# BGK and Fokker-Planck models of the Boltzmann equation for gases with discrete levels of vibrational energy

## TL;DR

This paper introduces BGK and Fokker-Planck models for the Boltzmann equation tailored to diatomic gases with discrete vibrational energy levels, crucial for high-temperature flow simulations such as atmospheric re-entry.

## Contribution

It develops and analyzes new kinetic models that incorporate discrete vibrational energy modes, ensuring conservation, entropy properties, and deriving Navier-Stokes asymptotics.

## Key findings

- Models satisfy conservation laws
- Models adhere to H-theorem (entropy increase)
- Derived Navier-Stokes asymptotics for these models

## Abstract

We propose two models of the Boltzmann equation (BGK and Fokker-Planck models) for rarefied flows of diatomic gases in vibrational non-equilibrium. These models take into account the discrete repartition of vibration energy modes, which is required for high temperature flows, like for atmospheric re-entry problems. We prove that these models satisfy conservation and entropy properties (H-theorem), and we derive their corresponding compressible Navier-Stokes asymptotics.