Global existence for the Vlasov–Darwin system in ℝ 3 for small initial data

We prove the global existence of weak solutions to the Vlasov–Darwin system in R3 for small initial data. The Vlasov–Darwin system is an approximation of the Vlasov–Maxwell model which is valid when the characteristic speed of the particles is smaller than the light velocity, but not too small. In contrast to the Vlasov–Maxwell system, the total energy conservation does not provide an L2‐bound on the transverse part of the electric field. This difficulty may be overcome by exploiting the underlying elliptic structure of the Darwin equations under a smallness assumption on the initial data. We finally investigate the convergence of the Vlasov–Darwin system towards the Vlasov–Poisson system. Copyright © 2003 John Wiley & Sons, Ltd.