Remarks on the $H$ Theorem for a non involutive Boltzmann like kinetic model

Abstract

In this paper, we consider a one-dimensional kinetic equation of Boltzmann type in which the binary collision process is described by the linear transformation $v^{\ast }=pv+qw$, $w^{\ast }=qv+pw$, where $(v,w)$ are the pre-collisional velocities and $(v^{\ast },w^{\ast })$ the post-collisional ones and $p\ge q>0$ are two positive parameters. This kind of model has been extensively studied by Pareschi and Toscani (in J. Stat. Phys., 124(2–4):747–779, 2006) with respect to the asymptotic behavior of the solutions in a Fourier metric. In the conservative case $p^2+q^2=1$, even if the transformation has Jacobian $J\ne 1$ and so it is not involutive, we remark that the $H$ Theorem holds true. As a consequence we prove exponential convergence in $L^1$ of the solution to the stationary state, which is the Maxwellian.