Global stability solution of the 2D MHD equations with mixed partial dissipation

Yana Guo; Yan Jia; Bo-Qing Dong

doi:10.3934/dcds.2021141

Article Contents

2022, Volume 42, Issue 2: 885-902. Doi: 10.3934/dcds.2021141

This issue Previous Article Pure strictly uniform models of non-ergodic measure automorphisms Next Article The nonlocal-interaction equation near attracting manifolds

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
^* Corresponding author: Bo-Qing Dong

Received: March 31, 2021

Revised: June 30, 2021

Early access: September 2021

Published: February 2022

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35A05, 35Q35; Secondary: 76D03.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[2]	H. Alfvén, Existence of electromagnetic-hydrodynamic waves, Nature, 150 (1942), 405-406.
[3]	D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511599965.
[4]	N. Boardman, H. 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.
[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.
[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.
[7]	P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, England, 2001. doi: 10.1017/CBO9780511626333.
[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.
[9]	B.-Q. Dong, Y. Jia, J. 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.
[10]	B.-Q. Dong, J. 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.
[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.
[12]	G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch, Ration. Mech. Anal., 46 (1972), 241-279. doi: 10.1007/BF00250512.
[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.
[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.
[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.
[16]	C. Li, J. 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.
[17]	F. Lin, L. 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.
[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.
[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.
[20]	R. Pan, Y. 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.
[21]	X. Ren, J. Wu, Z. 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.
[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.
[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.
[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.
[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.
[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.
[27]	J. Wu, Y. 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.
[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.
[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.