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The entropy conservation and energy conservation for the relativistic Vlasov–Klein–Gordon system

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Abstract

We are concerned with the properties of weak solutions for the relativistic Vlasov–Klein–Gordon system. Under some suitable regularity hypotheses on the density of particles and field, we show the renormalization property and global (local) entropy conservation laws. In addition, by virtue of the additional integrability condition \(\mathop {\int }\limits _{{\mathbb {R}}^3}\sqrt{1+|v|^{2}} fdv\in L^{\infty }((0, T); L^{2}({\mathbb {R}}_x^{3}))\), we prove that global (local) energy for the weak solutions are conserved.

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Acknowledgements

The authors would like to thank anonymous referees for their helpful comments and valuable suggestions concerning the presentation of this paper. This work is supported by the National Natural Science Foundation of China (Grant No. 12001406 and 11871024).

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Authors and Affiliations

  1. School of Mathematical and Physical Sciences, Wuhan Textile University, Wuhan, Hubei, 430200, People’s Republic of China

    Meixia Xiao

  2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, People’s Republic of China

    Xianwen Zhang

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  1. Meixia Xiao
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Correspondence to Meixia Xiao.

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Xiao, M., Zhang, X. The entropy conservation and energy conservation for the relativistic Vlasov–Klein–Gordon system. Z. Angew. Math. Phys. 73, 95 (2022). https://doi.org/10.1007/s00033-022-01737-5

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