  We prove well-posedness for a transport-diffusion problem coupled with a wave
equation for the potential. We assume that the initial data are small. A
bilinear form in the spirit of Kato's proof for the Navier-Stokes equations is
used, coupled with suitable estimates in Chemin-Lerner spaces. In the one
dimensional case, we get well-posedness for arbitrarily large initial data by
using Gagliardo-Nirenberg inequalities.
