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)

–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.