Classical limit of the relativistic Vlasov-Maxwell-Landau system
Chuqi Cao, Ling-Bing He, Yuanjie Lei, Qinghua Xiao
TL;DR
The paper rigorously justifies the non-relativistic (classical) limit of the relativistic Vlasov-Maxwell-Landau system to the Vlasov-Poisson-Landau system in a periodic box by establishing uniform-in-$c$ coercivity for the relativistic collision operator, introducing a weighted energy framework to counteract weak field dissipation, and proving global well-posedness independent of $c$. It formalizes a $c^{-1}$ expansion up to order three, showing that the leading-order dynamics recover the VPL system while higher-order terms satisfy controlled, decaying equations. The main results include global existence and uniform energy bounds for the RVML system, a precise $O(1/c)$ convergence rate to VPL, and the convergence of the self-consistent magnetic field to a fixed background, clarifying the Maxwell-to-Poisson reduction in the presence of collisions. This work advances the mathematical understanding of kinetic limits with self-consistent fields and highlights how collisional damping interacts with long-range electromagnetic effects in singular limits.
Abstract
The physical essence of the non-relativistic limit, from the relativistic Vlasov-Maxwell-Landau system to the Vlasov-Poisson-Landau system, lies in the transition from finite-speed electromagnetic waves to instantaneous Coulomb interactions, and from relativistic to Newtonian particle dynamics. We rigorously justify this limit (mathematically corresponding to the light speed $c \to \infty$) in a periodic box via three key technical advances: establishing a uniform-in-$c$ coercivity estimate for the relativistic Landau collision operator, constructing a novel weighted energy functional to overcome the weakening dissipation of the electromagnetic field at large $c$, and proving a corresponding global well-posedness result.