Controlling the propagation of the support for the relativistic Vlasov equation with a selfconsistent Lorentz invariant field

Successful techniques have been developed for controlling the propagation of the support (preventing blow up) for the classical Vlasov-Poisson equation, leading to global existence of smooth solutions.It is well known that these techniques all fail for the relativistic Vlasov-Poisson equation.This equation is a hybrid of a relativistic transport equation and a classical, Galilean invariant, field equation.In this paper we introduce a new equation for the field making it Lorentz invariant.We show that the propagation of the support, for solutions satisfying this equation and the relativistic Vlasov equation, may be controlled. 1 Introduction.Consider solutions to the Vlasov-Poisson equation, corresponding to smooth initial data with compact support.It follows by classical results (see [B],[H2]) that such solutions remain smooth for all time if and only if the propagation of the support of the phase space density may be controlled (preventing blow up).It is by now well known, that the support remains bounded on finite time intervals (see Pfaffelmoser [P], Lions and Perthame [LP], Schaeffer [Sc1], Horst [H1], Glassey [G] and Wollman [W]).However, none of these techniques, developed to control the propagation of the support, is applicable to the relativistic Vlasov-Poisson equation.The relativistic equation is obtained from the Vlasov-Poisson equation by substituting the velocity (v/m, v denotes momenta and m is the mass of the particle) by the relativistic velocity Indiana