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Weak Serrin-type finite time blowup and global strong solutions for three-dimensional density-dependent heat conducting magnetohydrodynamic equations with vacuum

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Abstract

This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible heat conducting magnetohydrodynamic (MHD) equations in the whole space. First of all, we establish a weak Serrin-type blowup criterion for strong solutions. It is shown that for the Cauchy problem of the 3D nonhomogeneous heat conducting MHD equations, the strong solution exists globally if the velocity satisfies the weak Serrin’s condition. In particular, this criterion is independent of the absolute temperature and magnetic field. Then as an immediate application, we prove the global existence and uniqueness of strong solution to the 3D nonhomogeneous heat conducting MHD equations under a smallness condition on the initial data. In addition, the initial vacuum is allowed.

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Acknowledgments

The author would like to thank the reviewers for careful reading and helpful suggestions on the original manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, No. 100, Science Avenue, Zhengzhou, 450001, P. R. China

    Huanyuan Li

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  1. Huanyuan Li
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Correspondence to Huanyuan Li.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 12001495).

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Li, H. Weak Serrin-type finite time blowup and global strong solutions for three-dimensional density-dependent heat conducting magnetohydrodynamic equations with vacuum. Appl Math 68, 593–621 (2023). https://doi.org/10.21136/AM.2022.0141-22

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