On the three-dimensional finite Larmor radius approximation: the case of electrons in a fixed background of ions

This paper is concerned with the analysis of a mathematical model arising in plasma physics, more specifically in fusion research. It directly follows [17], where the tri-dimensional analysis of a Vlasov-Poisson equation with finite Larmor radius scaling was led, corresponding to the case of ions with massless electrons whose density follows a linearized Maxwell-Boltzmann law. We now consider the case of electrons in a background of fixed ions, which was only sketched in [17]. Unfortunately, there is evidence that the formal limit is false in general. Nevertheless, we formally derive a fluid system for particular monokinetic data. We prove the local in time existence of analytic solutions and rigorously study the limit (when the Debye length vanishes) to a new anisotropic fluid system. This is achieved thanks to Cauchy-Kovalevskaya type techniques, as introduced by Caflisch [7] and Grenier [13]. We finally show that this approach fails in Sobolev regularity, due to multi-fluid instabilities.