# 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

## References

- Cercignani, C. Rarefied Gas Dynamics, from Basic Concepts to Actual Calculations; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Cercignani, C. The Boltzmann Equation and Its Applications; Springer: Berlin/Heidelberg, Germany, 1975. [Google Scholar]
- Goldmann, E.; Sirovich, L. Equations for Gas Mixtures. Phys. Fluids
**1967**, 10, 1928–1940. [Google Scholar] [CrossRef] - Aoki, K.; Bardos, C.; Takata, S. Knudsen Layer for Gas Mixtures. J. Stat. Phys.
**2003**, 112, 629–655. [Google Scholar] [CrossRef] - Pirner, M. Kinetic Modelling of Gas Mixtures; Würzburg University Press: Würzburg, Germany, 2018. [Google Scholar]
- Boscarino, S.; Cho, S.Y.; Groppi, M.; Russo, G. BGK models for inert mixtures: Comparison and applications. arXiv
**2021**, arXiv:2102.12757. [Google Scholar] - Struchtrup, H. The BGK-model with velocity-dependent collision frequency. Contin. Mech. Thermodyn.
**1997**, 9, 23–31. [Google Scholar] [CrossRef] - Haack, J.; Hauck, C.; Klingenberg, C.; Pirner, M.; Warnecke, S. A consistent BGK model with velocity-dependent collision frequency for gas mixtures. J. Stat. Phys.
**2021**, 184, 31. [Google Scholar] [CrossRef] - Pieraccini, S.; Puppo, G. Implicit-explicit schemes for BGK kinetic equations. J. Sci. Comput.
**2007**, 32, 1–28. [Google Scholar] [CrossRef] [Green Version] - Filbet, F.; Jin, S. A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys.
**2010**, 20, 7625–7648. [Google Scholar] [CrossRef] [Green Version] - Dimarco, G.; Pareschi, L. Numerical methods for kinetic equations. Acta Numer.
**2014**, 23, 369–520. [Google Scholar] [CrossRef] [Green Version] - Bennoune, M.; Lemou, M.; Mieussens, L. Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics. J. Comput. Phys.
**2008**, 227, 3781–3803. [Google Scholar] [CrossRef] - Bernard, F.; Iollo, A.; Puppo, G. Accurate asymptotic preserving boundary conditions for kinetic equations on Cartesian grids. J. Sci. Comput.
**2015**, 65, 735–766. [Google Scholar] [CrossRef] [Green Version] - Crestetto, A.; Klingenberg, C.; Pirner, M. Kinetic/fluid micro-macro numerical scheme for a two component gas mixture. SIAM Multiscale Model. Simul.
**2020**, 18, 970–998. [Google Scholar] [CrossRef] - Holway, L.H. New statistical models for kinetic theory: Methods of construction. Phys. Fluids
**1966**, 9, 1658–1673. [Google Scholar] [CrossRef] - Perthame, B.; Pulvirenti, M. Weighted L
^{∞}Bounds and Uniqueness for the Boltzmann BGK Model. Arch. Ration. Mech. Anal.**1993**, 125, 289–295. [Google Scholar] [CrossRef] - Yun, S.-B. Classical solutions for the ellipsoidal BGK model with fixed collision frequency. J. Differ. Equ.
**2015**, 259, 6009–6037. [Google Scholar] [CrossRef] - Shakhov, E.M. Generalization of the Krook kinetic relaxation equation. Fluid Dyn.
**1968**, 3, 95–96. [Google Scholar] [CrossRef] - Asinari, P. Asymptotic analysis of multiple-relaxation-time lattice Boltzmann schemes for mixture modeling. Comput. Math. Appl.
