Fluid dynamic limit of Boltzmann equation for granular hard--spheres in a nearly elastic regime

TL;DR

This paper rigorously derives a hydrodynamic limit from the Boltzmann equation for nearly elastic inelastic hard spheres, resulting in a new incompressible Navier-Stokes-Fourier system for granular flows.

Contribution

It provides the first rigorous derivation of hydrodynamic equations for inelastic hard spheres with small inelasticity, connecting microscopic and macroscopic descriptions.

Findings

Established the nearly elastic regime relating restitution coefficient and Knudsen number.

Proved exponential stability of the rescaled Boltzmann solution in a near-equilibrium regime.

Demonstrated convergence of the Boltzmann solution to a hydrodynamic system depending on macroscopic quantities.

Abstract

In this paper, we provide the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres with small inelasticity. The hydrodynamic system that we obtain is an incompressible Navier-Stokes-Fourier system with self-consistent forcing terms and is thus the first hydrodynamic system that properly describes rapid granular flows. To do that, we write our Boltzmann equation in nondimensional form introducing the dimensionless Knudsen number which is intended to tend to 0. The difficulties are then manyfold, the first one coming from the fact that the original Boltzmann equation is free-cooling and thus requires a self-similar change of variables to work with an equation that has an homogeneous steady state. The latter is not explicit and is heavy-tailed, which is a major obstacle to adapt energy estimates and spectral analysis. One of the main challenges here is to understand the relation between the restitution coefficient (which quantifies the loss of energy at the microscopic level) and the Knudsen number. This is done identifying the correct nearly elastic regime to capture nontrivial hydrodynamic behavior. We are then able to prove exponential stability uniformly with respect to the Knudsen number of the solution of our rescaled Boltzmann equation in a close to equilibrium regime. Finally, we prove that our solution to the Boltzmann equation converges in some very specific weak sense towards some hydrodynamic solution which depends on time and space variables only through macroscopic quantities. Such macroscopic quantities are solutions to a suitable modification of the incompressible Navier-Stokes-Fourier system which appears to be new in this context.