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Well-posedness for the free-boundary ideal compressible magnetohydrodynamic equations with surface tension

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Abstract

We establish the local existence and uniqueness of solutions to the free-boundary ideal compressible magnetohydrodynamic equations with surface tension in three spatial dimensions by a suitable modification of the Nash–Moser iteration scheme. The main ingredients in proving the convergence of the scheme are the tame estimates and unique solvability of the linearized problem in the anisotropic Sobolev spaces \(H_*^m\) for m large enough. In order to derive the tame estimates, we make full use of the boundary regularity enhanced from the surface tension. The unique solution of the linearized problem is constructed by designing some suitable \(\varepsilon \)–regularization and passing to the limit \(\varepsilon \rightarrow 0\).

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Acknowledgements

The authors would like to thank the anonymous referee for helpful comments and suggestions to improve the quality of redaction.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Koptyug av. 4, 630090, Novosibirsk, Russia

    Yuri Trakhinin

  2. Novosibirsk State University, Pirogova str. 1, 630090, Novosibirsk, Russia

    Yuri Trakhinin

  3. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Tao Wang

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  1. Yuri Trakhinin
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  2. Tao Wang
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Correspondence to Tao Wang.

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Communicated by Y. Giga.

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The research of Yuri Trakhinin was supported by Mathematical Center in Akademgorodok under Agreement No. 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation. The research of Tao Wang was partially supported by the National Natural Science Foundation of China under Grants 11971359 and 11731008.

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Trakhinin, Y., Wang, T. Well-posedness for the free-boundary ideal compressible magnetohydrodynamic equations with surface tension. Math. Ann. 383, 761–808 (2022). https://doi.org/10.1007/s00208-021-02180-z

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  • DOI: https://doi.org/10.1007/s00208-021-02180-z

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