Stability of the 2D MHD system with fractional vertical dissipation and magnetic damping

Qunyi Bie; Qiaoqi Yang; Xianbao Yuan; Qunyi Bie; Qiaoqi Yang; Xianbao Yuan

doi:10.3934/cam.2026024

Communications in Analysis and Mechanics

2026, Volume 18, Issue 3: 593-609. doi: 10.3934/cam.2026024

Previous Article Next Article

Research article

Stability of the 2D MHD system with fractional vertical dissipation and magnetic damping

1.
College of Mathematics and Physics & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, P. R. China
2.
College of Mathematics and Physics, China Three Gorges University, Yichang 443002, P. R. China

Received: 27 December 2025 Revised: 21 February 2026 Accepted: 29 April 2026 Published: 23 July 2026
35B35, 35B40, 76E25

Due to the importance in physical applications and mathematics, the stability problems of partially dissipative hydrodynamic system have attracted considerable attention in recent years and achieved significant progress. However, research on the stability analysis for the partial fractional dissipative system is relatively limited. The present study demonstrates the stability of a two-dimensional (2D) incompressible magnetohydrodynamic (MHD) system with fractional vertical dissipation and magnetic damping within the Sobolev space $ H^2(\mathbb{R}^2) $. In view of the presence of fractional order operators, classical tools and techniques designed for the fully dissipated MHD system no longer work; therefore, we develop new anisotropic inequalities that involve fractional derivatives to address this issue. Our result extends and improves the work in [Nonlinearity, 2021, 34(4): 2527].
- MHD system,
- partial and fractional dissipation,
- magnetic damping,
- stability
Citation: Qunyi Bie, Qiaoqi Yang, Xianbao Yuan. Stability of the 2D MHD system with fractional vertical dissipation and magnetic damping[J]. Communications in Analysis and Mechanics, 2026, 18(3): 593-609. doi: 10.3934/cam.2026024

Related Papers:

Abstract

Due to the importance in physical applications and mathematics, the stability problems of partially dissipative hydrodynamic system have attracted considerable attention in recent years and achieved significant progress. However, research on the stability analysis for the partial fractional dissipative system is relatively limited. The present study demonstrates the stability of a two-dimensional (2D) incompressible magnetohydrodynamic (MHD) system with fractional vertical dissipation and magnetic damping within the Sobolev space $ H^2(\mathbb{R}^2) $. In view of the presence of fractional order operators, classical tools and techniques designed for the fully dissipated MHD system no longer work; therefore, we develop new anisotropic inequalities that involve fractional derivatives to address this issue. Our result extends and improves the work in [Nonlinearity, 2021, 34(4): 2527].

