 2D magnetohydrodynamic equations with fractional dissipation and partial magnetic diffusion, J. Math. Fluid Mech., 20 (2018), 1541-1565.  doi: 10.1007/s00021-018-0376-3.  Google Scholar

[10]

B.-Q. DongJ. Li and J. Wu, Global regularity for the 2D MHD equation with partial hyper-resistivity, Int. Math. Res. Not., 14 (2019), 4261-4280.  doi: 10.1093/imrn/rnx240.  Google Scholar

[11]

L. Du and D. Zhou, Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion, SIAM J. Math. Anal., 47 (2015), 1562-1589.  doi: 10.1137/140959821.  Google Scholar

[12]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch, Ration. Mech. Anal., 46 (1972), 241-279.  doi: 10.1007/BF00250512.  Google Scholar

[13]

L.-B. He, L. Xu and P. Yu, On global dynamics of three dimensional magnetohydrodynamics: Nonlinear stability of Alfvén waves, Ann. PDE, 4 (2018), Paper No. 5,105 pp. doi: 10.1007/s40818-017-0041-9.  Google Scholar

[14]

R. Ji and J. Wu, The resistive magnetohydrodynamic equation near an equilibrium, J. Differential Equations, 268 (2020), 1854-1871.  doi: 10.1016/j.jde.2019.09.027.  Google Scholar

[15]

Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25 (2009), 575-583.  doi: 10.3934/dcds.2009.25.575.  Google Scholar

[16]

C. LiJ. Wu and X. Xu, Smoothing and stabilization effects of magnetic field on electrically conducting fluids, J. Differential Equations, 276 (2021), 368-403.  doi: 10.1016/j.jde.2020.12.012.  Google Scholar

[17]

F. LinL. Xu and P. Zhang, Global small solutions to 2-d incompressible mhd system, J. Differential Equations, 259 (2015), 5440-5485.  doi: 10.1016/j.jde.2015.06.034.  Google Scholar

[18]

F. Lin and P. Zhang, Global small solutions to an MHD-type system: The three-dimensional case, Comm. Pure Appl. Math., 67 (2014), 531-580.  doi: 10.1002/cpa.21506.  Google Scholar

[19]

H. Lin, R. Ji, J. Wu and L. Yan, Stability of perturbations near a background magnetic field of the 2D incompressible MHD equations with mixed partial dissipation, J. Funct. Anal., 279 (2020), 108519, 39 pp. doi: 10.1016/j.jfa.2020.108519.  Google Scholar

[20]

R. PanY. Zhou and Y. Zhu, Global classical solutions of three dimensional viscous mhd system without magnetic diffusion on periodic boxes, Arch. Ration. Mech. Anal., 227 (2018), 637-662.  doi: 10.1007/s00205-017-1170-8.  Google Scholar

[21]

X. RenJ. WuZ. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541.  doi: 10.1016/j.jfa.2014.04.020.  Google Scholar

[22]

M. Schonbek and T. Schonbek, Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows, Discrete Contin. Dyn. Syst., 13 (2005), 1277-1304.  doi: 10.3934/dcds.2005.13.1277.  Google Scholar

[23]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Commun. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.  Google Scholar

[24]

T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.  Google Scholar

[25]

D. Wei and Z. Zhang, Global well-posedness of the MHD equations in a homogeneous magnetic field, Anal. PDE, 10 (2017), 1361-1406.  doi: 10.2140/apde.2017.10.1361.  Google Scholar

[26]

J. Wu, The 2D magnetohydrodynamic equations with partial or fractional dissipation, Lectures on the Analysis of Nonlinear Partial Differential Equations, Part 5, MLM5, in Morningside Lectures on Mathematics, International Press, Somerville, MA, (2018), 283–332.  Google Scholar

[27]

J. WuY. Wu and X. Xu, Global small solution to the 2D mhd system with a velocity damping term, SIAM J. Math. Anal., 47 (2015), 2630-2656.  doi: 10.1137/140985445.  Google Scholar

[28]

J. Wu and Y. Zhu, Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium, Adv. Math., 377 (2021), 107466, 26 pp. doi: 10.1016/j.aim.2020.107466.  Google Scholar

[29]

T. Zhang, Global solutions to the 2D viscous, non-resistive MHD system with large background magnetic field, J. Differential Equations, 260 (2016), 5450-5480.  doi: 10.1016/j.jde.2015.12.005.  Google Scholar

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