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Stabilizing effect of magnetic field on the 2D ideal magnetohydrodynamic flow with mixed partial damping

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Abstract

This paper examines the stability of a 2D inviscid MHD system with anisotropic damping near a background magnetic field. It is well known that solutions of the incompressible Euler equations can grow rapidly in time and are thus unstable while solutions of the Euler equations with full damping are stable. Then naturally arises the question of whether solutions of the Euler equations with partial damping are stable. The main purpose of this paper is to give an affirmative answer to this question in the case when the fluid is coupled with the magnetic field through the MHD system with one-component damping. The result presented in this paper especially confirms the stabilizing effects of the magnetic field on the electrically conducting fluids, a phenomenon that has been observed in physical experiments and numerical simulations.

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Acknowledgements

S. Lai was partially supported by the National Natural Science Foundation of China (Nos. 11871407, 12071390). J. Wu was partially supported by the National Science Foundation of the United States under the Grant DMS 2104682, the Simons Foundation grant (Award No. 708968), and the AT &T Foundation at Oklahoma State University. J. Zhang was partially supported by the National Natural Science Foundation of China (Nos. 12071390, 12131007).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, 330022, Jiangxi, People’s Republic of China

    Suhua Lai

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Suhua Lai & Jianwen Zhang

  3. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, USA

    Jiahong Wu

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  1. Suhua Lai
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  2. Jiahong Wu
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  3. Jianwen Zhang
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Correspondence to Jianwen Zhang.

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Communicated by J. M. Ball.

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Lai, S., Wu, J. & Zhang, J. Stabilizing effect of magnetic field on the 2D ideal magnetohydrodynamic flow with mixed partial damping. Calc. Var. 61, 126 (2022). https://doi.org/10.1007/s00526-022-02230-7

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