# A Review on BGK Models for Gas Mixtures of Mono and Polyatomic Molecules

## Abstract

**:**

## 1. Introduction

## 2. BGK Models for Gas Mixtures

#### 2.1. Overview on Existing BGK Models for Gas Mixtures in the Literature

#### 2.1.1. BGK Models for Gas Mixtures with One Collision Term

#### 2.1.2. BGK Models for Gas Mixtures with Two Collision Terms

#### 2.2. Theoretical Results of BGK Models for Gas Mixtures

#### 2.2.1. Existence of Solutions

- 1.
- We assume periodic boundary conditions in x. Equivalently, we can construct solutions satisfying$${f}_{k}(t,{x}_{1},...,{x}_{d},{v}_{1},...,{v}_{d})={f}_{k}(t,{x}_{1},...,{x}_{i-1},{x}_{i}+{a}_{i},{x}_{i+1},...{x}_{d},{v}_{1},...{v}_{d})$$
- 2.
- We require that the initial values ${f}_{k}^{0},i=1,2$ satisfy assumption 1.
- 3.
- We are on the bounded domain in space $\mathsf{\Lambda}=\{x\in {\mathbb{R}}^{N}|{x}_{i}\in (0,{a}_{i})\}$.
- 4.
- Suppose that ${f}_{k}^{0}$ satisfies ${f}_{k}^{0}\ge 0$, ${(1+|v|}^{2}){f}_{k}^{0}\in {L}^{1}(\mathsf{\Lambda}\times {\mathbb{R}}^{d})$ with$\int {f}_{k}^{0}dxdv=1,k=1,2$.
- 5.
- Suppose ${N}_{q}\left({f}_{k}^{0}\right):=sup{f}_{k}^{0}(x,v){(1+|v|}^{q})=\frac{1}{2}{A}_{0}<\infty $ for some $q>d+2$.
- 6.
- Suppose ${\gamma}_{k}(x,t):=\int {f}_{k}^{0}(x-vt,v)dv\ge {C}_{0}>0$ for all $t\in \mathbb{R}.$
- 7.
- Assume that the collision frequencies are written as$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\nu}_{jk}(x,t){n}_{k}(x,t)={\tilde{\nu}}_{jk}\frac{{n}_{k}(x,t)}{{n}_{j}(x,t)+{n}_{k}(x,t)},\phantom{\rule{1.em}{0ex}}j,k=1,2,\end{array}\end{array}$$

**Definition**

**1.**

**Theorem**

**1.**

#### 2.2.2. Large-Time Behaviour

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

## 3. BGK Models for Gas Mixtures of Polyatomic Molecules

- Discrete dependency on the degrees of freedom in internal energy:

- A continuous scalar dependency on the degrees of freedom in internal energy:

- A discrete and continuous dependency of the degrees of freedom in internal energy:

- A vector-valued continuous dependency of the degrees of freedom in internal energy:

- Relaxation to an equilibrium distribution with equal temperature:

- Relaxation of the temperature due to a convex combination of temperatures in the Maxwell distribution:

**Theorem**

**6.**

- Relaxation of the temperatures with an additional kinetic equation:

**Theorem**

**7.**

- Relaxation of the temperatures with an additional relaxation term:

#### 3.1. Summary of Existing BGK Models for Gas Mixtures of Polyatomic Molecules in the Literature

#### 3.1.1. A BGK Model for Mixtures of Polyatomic Gases with One Relaxation Term

#### 3.1.2. A BGK Model for Mixtures of Polyatomic Gases with Two Relaxation Terms

**Theorem**

**8.**

#### 3.2. BGK Model for Mixtures of Polyatomic Gases with Intermediate Relaxation Terms

## 4. Conclusions

## Funding

## Conflicts of Interest

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Pirner, M. A Review on BGK Models for Gas Mixtures of Mono and Polyatomic Molecules. *Fluids* **2021**, *6*, 393.
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Pirner M. A Review on BGK Models for Gas Mixtures of Mono and Polyatomic Molecules. *Fluids*. 2021; 6(11):393.
https://doi.org/10.3390/fluids6110393

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Pirner, Marlies. 2021. "A Review on BGK Models for Gas Mixtures of Mono and Polyatomic Molecules" *Fluids* 6, no. 11: 393.
https://doi.org/10.3390/fluids6110393