Stability of the 3D MHD equations without vertical dissipation near an equilibrium

Ruihong Ji; Liya Jiang; Wen Luo; Ruihong Ji; Liya Jiang; Wen Luo

doi:10.3934/math.2023612

AIMS Mathematics

2023, Volume 8, Issue 5: 12143-12167. doi: 10.3934/math.2023612

Previous Article Next Article

Research article

Stability of the 3D MHD equations without vertical dissipation near an equilibrium

College of Mathematics and Physics, Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, China

Received: 02 November 2022 Revised: 06 March 2023 Accepted: 12 March 2023 Published: 22 March 2023
MSC : 35A05, 35Q35, 76D03

Important progress has been made on the standard Laplacian case with mixed partial dissipation and diffusion. The stability problem of the 3D incompressible magnetohydrodynamic (MHD) equations without vertical dissipation but with the fractional velocity dissipation $ (-\Delta)^\alpha u $ and magnetic diffusion $ (-\Delta)^\beta b $ is unfortunately not often well understood for many ranges of fractional powers. This paper discovers that there are new phenomena with the case $ \alpha, \beta \leq 1 $. We establish that, if an initial datum ($ u_0, b_0 $) in the Sobolev space $ H^3(\mathbb{R}^3) $ is close enough to the equilibrium state, and we replace the terms $ (-\Delta)^\alpha u $ and $ (-\Delta)^\beta b $ by $ (-\Delta_h)^\alpha u $ and $ (-\Delta_h)^\beta b $, respectively, the resulting equations with $ \alpha, \beta \in(\frac{1}{2}, 1] $ then always lead to a steady solution, where $ \Delta_h = \partial_{x_1}^2+\partial_{x_2}^2 $.
- magnetohydrodynamic equations,
- background magnetic field,
- partial dissipation,
- stability
Citation: Ruihong Ji, Liya Jiang, Wen Luo. Stability of the 3D MHD equations without vertical dissipation near an equilibrium[J]. AIMS Mathematics, 2023, 8(5): 12143-12167. doi: 10.3934/math.2023612

Related Papers:

Abstract

Important progress has been made on the standard Laplacian case with mixed partial dissipation and diffusion. The stability problem of the 3D incompressible magnetohydrodynamic (MHD) equations without vertical dissipation but with the fractional velocity dissipation $ (-\Delta)^\alpha u $ and magnetic diffusion $ (-\Delta)^\beta b $ is unfortunately not often well understood for many ranges of fractional powers. This paper discovers that there are new phenomena with the case $ \alpha, \beta \leq 1 $. We establish that, if an initial datum ($ u_0, b_0 $) in the Sobolev space $ H^3(\mathbb{R}^3) $ is close enough to the equilibrium state, and we replace the terms $ (-\Delta)^\alpha u $ and $ (-\Delta)^\beta b $ by $ (-\Delta_h)^\alpha u $ and $ (-\Delta_h)^\beta b $, respectively, the resulting equations with $ \alpha, \beta \in(\frac{1}{2}, 1] $ then always lead to a steady solution, where $ \Delta_h = \partial_{x_1}^2+\partial_{x_2}^2 $.

