On the asymptotic analysis of the Dirac-Maxwell system in the nonrelativistic limit

We study the behavior of solutions of the Dirac-Maxwell system (DM) in the nonrelativistic limit c -> infinity, where c is the speed of light. DM is a nonlinear system of PDEs obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the moving charge of the spinor. The limit c -> infinity, sometimes also called post-Newtonian, yields a Schrodinger-Poisson system, where the spin and magnetic field no longer appear. We prove that DM is locally well-posed for H-1 data (for fixed c), and that as c -> infinity the existence time grows at least as fast as, log(c), provided the data are uniformly bounded in H-1. Moreover, if the datum for the Dirac spinor converges in H-1, then the solution of DM converges, modulo a phase correction, in C([O, T]; H-1) to a solution of a Schrodinger-Poisson system. Our results also apply to a mixed state formulation of DM, and give also a convergence result for the Pauli equation as the "semi-nonrelativistic" limit, The proof relies on modifications of the bilinear null form estimates of Klainerman and Machedon, and extends our previous work on the nonrelativistic limit of the Mein-Gordon-Maxwell system.