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Global well-posedness to three-dimensional full compressible magnetohydrodynamic equations with vacuum

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Abstract

This paper studies the Cauchy problem for three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic equations with vacuum as far field density. We prove the global existence and uniqueness of strong solutions provided that the quantity \(\Vert \rho _0\Vert _{L^\infty }+\Vert b_0\Vert _{L^3}\) is suitably small and the viscosity coefficients satisfy \(3\mu >\lambda \). Here, the initial velocity and initial temperature could be large. The assumption on the initial density does not exclude that the initial density may vanish in a subset of \({\mathbb {R}}^3\) and that it can be of a nontrivially compact support. Our result is an extension of the works of Fan and Yu (Nonlinear Anal Real World Appl 10:392–409, 2009) and Li et al. (SIAM J Math Anal 45:1356–1387, 2013), where the local strong solutions in three dimensions and the global strong solutions for isentropic case were obtained, respectively. The analysis is based on some new mathematical techniques and some new useful energy estimates. This paper can be viewed as the first result concerning the global existence of strong solutions with vacuum at infinity in some classes of large data in higher dimension.

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Notes

  1. From (1.3) and \(3\mu >\lambda \), we have \(5\mu +2\lambda >0\). Then, by (1.3) again one gets \(7\mu +5\lambda >0\), which combined with (1.3) again implies \(9\mu +8\lambda >0\). This together with (1.3) once more gives \(11\mu +11\lambda >0\). Thus, the result follows.

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Acknowledgements

The authors would like to express their gratitude to the reviewers for careful reading and helpful suggestions which led to an improvement in the original manuscript.

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Correspondence to Xin Zhong.

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Corresponding author: Xin Zhong. Yang Liu was partially supported by National Natural Science Foundation of China (No. 11901288). Xin Zhong was partially supported by National Natural Science Foundation of China (Nos. 11901474, 12071359).

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Liu, Y., Zhong, X. Global well-posedness to three-dimensional full compressible magnetohydrodynamic equations with vacuum. Z. Angew. Math. Phys. 71, 188 (2020). https://doi.org/10.1007/s00033-020-01408-3

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