References

[1]	D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1997. https://doi.org/10.1016/0021-9169(95)90020-9
[2]	P. A. Davidson, Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, 2017.
[3]	E. Priest, T. Forbes, Magnetic Reconnection, MHD Theory and Applications, Cambridge University Press, Cambridge, 2000.
[4]	D. Chen, F. Jian, Stabilization effects of magnetic field on a 2D anisotropic MHD system with partial dissipation, Acta Appl. Math., 187 (2023), 9. https://doi.org/10.1007/s10440-023-00602-5 doi: 10.1007/s10440-023-00602-5
[5]	Y. Guo, Y. Jia, B. Dong, Global stability solution of the 2D MHD equations with mixed partial dissipation, Discrete Contin. Dyn. Syst. Ser. A, 42 (2022), 885–902. https://doi.org/10.3934/dcds.2021141 doi: 10.3934/dcds.2021141
[6]	H. Lin, T. Chen, R. Bai, H. Zhang, Stability for a system of 2D incompressible anisotropic magnetohydrodynamic equations, Z. Angew. Math. Phys., 74 (2023), 53. https://doi.org/10.1007/s00033-023-01944-8 doi: 10.1007/s00033-023-01944-8
[7]	H. Qiu, M. Chen, Stability for the two-dimensional anisotropic magnetohydrodynamic equations, J. Math. Phys., 66 (2025), 021506. https://doi.org/10.1063/5.0223139 doi: 10.1063/5.0223139
[8]	W. Yang, C. Fang, Stability and optimal decay for the 3D anisotropic magnetohydrodynamic equations, Stud. Appl. Math., 153 (2024), e12731. https://doi.org/10.1111/sapm.12731 doi: 10.1111/sapm.12731
[9]	M. Jin, Q. Jiu, Y. Xie, Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion, Z. Angew. Math. Phys., 75 (2024), 73. https://doi.org/10.1007/s00033-024-02215-w doi: 10.1007/s00033-024-02215-w
[10]	X. Suo, Q. Jiu, Global well-posedness of 2D incompressible Magnetohydrodynamic equations with horizontal dissipation, Discrete Contin. Dyn. Syst. Ser. A, 42 (2022), 4523–4553. https://doi.org/10.3934/dcds.2022063 doi: 10.3934/dcds.2022063
[11]	X. Ren, J. Wu, Z. Xiang, Z. Zhang, Global existence and decay of smooth solution for the 2D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503–541. https://doi.org/10.1016/j.jfa.2014.04.020 doi: 10.1016/j.jfa.2014.04.020
[12]	W. Feng, F. Hafeez, J. Wu, Influence of a background magnetic field on a 2D magnetohydrodynamic flow, Nonlinearity, 34 (2021), 2527. https://doi.org/10.1088/1361-6544/abb928 doi: 10.1088/1361-6544/abb928
[13]	N. Boardman, H. Lin, J. Wu, Stabilization of a background magnetic field on a 2 dimensional magnetohydrodynamic flow, SIAM J. Math. Anal., 52 (2020), 5001–5035. https://doi.org/10.1137/20m1324776 doi: 10.1137/20m1324776
[14]	Z. Chen, W. Feng, D. Qin, Stability for the 2D MHD equations with horizontal dissipation, Asymptot. Anal., 129 (2022), 361–377. https://doi.org/10.3233/asy-211733 doi: 10.3233/asy-211733
[15]	R. Ji, H. Lin, J. Wu, L. Yan, Stability for a system of the 2D magnetohydrodynamic equations with partial dissipation, Appl. Math. Lett., 94 (2019), 244–249. https://doi.org/10.1016/j.aml.2019.03.013 doi: 10.1016/j.aml.2019.03.013
[16]	S. Lai, J. Wu, J. Zhang, Stabilizing phenomenon for 2D anisotropic magnetohydrodynamic system near a background magnetic field, SIAM J. Math. Anal., 53 (2021), 6073–6093. https://doi.org/10.1137/21m139791x doi: 10.1137/21m139791x
[17]	H. Lin, R. Ji, J. Wu, 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. https://doi.org/10.1016/j.jfa.2020.108519 doi: 10.1016/j.jfa.2020.108519
[18]	R. Ji, L. Tian, Stability of the 3D incompressible MHD equations with horizontal dissipation in periodic domain, AIMS Math., 6 (2021), 11837–11849. https://doi.org/10.3934/math.2021687 doi: 10.3934/math.2021687
[19]	H. Lin, J. Wu, Y. Zhu, Global solutions to 3D incompressible MHD system with dissipation in only one direction, SIAM J. Math. Anal., 55 (2023), 4570–4598. https://doi.org/10.1137/22m1471274 doi: 10.1137/22m1471274
[20]	H. Shang, J. Wu, Q. Zhang, Stability and optimal decay for the 3D magnetohydrodynamic equations with only horizontal dissipation, J. Evol. Equ., 24 (2024), 12. https://doi.org/10.1007/s00028-023-00940-9 doi: 10.1007/s00028-023-00940-9
[21]	J. Wu, Y. Zhu, Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium, Adv. Math., 377 (2021), 107466. https://doi.org/10.1016/j.aim.2020.107466 doi: 10.1016/j.aim.2020.107466
[22]	W. Yang, M. Ma, Stabilizing effect of the magnetic field and decay estimates for the 3D MHD equations with only one direction dissipation, J. Math. Anal. Appl., 543 (2025), 128894. https://doi.org/10.1016/j.jmaa.2024.128894 doi: 10.1016/j.jmaa.2024.128894
[23]	X. Deng, Y. Xiao, A. Zang, Stability and inviscid limit of the 3D anisotropic MHD system near a background magnetic field with mixed fractional partial dissipation, arXiv: 2308.07547.
[24]	W. Feng, W. Wang, J. Wu, Stability for a system of the 2D incompressible MHD equations with fractional dissipation, J. Math. Fluid Mech., 26 (2024), 57. https://doi.org/10.1007/s00021-024-00892-1 doi: 10.1007/s00021-024-00892-1
[25]	R. Ji, L. Jiang, W. Luo, Stability of the 3D MHD equations without vertical dissipation near an equilibrium, AIMS Math., 8 (2023), 12143–12167. https://doi.org/10.3934/math.2023612 doi: 10.3934/math.2023612
[26]	J. Li, H. Wang, D. Zheng, Stability and sharp decay for 3D incompressible MHD system with fractional horizontal dissipation and magnetic diffusion, Z. Angew. Math. Phys., 74 (2023), 44. https://doi.org/10.1007/s00033-023-01939-5 doi: 10.1007/s00033-023-01939-5
[27]	Y. Zhong, Stability of 2D inviscid MHD equations with only fractional magnetic diffusion in the horizontal direction, Appl. Math. Lett., 163 (2025), 109446. https://doi.org/10.1016/j.aml.2024.109446 doi: 10.1016/j.aml.2024.109446
[28]	H. Brezis, P. Mironescu, Where Sobolev interacts with Gagliardo-Nirenberg, J. Funct. Anal., 277 (2019), 2839–2864. https://doi.org/10.1016/j.jfa.2019.02.019 doi: 10.1016/j.jfa.2019.02.019

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)