**2008**, 55, 1392–1407. [Google Scholar] [CrossRef] [Green Version] - Garzó, V.; Santos, A.; Brey, J.J. A kinetic model for a multicomponent gas. Phyics Fluids A
**1989**, 1, 380–383. [Google Scholar] [CrossRef] - Greene, J. Improved Bhatnagar-Gross-Krook model of electron-ion collisions. Phys. Fluids
**1973**, 16, 2022–2023. [Google Scholar] [CrossRef] - Gross, E.P.; Krook, M. Model for collision processes in gases: Small-amplitude oscillations of charged two-component systems. Phys. Rev.
**1956**, 102, 593. [Google Scholar] [CrossRef] - Hamel, B. Kinetic model for binary gas mixtures. Phys. Fluids
**2004**, 8, 418–425. [Google Scholar] [CrossRef] - Sofonea, V.; Sekerka, R. BGK models for diffusion in isothermal binary fluid systems. Physica
**2001**, 3, 494–520. [Google Scholar] [CrossRef] - Bobylev, A.V.; Bisi, M.; Groppi, M.; Spiga, G.; Potapenko, I.F. A general consistent BGK model for gas mixtures. Kinet. Relat. Model.
**2018**, 11, 1377. [Google Scholar] [CrossRef] [Green Version] - Haack, J.R.; Hauck, C.D.; Murillo, M.S. A conservative, entropic multispecies BGK model. J. Stat. Phys.
**2017**, 168, 826–856. [Google Scholar] [CrossRef] - Klingenberg, C.; Pirner, M.; Puppo, G. A consistent kinetic model for a two-component mixture with an application to plasma. Kinet. Relat. Model.
**2017**, 10, 445–465. [Google Scholar] [CrossRef] [Green Version] - Andries, P.; Aoki, K.; Perthame, B. A consistent BGK-type model for gas mixtures. J. Stat. Phys.
**2002**, 106, 993–1018. [Google Scholar] [CrossRef] - Brull, S.; Pavan, V.; Schneider, J. Derivation of a BGK model for mixtures. Eur. J. Mech. B/Fluids
**2012**, 33, 74–86. [Google Scholar] [CrossRef] - Groppi, M.; Monica, S.; Spiga, G. A kinetic ellipsoidal BGK model for a binary gas mixture. EPL J.
**2011**, 96, 64002. [Google Scholar] [CrossRef] - Brull, S. An ellipsoidal statistical model for gas mixtures. Commun. Math. Sci.
**2015**, 8, 1–13. [Google Scholar] [CrossRef] - Todorova, B.; Steijl, R. Derivation and numerical comparison of Shakov and Ellipsoidal Statistical kinetic models for a monoatomic gas mixture. Eur. J. Mech.-B/Fluids
**2019**, 76, 390–402. [Google Scholar] [CrossRef] [Green Version] - Klingenberg, C.; Pirner, M.; Puppo, G. Kinetic ES-BGK models for a multicomponent gas mixture. In Proceedings in Mathematics and Statistics of the International Conference on Hyperbolic Problems: Theory, Numeric and Applications; Springer: Cham, Switzerland, 2016. [Google Scholar]
- Groppi, M.; Russo, G.; Stracquadanio, G. Semi-Lagrangian Approximation of BGK Models for Inert and Reactive Gas Mixtures. In Meeting on Particle Systems and PDE’s; Springer: Cham, Switzerland, 2018; pp. 53–80. [Google Scholar]
- Bellan, P.M. Fundamentals of Plasma Physics; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Perthame, B. Global existence to the BGK model of Boltzmann equation. J. Differ. Equ.
**1989**, 82, 191–205. [Google Scholar] [CrossRef] [Green Version] - DiPerna, R.J.; Lions, P.-L. On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math.
**1989**, 130, 321–366. [Google Scholar] [CrossRef] - Ukai, S. Stationary solutions of the BGK model equation on a finite interval with large boundary data. Transp. Theory Statist. Phys.
**1992**, 21, 487–500. [Google Scholar] [CrossRef] - Yun, S.-B. Cauchy problem for the Boltzmann-BGK model near a global Maxwellian. J. Math. Phys.
