论文标题
非相关库仑碰撞的动力学方程的希尔伯特扩展
Hilbert expansion for kinetic equations with non-relativistic Coulomb collision
论文作者
论文摘要
在本文中，我们研究了整个空间中Landau方程和Vlasov-Maxwell-Landau系统的流体动力限制。我们的主要目的是两个方面：第一个是通过Hilbert扩展从Landau方程式对可压缩的Euler方程进行严格的推导；尽管第二个仍在希尔伯特（Hilbert）扩展的环境中，但弗拉索夫 - 马克斯韦 - 兰道系统（Vlasov-Maxwell-Landau System）的独特经典解决方案收敛，该解决方案在全球范围内及时显示为欧拉·马克斯韦（Euler-Maxwell）系统的全球平滑解决方案，因为Knudsen数字为零。我们分析的主要成分是在Landau方程和Vlasov-Maxwell-Landau系统方面得出一些新颖的相互作用估计，它们分别是局部Maxwellian和全球Maxwellian的小扰动。我们的结果解决了具有库仑电势的Landau型方程的流体力学极限中的一个开放问题，本文中开发的方法可以无缝地处理有关Hilbert扩展对其他类型动力学方程的有效性的问题。
In this paper, we study the hydrodynamic limits of both the Landau equation and the Vlasov-Maxwell-Landau system in the whole space. Our main purpose is two-fold: the first one is to give a rigorous derivation of the compressible Euler equations from the Landau equation via the Hilbert expansion; while the second one is to prove, still in the setting of Hilbert expansion, that the unique classical solution of the Vlasov-Maxwell-Landau system converges, which is shown to be globally in time, to the resulting global smooth solution of the Euler-Maxwell system, as the Knudsen number goes to zero. The main ingredient of our analysis is to derive some novel interplay energy estimates on the solutions of the Landau equation and the Vlasov-Maxwell-Landau system which are small perturbations of both a local Maxwellian and a global Maxwellian, respectively. Our result solves an open problem in the hydrodynamic limit for the Landau-type equations with Coulomb potential and the approach developed in this paper can seamlessly be used to deal with the problem on the validity of the Hilbert expansion for other types of kinetic equations.
