Hydrodynamics from the Dissipative Boltzmann Equation

Abstract

This paper deals with some questions related with the modelling of hydrodynamic equations for granular gases, at the light of recent mathematical results on the large-time behavior of the dissipative Boltzmann equation. This subject is relatively new, and the relevant mathematical theory is still restricted. In the pertinent literature [Duf01], rapid granular flows were frequently described at the macroscopic level by means of equations for fluid dynamics, modified to account for dissipation due to collisions among particles. This was the approach of Haff, which, in his pioneering paper [Haf83], gave a macroscopic description of the behavior of a granular material treating the individual grains as the molecules of a granular fluid, without resorting to the mesoscopic picture (the Boltzmann or Enskog kinetic equations). In more recent years it became clear that, in agreement with the well established derivation of conservative fluid dynamics from the Boltzmann equation [BGL91, BGL93], kinetic theory was the basis for a deeper understanding of macroscopic equations even for dissipative flows. Kinetic theory is suitable to describe the evolution of materials composed of many small discrete grains, in which the mean free path of the grains is much larger than the typical particle size. In this regime, granular gases can be described within the concepts of classical statistical mechanics, by adapting methods of the kinetic theory of ideal gases [Kog69, CIP94]. Many authors (see Refs. [NY93, DLK95, BCP97, EP97, BCG00, BP00a] and the references therein) adopted this line of thought, by introducing and discussing the evolution of a system of partially inelastic rigid spheres through Boltzmann-like equations. A typical kinetic model for the study of the evolution of a granular material takes the following form: the unknown is a time dependent density in phase space f(x, v, t) satisfying a Boltzmann–Enskog equation for inelastic hard–spheres, which for the force–free case reads