  We will consider a modified Boltzmann equation which contains together with
the collision operator an additional drift term that is characterized by a
matrix A. Furthermore, we consider a Maxwell gas where the collision kernel has
an angular singularity. Such an equation is used in the study of homoenergetic
solutions of the Boltzmann equation. Our goal is to prove that under smallness
assumptions on the drift term the long-time asymptotics is given by
self-similar solutions. We will work in the framework of measure-valued
solutions with finite moments of order p > 2 and show existence, uniqueness and
stability of these self-similar solutions for sufficiently small A.
Furthermore, we will prove that they have finite moments of arbitrary order by
decreasing the norm of A, depending on that order. In addition, the singular
collision operator allows to prove smoothness of these self-similar solutions.
At the end we apply the results to study the asymptotics of homoenergetic
solutions in particular cases. This extends previous results from the cutoff
case to non cutoff Maxwell gases.
