Consider the relativistic Vlasov–Maxwell–Boltzmann system describing the dynamics of an electron gas in the presence of a fixed ion background. Thanks to recent works Germain and Masmoudi (Ann Sci Éc Norm Supér 47(3):469–503, 2014), Guo et al. (J Math Phys 55(12):123102, 2014) and Deng et al. (Arch Ration Mech Anal 225(2):771–871, 2017), we establish the global-in-time validity of its Hilbert expansion and derive the limiting relativistic Euler–Maxwell system as the mean free path goes to zero. Our method is based on the \(L^2-L^{\infty }\) framework and the Glassey–Strauss Representation of the electromagnetic field, with auxiliary \(H^1\) estimates and \(W^{1,\infty }\) estimates to control the characteristic curves and corresponding \(L^{\infty }\) norm.

考虑相对论的Vlasov-Maxwell-Boltzmann系统，该系统描述了在固定离子背景下电子气的动力学。感谢Guermain和Masmoudi（Ann SciÉcNormSupér47（3）：469–503，2014）的最新著作，Guo等。（J Math Phys 55（12）：123102，2014）和Deng等人。（Arch Ration Mech Anal 225（2）：771–871，2017），我们建立了其希尔伯特展开的全局时间有效性，并在平均自由程变为零时推导了相对论极限欧拉-麦克斯韦系统。我们的方法基于\（L ^ 2-L ^ {\ infty} \）框架和电磁场的Glassey-Strauss表示，具有辅助的（H ^ 1 \）估计和\（W ^ {1， \ infty} \）估计值以控制特征曲线和相应的\（L ^ {\ infty} \）规范。