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Stability and exponential decay for magnetohydrodynamic equations

Part of: Incompressible viscous fluids Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  25 April 2022

Wen Feng ,
Farzana Hafeez ,
Jiahong Wu  and
Dipendra Regmi
Wen Feng
Affiliation:
Department of Mathematics, Niagara University, NY 14109, United States, (wfeng@niagara.edu)
Farzana Hafeez
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, United States, (farzana.hafeez@okstate.edu; jiahong.wu@okstate.edu)
Jiahong Wu
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, United States, (farzana.hafeez@okstate.edu; jiahong.wu@okstate.edu)
Dipendra Regmi
Affiliation:
Department of Mathematics, University of North Georgia-Gainesville Campus, Oakwood, GA 30566, United States, (dipendra.regmi@ung.edu)
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Abstract

This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain $\Omega = \mathbb {T}\times \mathbb {R}$ with $\mathbb {T}=[0,\,1]$ being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field $(1,\,0)$. Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in $H^{1}$ as $t\to \infty$. As a consequence, the solution converges to its horizontal average asymptotically.

Keywords

Magnetohydrodynamic equationspartial dissipationstabilitywave equations

MSC classification

Primary: 35A01: Existence problems
Secondary: 35B35: Stability 35Q35: PDEs in connection with fluid mechanics 76D09: Viscous-inviscid interaction
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 3 , June 2023 , pp. 853 - 880
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Alexakis, A.. Two-dimensional behavior of three-dimensional magnetohydrodynamic flow with a strong guiding field. Phys. Rev. E 84 (2011), 056330.10.1103/PhysRevE.84.056330CrossRefGoogle Scholar
Alfvén, H.. Existence of electromagnetic-hydrodynamic waves. Nature 150 (1942), 405–406.10.1038/150405d0CrossRefGoogle Scholar
Bardos, C., Sulem, C. and Sulem, P. L.. Longtime dynamics of a conductive fluid in the presence of a strong magnetic field. Trans. Am. Math. Soc. 305 (1988), 175–191.10.1090/S0002-9947-1988-0920153-5CrossRefGoogle Scholar
Biskamp, D.. Nonlinear Magnetohydrodynamics (Cambridge: Cambridge University Press, 1993).10.1017/CBO9780511599965CrossRefGoogle Scholar
Boardman, N., Lin, H. and Wu, J.. Stabilization of a background magnetic field on a 2D magnetohydrodynamic flow. SIAM J. Math. Anal. 52 (2020), 5001–5035.10.1137/20M1324776CrossRefGoogle Scholar
Burattini, P., Zikanov, O. and Knaepen, B.. Decay of magnetohydrodynamic turbulence at low magnetic Reynolds number. J. Fluid Mech. 657 (2010), 502–538.CrossRefGoogle Scholar
Cai, Y. and Lei, Z.. Global well-posedness of the incompressible magnetohydrodynamics. Arch. Rational Mech. Anal., 228 (2018), 969–993.10.1007/s00205-017-1210-4CrossRefGoogle Scholar
Cao, C., Regmi, D. and Wu, J.. The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. J. Differ. Eq. 254 (2013), 2661–2681.10.1016/j.jde.2013.01.002CrossRefGoogle Scholar
Cao, C. and Wu, J.. Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math. 226 (2011), 1803–1822.10.1016/j.aim.2010.08.017CrossRefGoogle Scholar
Cao, C., Wu, J. and Yuan, B.. The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion. SIAM J. Math. Anal. 46 (2014), 588–602.CrossRefGoogle Scholar