References

[1]	P. A. Davidson, An introduction to magnetohydrodynamics, Cambridge: Cambridge University Press, 2001. https://doi.org/10.1017/CBO9780511626333
[2]	D. Biskamp, Nonlinear magnetohydrodynamics, Cambridge, New York: Cambridge University Press, 1993. https://doi.org/10.1017/CBO9780511599965
[3]	H. Alfvén, Existence of electromagnetic-hydrodynamic vaves, Nature, 150 (1942), 405–406. https://doi.org/10.1038/150405d0 doi: 10.1038/150405d0
[4]	B. Dong, Y. Jia, J. Li, J. Wu, Global regularity for the 2D magnetohydrodynamics equations with horizontal dissipation and horizontal 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
[5]	Y. Dai, R. Ji, J. Wu, Unique weak solutions of the magnetohydrodynamic equations with fractional dissipation, Z. Angew. Math. Mech., 100 (2020), e201900290. https://doi.org/10.1002/zamm.201900290 doi: 10.1002/zamm.201900290
[6]	Y. Dai, Z. Tan, J. Wu, A class of global large solutions to the magnetohydrodynamic equations with fractional dissipation, Z. Angew. Math. Phys., 70 (2019), 153. https://doi.org/10.1007/s00033-019-1193-0 doi: 10.1007/s00033-019-1193-0
[7]	M. Sermange, R. Temam, Some mathematical questions related to the MHD equations, Commun. Pure Appl. Math., 36 (1983), 635–664. https://doi.org/10.1002/cpa.3160360506 doi: 10.1002/cpa.3160360506
[8]	J. Wu, Generalized MHD equations, J. Differ. Equations, 195 (2003), 284–312. https://doi.org/10.1016/j.jde.2003.07.007 doi: 10.1016/j.jde.2003.07.007
[9]	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
[10]	J. Wu, The 2D magnetohydrodynamic equations with partial or fractional dissipation, In: Lectures on the analysis of nonlinear partial differential equations: part 5, International Press of Boston, Inc., 2018,283–332.
[11]	K. Yamazaki, On the global well-posedness of N-dimensional generalized MHD system in anisotropic spaces, Adv. Differential Equations, 19 (2014), 201–224. https://doi.org/10.57262/ade/1391109084 doi: 10.57262/ade/1391109084
[12]	W. Yang, Q. Jiu, J. Wu, The 3D incompressible magnetohydrodynamic equations with fractional partial dissipation, J. Differ. Equations, 266 (2019), 630–652. https://doi.org/10.1016/j.jde.2018.07.046 doi: 10.1016/j.jde.2018.07.046
[13]	S. Abe, S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403–407. https://doi.org/10.1016/j.physa.2005.03.035 doi: 10.1016/j.physa.2005.03.035
[14]	M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Commun. Pure Appl. Math., 62 (2009), 198–214. https://doi.org/10.1002/cpa.20253 doi: 10.1002/cpa.20253
[15]	U. Frisch, S. Kurien, R. Pandit, W. Pauls, S. Ray, A. Wirth, et al., Hyperviscosity, Galerkin truncation, and bottlenecks in turbulence, Phys. Rev. Lett., 101 (2008), 144501. https://doi.org/10.1103/PhysRevLett.101.144501 doi: 10.1103/PhysRevLett.101.144501
[16]	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
[17]	K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity, Appl. Math. Lett., 29 (2014), 46–51. https://doi.org/10.1016/j.aml.2013.10.014 doi: 10.1016/j.aml.2013.10.014
[18]	K. Yamazaki, Global regularity of logarithmically supercritical MHD system with improved logarithmic powers, Dynam. Part. Differ. Eq., 15 (2018), 147–173. https://doi.org/10.4310/DPDE.2018.v15.n2.a4 doi: 10.4310/DPDE.2018.v15.n2.a4
[19]	Z. Ye, X. Xu, Global regularity of the two-dimensional incompressible generalized magnetohydrodynamics system, Nonlinear Anal., 100 (2014), 86–96. https://doi.org/10.1016/j.na.2014.01.012 doi: 10.1016/j.na.2014.01.012
[20]	Z. Ye, Remark on the global regularity of 2D MHD equations with almost Laplacian magnetic diffusion, J. Evol. Equ., 18 (2018), 821–844. https://doi.org/10.1007/s00028-017-0421-3 doi: 10.1007/s00028-017-0421-3
[21]	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
[22]	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
[23]	C. Tran, X. Yu, Z. Zhai, On global regularity of 2D generalized magnetohydrodynamic equations, J. Differ. Equations, 254 (2013), 4194–4216. https://doi.org/10.1016/j.jde.2013.02.016 doi: 10.1016/j.jde.2013.02.016
