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We consider a multi component gas mixture with translational and internal energy degrees of freedom assuming that the number of particles of each species remains constant. We will illustrate the derived model in the case of two species, but…

Analysis of PDEs · Mathematics 2018-07-02 Christian Klingenberg , Marlies Pirner , Gabriella Puppo

The phenomena of diffusion in multicomponent (more than two components) mixtures are very universal in both science and engineering, and from mathematical point of view, they are usually described by the Maxwell-Stefan (MS) based continuum…

Computational Physics · Physics 2019-02-27 Zhenhua Chai , Xiuya Guo , Lei Wang , Baochang Shi

Shakhov model is a relaxation approximation of the Boltzmann equation proposed to overcome the deficiency of the original BGK model, namely, the incorrect production of the Prandtl number. In this paper, we address the existence and the…

Analysis of PDEs · Mathematics 2021-11-02 Gi-Chan Bae , Seok-Bae Yun

We study the linearized kinetic equation of relaxation model which was proposed by Bhatnagar, Gross and Krook (also called BGK model) and solve the dispersion relation. Using the solution of the dispersion relation, we analyze the…

Solar and Stellar Astrophysics · Physics 2010-10-27 Makoto Takamoto , Shu-ichiro Inutsuka

This minicourse contains a description of recent results on the modelling of rarefied gases in weakly non equilibrium regimes, and the numerical methods used to approximate the resulting equations. Therefore this work focuses on BGK type…

Computational Physics · Physics 2019-02-25 Gabriella Puppo

Kinetic models based on the Bhatnagar-Gross-Krook (BGK) framework provide an efficient alternative to the Boltzmann equation for rarefied gas flows; however, existing formulations for gas mixtures remain limited in representing…

Fluid Dynamics · Physics 2026-04-28 Inchan Kim , Joonbeom Kim , Woonghwi Park , Eunji Jun

The Bhatnagar-Gross-Krook (BGK) single-relaxation-time collision model for the Boltzmann equation serves as the foundation of the lattice BGK (LBGK) method developed in recent years. The description of the collision as a uniform relaxation…

Numerical Analysis · Mathematics 2021-10-04 Xiaowen Shan , Yangyang Shi , Xuhui Li

The paper presents a solution to the Boltzmann kinetic equation based on the construction of its discrete conservative model. Discrete analogue of the collision integral is presented as a contraction of a tensor, which is independent from…

Statistical Mechanics · Physics 2017-07-04 George Arabuli

We propose a general multi-species Fokker-Planck model. We prove consistency of our model: conservation properties, positivity of all temperatures, H-Theorem and the shape of equilibrium as Maxwell distributions with the same mean velocity…

Analysis of PDEs · Mathematics 2024-09-25 Marlies Pirner

This paper considers a system of Boltzmann equations modelling the mixture of monatomic and polyatomic gases in an $L^{2}-L^{\infty}$ perturbation theory around global modified Maxwellians accounting for the internal energy of the mixture…

Analysis of PDEs · Mathematics 2025-11-20 Ricardo Alonso , Zongguang Li

In this paper, we are interested in the Cauchy problem for the Boltzmann-BGK model for a general class of collision frequencies. We prove that the Boltzmann-BGK model linearized around a global Maxwellian admits a unique global smooth…

Analysis of PDEs · Mathematics 2015-05-20 Seok-Bae Yun

The model of inelastic Maxwell particles (IMP) allows one to derive some exact results which show the strong influence of inelasticity on the nonequilibrium properties of a granular gas. The aim of this work is to propose a simple model…

Statistical Mechanics · Physics 2007-08-30 Andres Santos

In this paper we present a new ultra efficient numerical method for solving kinetic equations. In this preliminary work, we present the scheme in the case of the BGK relaxation operator. The scheme, being based on a splitting technique…

Mathematical Physics · Physics 2015-06-04 Giacomo Dimarco , Raphaël Loubere

Consistent BGK models for inert mixtures are compared, first in their kinetic behavior and then versus the hydrodynamic limits that can be derived in different collision-dominated regimes. The comparison is carried out both analytically and…

Mathematical Physics · Physics 2021-02-26 Sebastiano Boscarino , Seung Yeon Cho , Maria Groppi , Giovanni Russo

We consider a kinetic description of a multi-species gas mixture modeled with Bhatnagar-Gross-Krook (BGK) collision operators, in which the collision frequency varies not only in time and space but also with the microscopic velocity. In…

Numerical Analysis · Mathematics 2022-11-14 Jeffrey Haack , Cory Hauck , Christian Klingenberg , Marlies Pirner , Sandra Warnecke

Ideal gases can be modeled by the Boltzmann equation from statistical physics. Instead of trying to track the position and velocity of a large number of gas molecules, it is possible to describe the particles with a particle distribution…

Numerical Analysis · Mathematics 2025-11-27 Vienna B. Rossmanith

We study spatially non-homogeneous kinetic models for vehicular traffic flow. Classical formulations, as for instance the BGK equation, lead to unconditionally unstable solutions in the congested regime of traffic. We address this issue by…

Physics and Society · Physics 2019-07-22 Michael Herty , Gabriella Puppo , Sebastiano Roncoroni , Giuseppe Visconti

We consider the socalled Bathnagar-Gross-Krook (BGK) model, an approximation of the Boltzmann equation, describing the time evolution of a single momoatomic rarefied gas and satisfying the same two main properties (conservation properties…

Analysis of PDEs · Mathematics 2021-10-26 Marlies Pirner

Strong electric fields in the auroral and equatorial electrojets can distort the background ion distribution function away from Maxwellian. We developed a collisional plasma kinetic model using the Boltzmann equation and a simple BGK…

Plasma Physics · Physics 2024-08-13 Rattanakorn Koontaweepunya , Yakov S. Dimant , Meers M. Oppenheim

Dynamics of inelastic gases are studied within the framework of random collision processes. The corresponding Boltzmann equation with uniform collision rates is solved analytically for gases, impurities, and mixtures. Generally, the energy…

Statistical Mechanics · Physics 2007-05-23 E. Ben-Naim , P. L. Krapivsky