Exact Solutions for Some Discrete Models of the Boltzmann Equation

Abs tra ct. For the simplest of the disc rete mod els of the Boltzmann equations , the Broad well model, exact solutions have been obtained by Co rnille [15.16] in the form of bisolitons. In the present paper, we build exact solutions for more co mplex mod els. 1. Introduction For t he las t twenty years, t he st udy of disc rete mo de ls of t he Bolt zm ann equat ion h as attract ed t he attent ion of m any scient ists. The first mo de ls, with six or eight velociti es , were propose d in 1964 by J. Bro adwell [1,2]. After Br oadwell, R. Gatignol has writ ten the genera l form of equat ions wh ich repr esent the discrete models of the Bo ltzmann equ at ion [3,4]. T hose m odels are obtained by ass uming t hat the mo lecules of a gas can have on ly a fin ite number of velocit ies, iii' With t his assumption , the Bo lt zmann equation is rep laced by a semi-linear hyperb olic system of par t ial di fferent ial equations. We denote the density of mo lecules with velocity iii by Ni(t,X) (t t ime,:i p osit ion), and the discrete mo dels of the Boltzmann equation ar e also written in t he following form: aN·-The coe fficients A;1\ t ransit ion pr obabiliti es, are cons tants, po siti ve or zero; they dep end on three of the ind ices i, k,l , m. T he equations of R. Gatignol, (1.1), are the kin eti c equat ions. Since 1974 , the global ex iste nce of solu-ti ons of equ at ion (1.1) has b een proved for mo re and mo re complex models by Nish ida and Mimu ra [5], Cranda ll and Tar t ar [6], Ca ban nes [7,8],