Stability and exponential decay for the 2D incompressible MHD system with fractional horizontal dissipation

Mingxi Zhang; Ling Gao; Wenjing Yang; Mingxi Zhang; Ling Gao; Wenjing Yang

doi:10.3934/cam.2026017

Communications in Analysis and Mechanics

2026, Volume 18, Issue 2: 419-443. doi: 10.3934/cam.2026017

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Research article

Stability and exponential decay for the 2D incompressible MHD system with fractional horizontal dissipation

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

Received: 02 September 2025 Revised: 20 January 2026 Accepted: 18 March 2026 Published: 13 May 2026
35B35, 35B40, 76E25

This paper focuses on two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with only fractional horizontal dissipation $ \Lambda_{1}^{2\alpha}u $ and $ \Lambda_{1}^{2\beta}b $ in the spatial domain $ \Omega = \mathbb{T\times R} $ (where $ \mathbb{T} = [0, 1] $ is a periodic box, and $ \mathbb{R} $ is the whole line). For this system, Feng, Wang, and Wu assessed the global stability of perturbations near the steady solution given by a background magnetic field $ A = (A_{1}, A_{2}) $ with $ A_{1}, A_{2}\in\mathbb{R} $ and established nonlinear stability in the Sobolev space $ H^{3}(\Omega) $. In this paper, we consider the stability problem in the lower regularity space $ H^{2}(\Omega) $. By applying the Hölder's inequality with various anisotropic fractional exponents and some special anisotropic interpolation inequalities, we obtain the global stability of the system when $ \alpha\in\left(0, 1 \right] $ and $ \beta\in\left[ 0, 1 \right] $ in $ H^{2}(\Omega) $. In addition, we prove that the oscillation part $ (\widetilde{u}, \widetilde{b}) $ of the solution in $ H^{1}(\Omega) $ decays to zero exponentially in time.
- magnetohydrodynamic equations,
- fractional horizontal dissipation,
- anisotropic interpolation inequalities,
- global stability,
- exponential decay
Citation: Mingxi Zhang, Ling Gao, Wenjing Yang. Stability and exponential decay for the 2D incompressible MHD system with fractional horizontal dissipation[J]. Communications in Analysis and Mechanics, 2026, 18(2): 419-443. doi: 10.3934/cam.2026017

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Abstract

This paper focuses on two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with only fractional horizontal dissipation $ \Lambda_{1}^{2\alpha}u $ and $ \Lambda_{1}^{2\beta}b $ in the spatial domain $ \Omega = \mathbb{T\times R} $ (where $ \mathbb{T} = [0, 1] $ is a periodic box, and $ \mathbb{R} $ is the whole line). For this system, Feng, Wang, and Wu assessed the global stability of perturbations near the steady solution given by a background magnetic field $ A = (A_{1}, A_{2}) $ with $ A_{1}, A_{2}\in\mathbb{R} $ and established nonlinear stability in the Sobolev space $ H^{3}(\Omega) $. In this paper, we consider the stability problem in the lower regularity space $ H^{2}(\Omega) $. By applying the Hölder's inequality with various anisotropic fractional exponents and some special anisotropic interpolation inequalities, we obtain the global stability of the system when $ \alpha\in\left(0, 1 \right] $ and $ \beta\in\left[ 0, 1 \right] $ in $ H^{2}(\Omega) $. In addition, we prove that the oscillation part $ (\widetilde{u}, \widetilde{b}) $ of the solution in $ H^{1}(\Omega) $ decays to zero exponentially in time.

