The fixed-point proof of the Darwin vector potential in the Vlasov-Darwin system with generalized variables

Abstract

In this work, we study the well-posedness of the integral equation for the Darwin vector potential in the Vlasov-Darwin system with generalized variables (VDG system). As for this system, local existence, continuation criterion and global existence with small initial data have been established. Due to the effect of the non-relativistic term, a difficulty arises in dealing with the current density. All of the results for the VDG system impose restrictions on the speed of light c. In this work, we overcome this difficulty and use the Schauder fixed point theorem to prove that the integral equation of the Darwin vector potential is well-posed. Furthermore, the well-posedness of the Darwin vector potential does not require any restrictions.