Global well-posedness for the non-linear Maxwell-Schrödinger system

In this paper we study the Cauchy problem associated to the Maxwell-Schrodinger system with a defocusing pure-power non-linearity. This system has many applications in physics, for instance in the description of a charged non-relativistic quantum plasma, interacting with its self-generated electro-magnetic potential. We show the global well-posedness at high regularity for the sub-cubic case, and we provide polynomial bounds for the growth of the Sobolev norm of the solutions, for a certain range of non-linearities. The main tools are suitable a priori estimates, obtained by means of Koch-Tzvetkov type bounds for the non-homogeneous Schrodinger equation, which overcome the lack of Strichartz estimates for the magnetic-Schrodinger flow. Then we use a classical argument from the Kato school involving modified energies, which combined with the a priori estimates allows us to control the non-linearity globally in time. As a byproduct of our analysis, we show that the Lorentz force associated to the electro-magnetic field is well defined for solutions slightly more regular than being finite energy. This aspect is of fundamental importance since all the related physical models require the observability of electro-magnetic effects. The well posedness of the Lorentz force still appears to be an open problem in the case of solutions of finite energy only.