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

Abstract

In this paper we continue the formal analysis of the long-time asymptotics of the homoenergetic solutions for the Boltzmann equation that we began in [18]. They haveHomoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. 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]. All the solutions obtained in this case can be approximated by Maxwellian distributions with changing temperature.In this paper we focus in the case where the hyperbolic terms are much larger than the collision term for large times (hyperbolic-dominated behavior). In the hyperbolicdominated case it does not seem to be possible to describe in a simple way all the long time asymptotics of the solutions, but we discuss several physical situations and formulate precise conjectures. We give explicit formulas for the relationship between density, temperature and entropy for these solutions. These formulas differ greatly from the ones at equilibrium. *

Rescaled Objective Solutions of Fokker--Planck and Boltzmann equations

“…Results on the existence of self-similar profiles (i.e. equilibria for the shape equation) and long-time behaviour in the case of soft interaction potentials have been obtained in [23] and [24]. In [23] the existence of stationary self-similar solutions is established rigorously for Maxwellian molecules (where the repulsive force between particles at distance r is r −5 ) after isotropic rescaling (where η is a multiple of the identity).…”

“…Detailed information about energy flux can then be derived. Moreover, in [24] formal calculations covering the supercritical case where the force decays faster than r −5 are being presented.…”

“…It is noteworthy that the analysis in [23] and [24] also covers other objectivity conditions than the simple shear, for example homogeneous dilations where S = (Id, −Id), and other choices. In view of these results it will be worthwhile to extend our approach to other objectivity conditions and explore different choices for the collision operator in the Boltzmann equation.…”

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

“…The case in which the analogy between the entropy formulas for the equilibrium case and the considered solutions is the largest, nonsurprisingly, if the particle distribution is given by a Hilbert expansion (see details in [19]). However, there is also a large analogy between the entropy formulas of equilibrium distributions and self-similar solutions.…”

“…In this gap the collision rate is small but it still plays a significant role in the formal asymptotic behavior of the Boltzmann equation. A detailed justification for these conjectures can be found in [19].…”