From Boltzmann equation for granular gases to a modified Navier-Stokes-Fourier system

TL;DR

This paper rigorously derives a modified Navier-Stokes-Fourier system from the Boltzmann equation for inelastic granular gases, establishing existence, stability, and convergence of solutions in a specific hydrodynamic regime.

Contribution

It provides the first rigorous derivation of hydrodynamic equations for granular flows from the Boltzmann equation, including a new system and stability analysis.

Findings

Existence of classical solutions to the modified hydrodynamic system.

Uniform exponential stability of solutions near thermal equilibrium.

Convergence of solutions to a hydrodynamic limit described by a modified Navier-Stokes-Fourier system.

Abstract

In this paper, we give an overview of the results established in [3] which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system of hydrodynamic equations describing granular flows and prove existence of classical solutions to the aforementioned system. One of the main issue is to identify the correct relation between the restitution coefficient (which quantifies the rate of energy loss at the microscopic level) and the Knudsen number which allows us to obtain non trivial hydrodynamic behavior. In such a regime, we construct strong solutions to the inelastic Boltzmann equation, near thermal equilibrium whose role is played by the so-called homogeneous cooling state. We prove then the uniform exponential stability with respect to the Knudsen number of such solutions, using a spectral analysis of the linearized problem combined with technical a priori nonlinear estimates. Finally, we prove that such solutions converge, in a specific weak sense, towards some hydrodynamic limit that depends on time and space variables only through macroscopic quantities that satisfy a suitable modification of the incompressible Navier-Stokes-Fourier system.