Classical Limit and Asymptotic Behaviour of Relativistic Vlasov-Darwin System with Generalized Variables

We investigate the classical limit of global classical solutions to the relativistic Vlasov-Darwin system with generalized variables. Under the assumption of sufficiently small and compactly supported initial data, we prove that the relativistic dynamics converge to those of the Vlasov-Poisson system as the speed of light tends to infinity. In particular, we establish pointwise convergence of the distribution function with an explicit rate of order $$O(C^{-1})$$ together with quantitative estimates for the associated scalar and vector potentials. Moreover, sharp decay estimates describing the long-time asymptotic behavior of the solutions are derived. The analysis is based on characteristic methods and refined comparison arguments between relativistic and non-relativistic flows, and provides the first rigorous justification of the classical limit for the generalized Vlasov-Darwin framework.