References

[1]	P. A. Davidson, Introduction to magnetohydrodynamics, Cambridge University Press, Cambridge, 2017.
[2]	J. Philidet, C. Gissinger, F. Lignières, L. Petitdemange, Magnetohydrodynamics of stably stratified regions in planets and stars, Geophys. Astrophys. Fluid Dyn., 114 (2020), 336–355. https://doi.org/10.1080/03091929.2019.1670827 doi: 10.1080/03091929.2019.1670827
[3]	D. Riashchikov, E. Scoptsova, D. Zavershinskii, Diagnostics of solar corona heating parameters using the observed gravitational stratification of the medium, Solar Phys., 299 (2024), 136. https://doi.org/10.1007/s11207-024-02383-y doi: 10.1007/s11207-024-02383-y
[4]	S. Abe, S. Thurner, Anomalous diffusion in view of Einstein's 1905 theory of Brownian motion, Phys. A: Stat. Mech. Appl., 356 (2005), 403–407. https://doi.org/10.1016/j.physa.2005.03.035 doi: 10.1016/j.physa.2005.03.035
[5]	A. E. Gill, Atmosphere-ocean dynamics, Elsevier, 2016. https://doi.org/10.1016/0141-1187(83)90039-1
[6]	B. Dong, Y. Jia, J. Li, 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. https://doi.org/10.1007/s00021-018-0376-3 doi: 10.1007/s00021-018-0376-3
[7]	B. Dong, J. Li, J. Wu, Global regularity for the 2D MHD equations with partial hyper-resistivity, Int. Math. Res. Not., 14 (2019), 4261–4280. https://doi.org/10.1093/imrn/rnx240 doi: 10.1093/imrn/rnx240
[8]	Q. Jiu, J. Zhao, A remark on global regularity of 2D generalized magnetohydrodynamic equations, J. Math. Anal. Appl., 412 (2014), 478–484. https://doi.org/10.1016/j.jmaa.2013.10.074 doi: 10.1016/j.jmaa.2013.10.074
[9]	Q. Jiu, J. Zhao, Global regularity of 2D generalized MHD equations with magnetic diffusion, Z. Angew. Math. Phys., 66 (2015), 677–687. https://doi.org/10.1007/s00033-014-0415-8 doi: 10.1007/s00033-014-0415-8
[10]	C. V. Tran, X. Yu, Z. Zhai, On global regularity of 2D generalized magnetohydrodynamic equations, J. Differ. Equ., 254 (2013), 4194–4216. https://doi.org/10.1016/j.jde.2013.02.016 doi: 10.1016/j.jde.2013.02.016
[11]	C. Cao, J. Wu, B. Yuan, The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion, SIAM J. Math. Anal., 46 (2014), 588–602. https://doi.org/10.1137/130937718 doi: 10.1137/130937718
[12]	J. Fan, H. Malaikah, S. Monaquel, G. Nakamura, Y. Zhou, Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math., 175 (2014), 127–131. https://doi.org/10.1007/s00605-014-0652-0 doi: 10.1007/s00605-014-0652-0
[13]	X. Ren, J. Wu, Z. Xiang, 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. https://doi.org/10.1016/j.jfa.2014.04.020 doi: 10.1016/j.jfa.2014.04.020
[14]	J. Wu, Generalized MHD equations, J. Differ. Equ., 195 (2003), 284–312. https://doi.org/10.1016/j.jde.2003.07.007 doi: 10.1016/j.jde.2003.07.007
[15]	J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295–305. https://doi.org/10.1007/s00021-009-0017-y doi: 10.1007/s00021-009-0017-y
[16]	J. Wu, The 2D magnetohydrodynamic equations with partial or fractional dissipation, Lect. Anal. Nonlinear Partial Differ. Equ., 5(2018), 283–332.
[17]	K. Yamazaki, On the global regularity of two-dimensional generalized magnetohydrodynamics system, J. Math. Anal. Appl., 416 (2014), 99–111. https://doi.org/10.1016/j.jmaa.2014.02.027 doi: 10.1016/j.jmaa.2014.02.027
[18]	Z. Ye, X. Xu, Global regularity of the two-dimensional incompressible generalized magnetohydrodynamics system, Nonlinear Anal., Theory Methods Appl., 100 (2014), 86–96. https://doi.org/10.1016/j.na.2014.01.012 doi: 10.1016/j.na.2014.01.012
[19]	B. Yuan, J. Zhao, Global regularity of 2D almost resistive MHD equations, Nonlinear Anal. Real World Appl., 41 (2018), 53–65. https://doi.org/10.1016/j.nonrwa.2017.10.006 doi: 10.1016/j.nonrwa.2017.10.006
[20]	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
[21]	W. Feng, F. Hafeez, J. Wu, D. Regmi, Stability and exponential decay for magnetohydrodynamic equations, Proc. R. Soc. Edinb. A, 153 (2023), 853–880. https://doi.org/10.1017/prm.2022.23 doi: 10.1017/prm.2022.23
[22]	W. Feng, W. Wang, J. Wu, Nonlinear stability for the 2D incompressible MHD system with fractional dissipation in the horizontal direction, J. Evol. Equ., 23 (2023), 32. https://doi.org/10.1007/s00028-023-00886-y doi: 10.1007/s00028-023-00886-y
[23]	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
[24]	F. Lin, L. Xu, P. Zhang, Global small solutions of 2-D incompressible MHD system, J. Differ. Equ., 259 (2015), 5440–5485. https://doi.org/10.1016/j.jde.2015.06.034 doi: 10.1016/j.jde.2015.06.034
[25]	T. Tao, Nonlinear dispersive equations: local and global analysis, American Mathematical Society, 2006.
[26]	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
[27]	A. J. Majda, A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. https://doi.org/10.1017/cbo9780511613203

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