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 p2 + q2 = 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 L1 of the solution to the stationary state, which is the Maxwellian.

Remarks on the H theorem for a non involutive Boltzmann like kinetic model / G. Furioli, E. Terraneo. - In: ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI. - ISSN 1120-6330. - 21:2(2010), pp. 193-213. [10.4171/RLM/567]

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

E. Terraneo
Ultimo
2010

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 p2 + q2 = 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 L1 of the solution to the stationary state, which is the Maxwellian.
Conservative Boltzmann equations ; asymptotic behavior of solutions ; kinetic equations
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/151426
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