Slow decay for a linearized model of the solar wind

The solar wind interacting with a magnetized obstacle is modelled with the steady Vlasov-Poisson system in the plane. The system is linearized for the (given) magnetic field of the obstacle being small. The main focus is on the rate of decay of the spatial charge density “downwind” of the obstacle. A special case that admits an explicit solution is presented. It is also shown that when the background particle distribution is compactly supported in velocity, that the spatial charge density cannot, in general, decay faster than $x^{-\frac {1}{2}}_1$, where $x_1$ is the downwind distance.