Chen, W., Zhang, Z. and Zhou, J.. Global well-posedness for the 3-D MHD equations with partial dissipation in periodic domain. Sci. China Math. in press.Google Scholar
Davidson, P. A.. Magnetic damping of jets and vortices. J. Fluid Mech. 299 (1995), 153–186.10.1017/S0022112095003466CrossRefGoogle Scholar
Davidson, P. A.. The role of angular momentum in the magnetic damping of turbulence. J. Fluid Mech. 336 (1997), 123–150.CrossRefGoogle Scholar
Davidson, P. A.. An Introduction to Magnetohydrodynamics (Cambridge, England: Cambridge University Press, 2001).10.1017/CBO9780511626333CrossRefGoogle Scholar
Deng, W. and Zhang, P.. Large time behavior of solutions to 3-D MHD system with initial data near equilibrium. Arch. Rational Mech. Anal. 230 (2018), 1017–1102.10.1007/s00205-018-1265-xCrossRefGoogle Scholar
Dong, B., Jia, Y., Li, J. and Wu, J.. Global regularity and time decay for the 2D magnetohydrodynamic equations with fractional dissipation and partial magnetic diffusion. J. Math. Fluid Mech. 20 (2018), 1541–1565.10.1007/s00021-018-0376-3CrossRefGoogle Scholar
Dong, B., Li, J. and Wu, J.. Global regularity for the 2D MHD equations with partial hyperresistivity. Intern. Math Res. Notices 14 (2019), 4261–4280.10.1093/imrn/rnx240CrossRefGoogle Scholar
Dong, B., Wu, J., Xu, X. and Zhu, N.. Stability and exponential decay for the 2D anisotropic Navier-Stokes equations with horizontal dissipation, submitted for publication.Google Scholar
Du, L. and Zhou, D.. Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion. SIAM J. Math. Anal. 47 (2015), 1562–1589.10.1137/140959821CrossRefGoogle Scholar
Fan, J., Malaikah, H., Monaquel, S., Nakamura, G. and Zhou, Y.. Global Cauchy problem of 2D generalized MHD equations. Monatsh. Math. 175 (2014), 127–131.CrossRefGoogle Scholar
Fefferman, C. L., McCormick, D. S., Robinson, J. C. and Rodrigo, J. L.. Higher order commutator estimates and local existence for the non-resistive MHD equations and related models. J. Funct. Anal. 267 (2014), 1035–1056.10.1016/j.jfa.2014.03.021CrossRefGoogle Scholar
Fefferman, C. L., McCormick, D. S., Robinson, J. C. and Rodrigo, J. L.. Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces. Arch. Ration. Mech. Anal. 223 (2017), 677–691.10.1007/s00205-016-1042-7CrossRefGoogle Scholar
F. Hafeez, W. Feng and Wu, J.. Influence of a background magnetic field on a 2D magnetohydrodynamic flow. Nonlinearity 34 (2021), 2527–2562.Google Scholar
Gallet, B., Berhanu, M. and Mordant, N.. Influence of an external magnetic field on forced turbulence in a swirling flow of liquid metal. Phys. Fluids 21 (2009), 085107.10.1063/1.3194304CrossRefGoogle Scholar
Gallet, B. and Doering, C. R.. Exact two-dimensionalization of low-magnetic-Reynolds-number flows subject to a strong magnetic field. J. Fluid Mech. 773 (2015), 154–177.10.1017/jfm.2015.232CrossRefGoogle Scholar
He, L., Xu, L. and Yu, P.. On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfvén waves. Ann. PDE 4 (2018), 105.10.1007/s40818-017-0041-9CrossRefGoogle Scholar
Hu, X.. Global existence for two dimensional compressible magnetohydrodynamic flows with zero magnetic diffusivity, arXiv: 1405.0274v1 [math.AP] 1 May 2014.Google Scholar
