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doi: 10.3934/dcds.2022063
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Global well-posedness of 2D incompressible Magnetohydrodynamic equations with horizontal dissipation

1. 

School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, Beijing 100875, China

2. 

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

*Corresponding author: Xiaoxiao Suo

Received  June 2021 Revised  March 2022 Early access May 2022

This paper focuses on two-dimensional incompressible non-resistive MHD equations with only horizontal dissipation in $ \mathbb{T}\times\mathbb{R} $. Invoking three Poincaré-type inequalities about the horizontal derivative, we study the global well-posedness of the system near a background magnetic via the structure of the perturbation MHD system and the symmetry condition imposed on the initial data. By a precise time-weighted energy estimate, we also establish the global well-posedness of the system with only horizontal magnetic damping. Here we overcome the difficulties brought by the absence of magnetic diffusion and the appearance of the boundary. We note that the stability of MHD equations with one-directional dissipation in $ \mathbb{R}^2 $ or a bounded domain appears to be unknown.

Citation: Xiaoxiao Suo, Quansen Jiu. Global well-posedness of 2D incompressible Magnetohydrodynamic equations with horizontal dissipation. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022063
References:
[1]

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

[2]

H. Alfv'en, Existence of electromagnetic-hydrodynamic waves, Nature, 150 (1942), 405-406. 

[3]

C. BardosC. Sulem and P. L. Sulem, Longtime dynamics of a conductive fluid in the presence of a strong magnetic field, Trans. Am. Math. Soc., 305 (1988), 175-191.  doi: 10.1090/S0002-9947-1988-0920153-5.

[4] D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511599965.
[5] H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, New York, 1970. 
[6]

X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041.

[7]

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

[8]

C. CaoD. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differential Equations, 254 (2013), 2661-2681.  doi: 10.1016/j.jde.2013.01.002.

[9]

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.

[10]

C. CaoJ. Wu and B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46 (2014), 588-602.  doi: 10.1137/130937718.

[11]

J. CheminD. McCormickJ. Robinson and J. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 286 (2016), 1-31.  doi: 10.1016/j.aim.2015.09.004.

[12]

Q. Chen and X. Ren, Global well-posedness for the 2D MHD non-resistive MHD equations in two kinds of periodic domains, Z. Angew. Math. Phys., 70 (2019), Paper No. 18, 13 pp. doi: 10.1007/s00033-018-1066-y.

[13]

B. 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.

[14]

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

[15]

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

[16]

L. 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.

[17]

X. Hu and F. Lin, Global existence for two dimensional incompressible magnetohydrodynamic flows with zero magnetic diffusivity, 2014, arXiv: 1405.0082v1.

[18]

Q. JiuX. SuoJ. Wu and H. Yu, Unique weak solutions of the non-resistive magnetohydrodynamic equations with fractional dissipation, Commun. Math. Sci., 18 (2020), 987-1022.  doi: 10.4310/CMS.2020.v18.n4.a5.

[19]

H. LinR. JiJ. 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.  doi: 10.1016/j.jfa.2020.108519.

[20]

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.

[21]

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.

[22]

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.

[23]

X. RenZ. Xiang and Z. Zhang, Global well-posedness for the 2D MHD equations without magnetic diffusion in a strip domain, Nonlinearity, 29 (2016), 1257-1291.  doi: 10.1088/0951-7715/29/4/1257.

[24]

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.

[25]

Z. Tan and Y. Wang, Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems, SIAM J. Math. Anal., 50 (2018), 1432-1470.  doi: 10.1137/16M1088156.

[26]

D. Wei and Z. Zhang, Global well-posedness for the 2-D MHD equations with magnetic diffusion, Commun. Math. Res., 36 (2020), 377-389.  doi: 10.4208/cmr.2020-0022.

[27]

J. Wu, The 2D magnetohydrodynamic equations with partial or fractional dissipation, Lectures on the analysis of nonlinear partial differential equations, Morningside Lect. Math., Int. Press, Somerville, MA, 5 (2018), 283–332.

[28]

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.

