Is the linearized Boltzmann-Enskog operator dissipative?

In this paper, it is shown that the linearized Boltzmann-Enskog collision operator cannot be dissipative in the L2-space setting contrarily to the linearized Boltzmann operator. Some estimates useful for the spectral theory are given. The Enskog equation is a modification of the Boltzmann kinetic equation, in which each particle is considered as a hard sphere with nonzero diameter a > 0 (and therefore, the collisions take place in a point at the distances a/2 from the centers of two colliding particles). The Enskog equation gives a quite good description of the transport phenomena in moderately dense gases [1,2]. The mathematical theory of various versions of the Enskog equation can be found in [3-13] (see also references therein). In the present paper, only the simplified case of the Enskog equation (referred to as the Boltzmann-Enskog equation), for which the pair correlation function is equal to 1, is considered. The Boltzmann-Enskog equation (in the dimensionless form) reads (Or + v. 0x) f = ~Ea(f, f), (1) Ea (fl, f2) (x, v) = ~ a 2 (fl (x + an, w') f2 (x, v') + f2 (x + an, w') fl (x, v')-fl (x-an, w)f2(x,v)-f2(x-an, w)fl(x,v)) (n. (w-v) V 0) dndw, where a is the (dimensionless) diameter of the (hard sphere) particles; ~ is the Knudsen number; f = f(t,x,v) is the one-particle distribution function; t >_ 0, x = (xl,x2,x3) c IRa , v = (vl, v2, v3) c R a are time, space, and velocity variables, respectively. The standard notation has been used-a particle with the center at x and the velocity v collides with a particle with the center at x-an and the velocity w; v' = v + ((w-v)-n)n and w' = w-((w-v) • n)n are the velocities after the collision; n E S 2 = {n E R 3 : Inl = 1}, al V a2 = max{al, a2}.