Hu, X. and Lin, F.. Global existence for two dimensional incompressible magnetohydrodynamic flows with zero magnetic diffusivity, arXiv: 1405.0082v1 [math.AP] 1 May 2014.Google Scholar
Ji, R. and Wu, J.. The resistive magnetohydrodynamic equation near an equilibrium. J. Differ. Eq. 268 (2020), 1854–1871.10.1016/j.jde.2019.09.027CrossRefGoogle Scholar
Jiu, Q., Niu, D., Wu, J., Xu, X. and Yu, H.. The 2D magnetohydrodynamic equations with magnetic diffusion. Nonlinearity 28 (2015), 3935–3956.10.1088/0951-7715/28/11/3935CrossRefGoogle Scholar
Li, C., Wu, J. and Xu, X.. Smoothing and stabilization effects of magnetic field on electrically conducting fluids. J. Differ. Eq. 276 (2021), 368–403.10.1016/j.jde.2020.12.012CrossRefGoogle Scholar
Li, J., Tan, W. and Yin, Z.. Local existence and uniqueness for the non-resistive MHD equations in homogeneous Besov spaces. Adv. Math. 317 (2017), 786–798.10.1016/j.aim.2017.07.013CrossRefGoogle Scholar
Lin, F., Xu, L. and Zhang, P.. Global small solutions to 2-D incompressible MHD system. J. Differ. Eq. 259 (2015), 5440–5485.10.1016/j.jde.2015.06.034CrossRefGoogle Scholar
Lin, F. and Zhang, P.. Global small solutions to an MHD-type system: the three-dimensional case. Comm. Pure Appl. Math. 67 (2014), 531–580.10.1002/cpa.21506CrossRefGoogle Scholar
Lin, H. and Du, L.. Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions. Nonlinearity 26 (2013), 219–239.10.1088/0951-7715/26/1/219CrossRefGoogle Scholar
Lin, H., Ji, R., Wu, J. and Yan, L.. Stability of perturbations near a background magnetic field of the 2D incompressible MHD equations with mixed partial dissipation. J. Funct. Anal. 279 (2020), 108519.10.1016/j.jfa.2020.108519CrossRefGoogle Scholar
Majda, A. and Bertozzi, A.. Vorticity and Incompressible Flow (Cambridge, UK: Cambridge University Press, 2002).Google Scholar
Pan, R., Zhou, Y. and Zhu, Y.. Global classical solutions of three dimensional viscous MHD system without magnetic diffusion on periodic boxes. Arch. Rational Mech. Anal. 227 (2018), 637–662.10.1007/s00205-017-1170-8CrossRefGoogle Scholar
Priest, E. and Forbes, T.. Magnetic Reconnection, MHD Theory and Applications (Cambridge: Cambridge University Press, 2000).CrossRefGoogle Scholar
Ren, X., Wu, J., Xiang, Z. and Zhang, Z.. Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion. J. Funct. Anal. 267 (2014), 503–541.10.1016/j.jfa.2014.04.020CrossRefGoogle Scholar
Ren, X., Xiang, Z. and Zhang, Z.. Global well-posedness for the 2D MHD equations without magnetic diffusion in a strip domain. Nonlinearity 29 (2016), 1257–1291.10.1088/0951-7715/29/4/1257CrossRefGoogle Scholar
Schonbek, M. E., Schonbek, T. P. and Süli, E.. Large-time behaviour of solutions to the magnetohydrodynamics equations. Math. Ann. 304 (1996), 717–756.10.1007/BF01446316CrossRefGoogle Scholar
Sermange, M. and Temam, R.. Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36 (1983), 635–664.10.1002/cpa.3160360506CrossRefGoogle Scholar
Tan, Z. and Wang, Y.. Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems. SIAM J. Math. Anal. 50 (2018), 1432–1470.CrossRefGoogle Scholar
Tao, T.. Nonlinear dispersive equations: local and global analysis, CBMS regional conference series in mathematics, providence, RI: American Mathematical Society, 2006.10.1090/cbms/106CrossRefGoogle Scholar
Wan, R.. On the uniqueness for the 2D MHD equations without magnetic diffusion. Nonlin. Anal. Real World Appl. 30 (2016), 32–40.CrossRefGoogle Scholar
