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

Abstract

This paper focuses on two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with only fractional horizontal dissipation Λ^2α₁ u and Λ^2β₁ b in the spatial domain Ω= T × R (where T = [0, 1] is a periodic box, and 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 fieldA = (A₁, A₂) with A₁, A₂ ∈ R and established nonlinear stability in the Sobolev space H³(Ω). In this paper, we consider the stability problem in the lower regularity space H²(Ω). 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 α ∈ (0, 1] and β ∈ [0, 1] inH²(Ω). In addition, we prove that the oscillation part (e u, e b) of the solution in H¹(Ω) decays to zero exponentially in time.

Keywords

magnetohydrodynamic equations / fractional horizontal dissipation / anisotropic interpolation inequalities / global stability / exponential decay

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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[14] J. Wu, Generalized MHD equations, J. Differ. Equ., 195 (2003), 284-312. https://doi.org/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

[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

[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

[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

[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

[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

[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

[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

[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

[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

[27] A. J. Majda, A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. https://doi.org/10.1017/cbo9780511613203