Solutions to the Boltzmann equation in the Boussinesq regime

TL;DR

This paper analyzes the Boltzmann equation in a Boussinesq regime, demonstrating the existence of solutions that approximate the Oberbeck-Boussinesq equations for small Knudsen numbers and temperature differences.

Contribution

It establishes the rigorous connection between the Boltzmann equation and Boussinesq fluid dynamics in a specific asymptotic regime.

Findings

Existence of solutions near a global Maxwellian.

Moments approximate density, velocity, and temperature from Oberbeck-Boussinesq equations.

Results valid up to the regularity time of the Boussinesq solution.

Abstract

We consider a gas in a horizontal slab, in which the top and bottom walls are kept at different temperatures. The system is described by the Boltzmann equation (BE) with Maxwellian boundary conditions specifying the wall temperatures. We study the behavior of the system when the Knudsen number $\epsilon$ is small and the temperature difference between the walls as well as the velocity field is of order $\epsilon$, while the gravitational force is of order $\epsilon^2$. We prove that there exists a solution to the BE for which is near a global Maxwellian, and whose moments are close, up to order $\epsilon^2$ to the density, velocity and temperature obtained from the smooth solution of the Oberbeck-Boussinesq equations, up to the time this one stays regular.