Wei, D. and Zhang, Z.. Global well-posedness of the MHD equations in a homogeneous magnetic field. Anal. PDE 10 (2017), 1361–1406.10.2140/apde.2017.10.1361CrossRefGoogle Scholar
Wei, D. and Zhang, Z.. Global well-posedness for the 2D MHD equations with magnetic diffusion. Commun. Math. Res. 36 (2020), 377–389.Google Scholar
Wu, J.. The 2D magnetohydrodynamic equations with partial or fractional dissipation, Lectures on the analysis of nonlinear partial differential equations, Morningside Lectures on Mathematics, Part 5, MLM5, pp. 283-332, International Press, Somerville, MA, 2018.Google Scholar
Wu, J. and Wu, Y.. Global small solutions to the compressible 2D magnetohydrodynamic system without magnetic diffusion. Adv. Math. 310 (2017), 759–888.10.1016/j.aim.2017.02.013CrossRefGoogle Scholar
Wu, J., Wu, Y. and Xu, X.. Global small solution to the 2D MHD system with a velocity damping term. SIAM J. Math. Anal. 47 (2015), 2630–2656.10.1137/140985445CrossRefGoogle Scholar
Wu, J. and Zhu, Y.. Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium. Adv. Math. 377 (2021), 107466.10.1016/j.aim.2020.107466CrossRefGoogle Scholar
Xiao, Y., Xin, Z.-P. and Wu, J.. Vanishing viscosity limit for the 3D magneto-hydrodynamic system with a slip boundary condition. J. Funct. Anal. 257 (2009), 3375–3394.10.1016/j.jfa.2009.09.010CrossRefGoogle Scholar
Xu, L. and Zhang, P.. Global small solutions to three-dimensional incompressible magnetohydrodynamical system. SIAM J. Math. Anal. 47 (2015), 26–65.10.1137/14095515XCrossRefGoogle Scholar
Yamazaki, K.. On the global well-posedness of N-dimensional generalized MHD system in anisotropic spaces. Adv. Differ. Eq. 19 (2014), 201–224.Google Scholar
Yamazaki, K.. Remarks on the global regularity of the two-dimensional magnetohydrodynamics system with zero dissipation. Nonlinear Anal. 94 (2014), 194–205.CrossRefGoogle Scholar
Yamazaki, K.. On the global regularity of two-dimensional generalized magnetohydrodynamics system. J. Math. Anal. Appl. 416 (2014), 99–111.10.1016/j.jmaa.2014.02.027CrossRefGoogle Scholar
Yamazaki, K.. Global regularity of logarithmically supercritical MHD system with zero diffusivity. Appl. Math. Lett. 29 (2014), 46–51.CrossRefGoogle Scholar
Yang, W., Jiu, Q. and Wu, J.. The 3D incompressible Navier-Stokes equations with partial hyperdissipation. Math. Nach. 292 (2019), 1823–1836.CrossRefGoogle Scholar
Yang, W., Jiu, Q. and Wu, J.. The 3D incompressible magnetohydrodynamic equations with fractional partial dissipation. J. Differ. Eq. 266 (2019), 630–652.10.1016/j.jde.2018.07.046CrossRefGoogle Scholar
Yuan, B. and Zhao, J.. Global regularity of 2D almost resistive MHD equations. Nonlin. Anal. Real World Appl. 41 (2018), 53–65.10.1016/j.nonrwa.2017.10.006CrossRefGoogle Scholar
Zhang, T.. An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system, (2014), arXiv:1404.5681.Google Scholar
Zhang, T.. Global solutions to the 2D viscous, non-resistive MHD system with large background magnetic field. J. Differ. Equ. 260 (2016), 5450–5480.10.1016/j.jde.2015.12.005CrossRefGoogle Scholar
Zhou, Y. and Zhu, Y.. Global classical solutions of 2D MHD system with only magnetic diffusion on periodic domain. J. Math. Phys. 59 (2018), 081505.CrossRefGoogle Scholar