# Remarks on the <i>H</i> Theorem for a non involutive Boltzmann like kinetic model

### Giulia Furioli

Università di Bergamo, Dalmine, Italy### Elide Terraneo

Università degli Studi di Milano, Italy

## 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** = *pv* + *qw*, *w** = *qv* + *pw*, where (*v, w*) are the pre-collisional velocities and (*v*, w**) the post-collisional ones and *p* ≥ *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* ≠ 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.

## Cite this article

Giulia Furioli, Elide Terraneo, Remarks on the <i>H</i> Theorem for a non involutive Boltzmann like kinetic model. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 21 (2010), no. 2, pp. 193–213

DOI 10.4171/RLM/567