In a recent paper Calogero and Alcántara [Kinet. Relat. Models, 4 (2011), pp. 401-426] derived a Lorentz-invariant Fokker-Planck equation, which corresponds to the evolution of a particle distribution associated with relativistic Brownian Motion. We study the “one and one-half” dimensional version of this problem with nonlinear electromagnetic interactions-the relativistic Vlasov-Maxwell-Fokker-Planck system-and obtain the first results concerning well-posedness of solutions. Specifically, we prove the global-in-time existence and uniqueness of classical solutions to the Cauchy problem and a gain in regularity of the distribution function in its momentum argument.

1. Introduction. A plasma is a partially or completely ionized gas. Matter exists in this state if the velocities of individual particles in a material achieve magnitudes approaching the speed of light. If a plasma is of sufficiently low density or the time scales of interest are small enough, it is deemed to be “collisionless”, as collisions between particles become extremely infrequent. Many examples of collisionless plasmas occur in nature, including the solar wind, the Van Allen radiations belts, and galactic nebulae.