Two fundamental models in plasma physics are given by the Vlasov–Maxwell–Boltzmann system and the compressible Euler–Maxwell system which both capture the complex dynamics of plasmas under the self-consistent electromagnetic interactions at the kinetic and fluid levels, respectively. It has remained a long-standing open problem to rigorously justify the hydrodynamic limit from the former to the latter as the Knudsen number $𝜀$ tends to zero. In this paper, we give an affirmative answer to the problem for smooth solutions to both systems near constant equilibrium in the whole space in case when only the dynamics of electrons is taken into account. The explicit rate of convergence in $𝜀$ over an almost global time interval is also obtained for well-prepared data. For the proof, one of main difficulties occurs to the cubic growth of large velocities due to the action of the classical transport operator on local Maxwellians and we develop new $𝜀$-dependent energy estimates basing on the macro–micro decomposition to characterize the asymptotic limit in the compressible setting.

弗拉索夫-麦克斯韦-玻尔兹曼系统和可压缩欧拉-麦克斯韦系统给出了等离子体物理学中的两个基本模型，它们分别捕获了在动力学和流体水平上自洽电磁相互作用下等离子体的复杂动力学。严格证明从前者到后者的流体动力学极限作为克努森数仍然是一个长期悬而未决的问题$𝜀$趋于零。在本文中，我们对在仅考虑电子动力学的情况下在整个空间中接近恒定平衡的两个系统的平滑解问题给出了肯定的答案。显式收敛速度$𝜀$对于准备充分的数据，还可以获得几乎全球时间间隔的数据。为了证明，由于经典输运算子对当地麦克斯韦方程的作用，主要困难之一是大速度的立方增长，我们开发了新的$𝜀$基于宏微观分解的依赖能量估计来表征可压缩设置中的渐近极限。