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Global Small Solutions to a Special \(2\frac{1}{2}\)-D Compressible Viscous Non-resistive MHD System

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Abstract

This paper solves the global well-posedness and stability problem on a special \(2\frac{1}{2}\)-D compressible viscous non-resistive MHD system near a steady-state solution. The steady-state here consists of a positive constant density and a background magnetic field. The global solution is constructed in \(L^p\)-based homogeneous Besov spaces, which allow general and highly oscillating initial velocity. The well-posedness problem studied here is extremely challenging due to the lack of the magnetic diffusion and remains open for the corresponding 3D MHD equations. Our approach exploits the enhanced dissipation and stabilizing effect resulting from the background magnetic field, a phenomenon observed in physical experiments. In addition, we obtain the solution’s optimal decay rate when the initial data is further assumed to be in a Besov space of negative index.

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Data Availability Statement

Our manuscript has no associated data.

Notes

  1. Note that for technical reasons, we need a small overlap between low and high frequency.

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Acknowledgements

Dong was partially supported by the National Natural Science Foundation of China under grant 11871346, the NSF of Guangdong Province under grant 2020A1515010530, NSF of Shenzhen City (Nos.JCYJ20180305125554234, 20200805101524001). Wu was partially supported by the National Science Foundation of the USA under grant DMS 2104682, the Simons Foundation grant (Award number 708968) and the AT &T Foundation at Oklahoma State University. Zhai was partially supported by the National Natural Science Foundation of China under grant11601533, the Guangdong Provincial Natural Science Foundation under grant 2022A1515011977 and the Science and Technology Program of Shenzhen under grant 20200806104726001.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China

    Boqing Dong

  2. Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK, 74078, USA

    Jiahong Wu

  3. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510520, China

    Xiaoping Zhai

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  1. Boqing Dong
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  2. Jiahong Wu
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Correspondence to Xiaoping Zhai.

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Dong, B., Wu, J. & Zhai, X. Global Small Solutions to a Special \(2\frac{1}{2}\)-D Compressible Viscous Non-resistive MHD System. J Nonlinear Sci 33, 21 (2023). https://doi.org/10.1007/s00332-022-09881-y

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