Self-Similar Profiles for Homoenergetic Solutions of the Boltzmann Equation: Particle Velocity Distribution and Entropy

Abstract

In this paper we study a class of solutions of the Boltzmann equation which have the−1 with the matrix A describing a shear flow or a dilatation or a combination of both. These solutions are known as homoenergetic solutions. We prove existence of homoenergetic solutions for a large class of initial data. For different choices for the matrix A and for different homogeneities of the collision kernel, we characterize the long time asymptotics of the velocity distribution for the corresponding homoenergetic solutions. For a large class of choices of A we then prove rigorously the existence of self-similar solutions of the Boltzmann equation. The latter are non Maxwellian distributions and describe far-from-equilibrium flows. For Maxwell molecules we obtain exact formulas for the H-function for some of these flows. These formulas show that in some cases, despite being very far from equilibrium, the relationship between density, temperature and entropy is exactly the same as in the equilibrium case. We make conjectures about the asymptotics of homoenergetic solutions that do not have self-similar profiles.

Long-Time Asymptotics for Homoenergetic Solutions of the Boltzmann Equation: Collision-Dominated Case

“…It is possible to find solutions of the Boltzmann equation (1.3) that have the same statistics as the molecular dynamics simulation for discrete systems described above (see (1.1)). We refer to [17] for details. Thus, the analogous of the ansatz (1.1) in terms of the particle velocities can be written as:…”

“…Under mild smoothness conditions, solutions with the form (1.8) exist if ξ(t, x) = A(I + tA) −1 x (cf. [17]). Formally, if f is a solution of the Boltzmann equation (1.3) of the form (1.7) the function g satisfies ∂ t g − L (t) w · ∂ w g = Cg (w) (1.9) where the collision operator C is defined as in (1.3).…”

“…The properties of the solutions of (1.9) for large times t depend greatly on the homogeneity of the kernel yielding the cross section of the collision operator Cg. In [17] we have focused on the analysis of solutions of (1.9) for which the terms L (t) w · ∂ w g and Cg (w) are of the same order of magnitude. We have rigorously proved in [17] the existence of self-similar solutions in the class of homoenergetic flows when the collision kernel describes the interaction between Maxwell molecules, which corresponds to homogeneity γ = 0.…”

“…In [17] we have focused on the analysis of solutions of (1.9) for which the terms L (t) w · ∂ w g and Cg (w) are of the same order of magnitude. We have rigorously proved in [17] the existence of self-similar solutions in the class of homoenergetic flows when the collision kernel describes the interaction between Maxwell molecules, which corresponds to homogeneity γ = 0. In all the cases when such selfsimilar solutions exist, the terms L (t) w · ∂ w g and Cg (w) have a comparable size as t → ∞.…”

“…Depending on the properties of the collision kernel the collision and the hyperbolic terms might be of the same order of magnitude as t → ∞, or the collision term could be the dominant one for large times, or the hyperbolic term could be the largest. The first case has been rigorously studied in [17]. Formal asymptotic expansions in the second case have been obtained in [18].…”