Long-time Asymptotics for the Coupled Maxwell-lorentz Equations

We determine the long time behavior of solutions to the Maxwell-Lorentz equations, which describe a charge coupled to the electromagnetic eld and subject to external time-independent potentials. The stationary solutions have vanishing magnetic eld and a Coulomb type electrostatic eld centered at the points of the set Z at which the external force vanishes. We prove that solutions of nite energy with bounded charge trajectory converge, in suitable local energy seminorms, to the set of stationary states in the long time limit t ! 1. If the set Z is discrete, this implies the convergence to a deenite stationary state. For an unbounded particle trajectory, at least, the acceleration vanishes as t ! 1 and the Maxwell eld as seen from the particle converges to the stationary Coulomb eld.