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On global solutions to the 3D viscous, compressible, and heat-conducting magnetohydrodynamic flows

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Abstract

This paper concerns the Cauchy problem of three-dimensional viscous, compressible, and heat-conducting magnetohydrodynamic flows. Both some new \(L^p\) gradient estimates and the “div-curl” decomposition of \(\Vert \nabla \text{ u }\Vert _{L^3}\) are established; the existence of global solutions to the Cauchy problem with small energy and lower regularity assumed on the initial data are obtained. Furthermore, we also prove that the global solution belongs to a new class of functions in which the uniqueness can be shown to hold.

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Acknowledgements

The author would like to thank the anonymous referee for his/her helpful comments, which improve the presentation of the paper.

Funding

This research is supported in part by the Natural Science Foundation of Shandong Province of China (Grant No. ZR2021QA049)

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  1. School of Mathematics and Information Science, Weifang University, Weifang, 261061, People’s Republic of China

    Mingyu Zhang

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  1. Mingyu Zhang
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This entire work has been completed by the author, Dr. MZ. The author read and approved the finial manuscript.

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Correspondence to Mingyu Zhang.

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This research is supported in part by the Natural Science Foundation of Shandong Province of China (Grant No. ZR2021QA049)

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Zhang, M. On global solutions to the 3D viscous, compressible, and heat-conducting magnetohydrodynamic flows. Z. Angew. Math. Phys. 73, 193 (2022). https://doi.org/10.1007/s00033-022-01833-6

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  • DOI: https://doi.org/10.1007/s00033-022-01833-6

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