**2010**, 51, 123514. [Google Scholar] [CrossRef] [Green Version] - Desvillettes, L. Convergence to equilibrium in large time for Boltzmann and B.G.K. equations. Arch. Ration. Mech. Anal.
**1990**, 110, 73–91. [Google Scholar] [CrossRef] - Saint-Raymond, L. From the BGK model to the Navier-Stokes equations. Ann. Sci. École Norm. Sup.
**2003**, 36, 271–317. [Google Scholar] [CrossRef] [Green Version] - Klingenberg, C.; Pirner, M. Existence, Uniqueness and Positivity of solutions for BGK models for mixtures. J. Differ. Equs.
**2018**, 264, 207–227. [Google Scholar] [CrossRef] [Green Version] - Liu, L.; Pirner, M. Hypocoercivity for a BGK model for gas mixtures. J. Differ. Equ.
**2019**, 267, 119–149. [Google Scholar] [CrossRef] [Green Version] - Achleitner, F.; Arnold, A.; Carlen, E. On multi-dimensional hypocoercive BGK models. Kinet. Relat. Models
**2018**, 11, 953–1009. [Google Scholar] [CrossRef] [Green Version] - Morse, T.F. Kinetic Model for Gases with Internal Degrees of Freedom. Phys. Fluids
**1964**, 7, 159–169. [Google Scholar] [CrossRef] - Andries, P.; Perthame, B. The ES-BGK model equation with correct Prandtl number. AIP Conf. Proc.
**2001**, 30, 30–36. [Google Scholar] - Mathiaud, J.; Mieussens, L. BGK and Fokker-Planck models of the Boltzmann equation for gases with discrete levels of vibrational energy. J. Stat. Phys.
**2020**, 494, 1076–1095. [Google Scholar] [CrossRef] [Green Version] - Bernard, F.; Iollo, A.; Puppo, G. BGK Polyatomic Model for Rarefied Flow. J. Sci. Comput.
**2019**, 78, 1893–1916. [Google Scholar] [CrossRef] [Green Version] - Park, S.J.; Yun, S.-B. Cauchy problem for the ellipsoidal BGK model for polyatomic particles. J. Differ. Equ.
**2019**, 266, 7678–7708. [Google Scholar] [CrossRef] [Green Version] - Park, S.; Yun, S. Entropy production estimates for the polyatomic ellipsoidal BGK model. Appl. Math. Lett.
**2016**, 58, 26–33. [Google Scholar] [CrossRef] [Green Version] - Brull, S.; Schneider, J. On the ellipsoidal statistical model for polyatomic gases. Contin. Mech. Thermodyn.
**2009**, 20, 489–508. [Google Scholar] [CrossRef] [Green Version] - Pirner, M. A BGK model for gas mixtures of polyatomic molecules allowing for slow and fast relaxation of the temperatures. J. Stat. Phys.
**2018**, 173, 1660–1687. [Google Scholar] [CrossRef] [Green Version] - Todorova, B.; White, C.; Steijl, R. Modeling of nitrogen and oxygen gas mixture with a novel diatomic kinetic model. AIP Adv.
**2020**, 10, 095218. [Google Scholar] [CrossRef] - Bisi, M.; Monaco, R.; Soares, A.J. A BGK model for reactive mixtures of polyatomic gases with continuous internal energy. J. Phys. A Math. Theor.
**2018**, 51, 125501. [Google Scholar] [CrossRef] [Green Version] - Bisi, M.; Cáceres, M. A BGK relaxation model for polyatomic gas mixtures. Commun. Math. Sci.
**2016**, 14, 297–325. [Google Scholar] [CrossRef] - Tantos, C.; Varoutis, S.; Day, C. Heat transfer in binary polyatomic gas mixtures over the whole range of the gas rarefaction based on kinetic deterministic modeling. Phys. Fluids
**2021**, 33, 022004. [Google Scholar] [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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

**Chicago/Turabian Style**

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