[24]	J. Wu, Regularity criteria for the generalized MHD equations, Commun. Part. Diff. Eq., 33 (2008), 285–306. https://doi.org/10.1080/03605300701382530 doi: 10.1080/03605300701382530
[25]	K. Yamazaki, Remarks on the global regularity of the two-dimensional magnetohydrodynamics system with zero dissipation, Nonlinear Anal., 94 (2014), 194–205. https://doi.org/10.1016/j.na.2013.08.020 doi: 10.1016/j.na.2013.08.020
[26]	B. Yuan, L. Bai, Remarks on global regularity of 2D generalized MHD equations, J. Math. Anal. Appl., 413 (2014), 633–640. https://doi.org/10.1016/j.jmaa.2013.12.024 doi: 10.1016/j.jmaa.2013.12.024
[27]	Y. Cai, Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics, Arch. Rational Mech. Anal., 228 (2018), 969–993. https://doi.org/10.1007/s00205-017-1210-4 doi: 10.1007/s00205-017-1210-4
[28]	L. He, L. Xu, P. Yu, On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfvén waves, Ann. PDE, 4 (2018), 5. https://doi.org/10.1007/s40818-017-0041-9 doi: 10.1007/s40818-017-0041-9
[29]	R. Pan, Y. Zhou, Y. Zhu, Global classical solutions of three dimensional viscous MHD system without magnetic diffusion on periodic boxes, Arch. Rational Mech. Anal., 227 (2018), 637–662. https://doi.org/10.1007/s00205-017-1170-8 doi: 10.1007/s00205-017-1170-8
[30]	D. Wei, Z. Zhang, Global well-posedness of the MHD equations in a homogeneous magnetic field, Anal. PDE, 10 (2017), 1361–1406. https://doi.org/10.2140/apde.2017.10.1361 doi: 10.2140/apde.2017.10.1361
[31]	J. Pedlosky, Geophysical fluid dynamics, New York: Springer, 1987. https://doi.org/10.1007/978-1-4612-4650-3
[32]	N. Boardman, H. Lin, J. Wu, Stabilization of a background magnetic field on a 2D magnetohydrodynamic flow, SIAM J. Math. Anal., 52 (2020), 5001–5035. https://doi.org/10.1137/20M1324776 doi: 10.1137/20M1324776
[33]	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
[34]	B. Dong, J. Li, J. Wu, Global regularity for the 2D MHD equations with partial hyperresistivity, Int. Math. Res. Notices, 14 (2019), 4261–4280. https://doi.org/10.1093/imrn/rnx240 doi: 10.1093/imrn/rnx240
[35]	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
[36]	R. Ji, J. Wu, W. Yang, Stability and optimal decay for the 3D Navier-Stokes equations with horizontal dissipation, J. Differ. Equations, 290 (2021), 57–77. https://doi.org/10.1016/j.jde.2021.04.026 doi: 10.1016/j.jde.2021.04.026
[37]	Q. Jiu, D. Niu, J. Wu, X. Xu, H. Yu, The 2D magnetohydrodynamic equations with magnetic diffusion, Nonlinearity, 28 (2015), 3935–3955. https://doi.org/10.1088/0951-7715/28/11/3935 doi: 10.1088/0951-7715/28/11/3935
[38]	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
[39]	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
[40]	Y. Zhou, Y. Zhu, Global classical solutions of 2D MHD system with only magnetic diffusion on periodic domain, J. Math. Phys., 59 (2018), 081505. https://doi.org/10.1063/1.5018641 doi: 10.1063/1.5018641
[41]	P. Mironescu, H. Brezis, Gagliardo-Nirenberg inequalities and non-inequalities: the full story, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1355–1376. https://doi.org/10.1016/j.anihpc.2017.11.007 doi: 10.1016/j.anihpc.2017.11.007
[42]	E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in pià variabili, Ricerche Mat., 8 (1959), 24–51.
[43]	L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115–162.
[44]	T. Tao, Nonlinear dispersive equations: local and global analysis, Providence, RI: American Mathematical Society, 2006.
[45]	C. Cao, D. Regmi, J. Wu, The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion, J. Differ. Equations, 254 (2013), 2661–2681. https://doi.org/10.1016/j.jde.2013.01.002 doi: 10.1016/j.jde.2013.01.002
[46]	C. Cao, J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803–1822. https://doi.org/10.1016/j.aim.2010.08.017 doi: 10.1016/j.aim.2010.08.017

Reader Comments

Your name:*

Email:*
© 2023 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)