The stability problem on the magnetohydrodynamics (MHD) equations with partial or no dissipation is not well-understood. This paper focuses on the 3D incompressible MHD equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the stability of perturbations near the steady solution given by a background magnetic field in periodic domain. The new stability result presented here is among few stability conclusions currently available for ideal or partially dissipated MHD equations.
Citation: Ruihong Ji, Ling Tian. Stability of the 3D incompressible MHD equations with horizontal dissipation in periodic domain[J]. AIMS Mathematics, 2021, 6(11): 11837-11849. doi: 10.3934/math.2021687
The stability problem on the magnetohydrodynamics (MHD) equations with partial or no dissipation is not well-understood. This paper focuses on the 3D incompressible MHD equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the stability of perturbations near the steady solution given by a background magnetic field in periodic domain. The new stability result presented here is among few stability conclusions currently available for ideal or partially dissipated MHD equations.
| [1] | H. Alfvén, Existence of electromagnetic-hydrodynamic vaves, Nature, 150 (1942), 405–406. |
| [2] | D. Biskamp, Nonlinear magnetohydrodynamics, Cambridge, New York: Cambridge University Press, 1993. |
| [3] | P. A. Davidson, An introduction to magnetohydrodynamics, Cambridge, England: Cambridge University Press, 2001. |
| [4] |
J. Bourgain, D. Li, Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math., 201 (2015), 97–157. doi: 10.1007/s00222-014-0548-6
|
| [5] |
W. Deng, P. Zhang, Large time behavior of solutions to 3D MHD system with initial data near equilibrium, Arch. Rational Mech. Anal., 230 (2018), 1017–1102. doi: 10.1007/s00205-018-1265-x
|
| [6] |
X. P. Hu, D. H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Rational Mech. Anal., 197 (2010), 203–238. doi: 10.1007/s00205-010-0295-9
|
| [7] |
F. H. Lin, L. Xu, P. Zhang, Global small solutions to 2D incompressible MHD system, J. Differ. Equations, 259 (2015), 5440–5485. doi: 10.1016/j.jde.2015.06.034
|
| [8] |
J. H. Wu, Y. Zhu, Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equitibrium, Adv. Math., 377 (2021), 107466. doi: 10.1016/j.aim.2020.107466
|
| [9] |
C. S. Cao, J. H. Wu, B. Q. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46 (2014), 588–602. doi: 10.1137/130937718
|
| [10] |
Q. S. Jiu, J. F. Zhao, Global regularity of 2D generalized MHD equations with magnetic diffusion, Z. Angew. Math. Phys., 66 (2015), 677–687. doi: 10.1007/s00033-014-0415-8
|
| [11] | A. J. Majda, A. L. Bertozzt, Vorticity and incompressible flow, Cambridge University Press, 2002. |
| [12] | K. Yamazaki, On the global well-posedness of N-dimensional generalized MHD system in anisotropic spaces, Adv. Differ. Equations, 19 (2014), 201–224. |
| [13] |
C. S. Cao, Y. H. Guo, E. S. Titi, Global strong solutions for the three-dimensional Hasegawa-Mima model with partial dissipation, SIAM J. Math. Phys., 59 (2018), 071503. doi: 10.1063/1.5022099
|
| [14] | T. Tao, Nonlinear dispersive equations: Local and global analysis, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 2006. |
| [15] | H. Bahouri, J. Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differential equations, Springer, 2011. |
Ruihong Ji, Ling Tian. Stability of the 3D incompressible MHD equations with horizontal dissipation in periodic domain[J]. AIMS Mathematics, 2021, 6(11): 11837-11849. doi: 10.3934/math.2021687
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