[29]

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), Paper No. 107466, 26 pp. doi: 10.1016/j.aim.2020.107466.

[30]

L. Xu and P. Zhang, Global small solutions to three-dimensional incompressible magnetohydrodynamical system, SIAM J. Math. Anal., 47 (2015), 26-65.  doi: 10.1137/14095515X.

[31]

Y. Zhou and Y. Zhu, Global classical solutions of 2D MHD system with only magnetic diffusion on periodic domain, J. Math. Phys., 59 (2018), 081505, 12 pp. doi: 10.1063/1.5018641.

show all references

References:
[1]

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

[2]

H. Alfv'en, Existence of electromagnetic-hydrodynamic waves, Nature, 150 (1942), 405-406. 

[3]

C. BardosC. Sulem and P. L. Sulem, Longtime dynamics of a conductive fluid in the presence of a strong magnetic field, Trans. Am. Math. Soc., 305 (1988), 175-191.  doi: 10.1090/S0002-9947-1988-0920153-5.

[4] D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511599965.
[5] H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, New York, 1970. 
[6]

X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809.  doi: 10.1016/j.jmaa.2008.01.041.

[7]

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

[8]

C. CaoD. Regmi and J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differential Equations, 254 (2013), 2661-2681.  doi: 10.1016/j.jde.2013.01.002.

[9]

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.

[10]

C. CaoJ. Wu and B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46 (2014), 588-602.  doi: 10.1137/130937718.

[11]

J. CheminD. McCormickJ. Robinson and J. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 286 (2016), 1-31.  doi: 10.1016/j.aim.2015.09.004.

[12]

Q. Chen and X. Ren, Global well-posedness for the 2D MHD non-resistive MHD equations in two kinds of periodic domains, Z. Angew. Math. Phys., 70 (2019), Paper No. 18, 13 pp. doi: 10.1007/s00033-018-1066-y.

[13]

B. 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.

[14]

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

[15]

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

[16]

L. 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.

[17]

X. Hu and F. Lin, Global existence for two dimensional incompressible magnetohydrodynamic flows with zero magnetic diffusivity, 2014, arXiv: 1405.0082v1.

[18]

Q. JiuX. SuoJ. Wu and H. Yu, Unique weak solutions of the non-resistive magnetohydrodynamic equations with fractional dissipation, Commun. Math. Sci., 18 (2020), 987-1022.  doi: 10.4310/CMS.2020.v18.n4.a5.

[19]

H. LinR. JiJ. 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.  doi: 10.1016/j.jfa.2020.108519.

[20]

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.

[21]

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.

[22]

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.

[23]

X. RenZ. Xiang and Z. Zhang, Global well-posedness for the 2D MHD equations without magnetic diffusion in a strip domain, Nonlinearity, 29 (2016), 1257-1291.  doi: 10.1088/0951-7715/29/4/1257.

[24]

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.

[25]

Z. Tan and Y. Wang, Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems, SIAM J. Math. Anal., 50 (2018), 1432-1470.  doi: 10.1137/16M1088156.

[26]

D. Wei and Z. Zhang, Global well-posedness for the 2-D MHD equations with magnetic diffusion, Commun. Math. Res., 36 (2020), 377-389.  doi: 10.4208/cmr.2020-0022.

[27]

J. Wu, The 2D magnetohydrodynamic equations with partial or fractional dissipation, Lectures on the analysis of nonlinear partial differential equations, Morningside Lect. Math., Int. Press, Somerville, MA, 5 (2018), 283–332.

[28]

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.

[29]

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), Paper No. 107466, 26 pp. doi: 10.1016/j.aim.2020.107466.

[30]

L. Xu and P. Zhang, Global small solutions to three-dimensional incompressible magnetohydrodynamical system, SIAM J. Math. Anal., 47 (2015), 26-65.  doi: 10.1137/14095515X.

[31]

Y. Zhou and Y. Zhu, Global classical solutions of 2D MHD system with only magnetic diffusion on periodic domain, J. Math. Phys., 59 (2018), 081505, 12 pp. doi: 10.1063/1.5018641.

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