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Nonlinear stability of rarefaction waves for the compressible MHD equations

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Abstract

This paper is concerned with time-asymptotic nonlinear stability of rarefaction waves to the Cauchy problem for one-dimensional compressible non-isentropic magnetohydrodynamics (MHD) equations (including its isentropic case), which describe the motion of a conducting fluid in a magnetic field. Through some elaborate and rigorous mathematical analysis, we can construct the rarefaction waves \( \left( v^r, u^r, \theta ^r, b^r \right) (x/t) \) where magnetic component \( b^r \left( x/t \right) \) is a nontrivial profile, namely a non-constant function. Then the solution of the compressible MHD equations is proved to tend towards the rarefaction waves time-asymptotically under small initial perturbations and weak wave strength, and also under a technical assumption that the parameter \( \beta = v_+ b_+ \) is bounded by a specific constant. The proof of the main result is based on elementary \(L^2\) energy methods.

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Acknowledgements

The authors would like to thank the referees for all valuable and helpful comments on the manuscript. The research was supported by the National Natural Science Foundation of China \(\#\)12171160, 11831003 and the Guangdong Provincial Key Laboratory of Human Digital Twin \(\#\)2022B1212010004.

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  1. College of Mathematics and Informatics, South China Agricultural University, Guangzhou, 510642, People’s Republic of China

    Huancheng Yao

  2. School of Mathematics, South China University of Technology, Guangzhou, 510641, People’s Republic of China

    Changjiang Zhu

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  1. Huancheng Yao
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Correspondence to Changjiang Zhu.

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Yao, H., Zhu, C. Nonlinear stability of rarefaction waves for the compressible MHD equations. Z. Angew. Math. Phys. 74, 137 (2023). https://doi.org/10.1007/s00033-023-02024-7

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