Cauchy problem for the ellipsoidal-BGK model of the Boltzmann equation

2016 ◽  
Vol 57 (8) ◽  
pp. 081512 ◽  
Author(s):  
Sa Jun Park ◽  
Seok-Bae Yun
Keyword(s):  
Cauchy Problem ◽  
Boltzmann Equation ◽  
Bgk Model ◽  
The Boltzmann Equation

THE CONTRIBUTION OF THE BHATNAGAR–GROSS–KROOK MODEL TO THE DEVELOPMENT OF RAREFIED GAS DYNAMICS IN THE EARLY YEARS OF THE SPACE AGE

2013 ◽  
Vol 25 (01) ◽  
pp. 1340025
Author(s):  
RODDAM NARASIMHA
Keyword(s):  
Boltzmann Equation ◽  
Gas Dynamics ◽  
Rarefied Gas ◽  
Rarefied Gas Dynamics ◽  
Early Years ◽  
Plane Shock ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Nonequilibrium Flows ◽  
Space Age

The advent of the space age in 1957 was accompanied by a sudden surge of interest in rarefied gas dynamics (RGD). The well-known difficulties associated with solving the Boltzmann equation that governs RGD made progress slow but the Bhatnagar–Gross–Krook (BGK) model, proposed three years before Sputnik, turned out to have been an uncannily timely, attractive and fruitful option, both for gaining insights into the Boltzmann equation and for estimating various technologically useful flow parameters. This paper gives a view of how BGK contributed to the growth of RGD during the first decade of the space age. Early efforts intended to probe the limits of the BGK model showed that, in and near both the continuum Euler limit and the collisionless Knudsen limit, BGK could provide useful answers. Attempts were therefore made to tackle more ambitious nonlinear nonequilibrium problems. The most challenging of these was the structure of a plane shock wave. The first exact numerical solutions of the BGK equation for the shock appeared during 1962 to 1964, and yielded deep insights into the character of transitional nonequilibrium flows that had resisted all attempts at solution through the Boltzmann equation. In particular, a BGK weak shock was found to be amenable to an asymptotic analysis. The results highlighted the importance of accounting separately for fast-molecule dynamics, most strikingly manifested as tails in the distribution function, both in velocity and in physical space — tails are strange versions or combinations of collisionless and collision-generated flows. However, by the mid-1960s Monte-Carlo methods of solving the full Boltzmann equation were getting to be mature and reliable and interest in the BGK waned in the following years. Interestingly, it has seen a minor revival in recent years as a tool for developing more effective algorithms in continuum computational fluid dynamics, but the insights derived from the BGK for strongly nonequilibrium flows should be of lasting value.

scholarly journals THE EINSTEIN–BOLTZMANN SYSTEM AND POSITIVITY

2013 ◽  
Vol 10 (01) ◽  
pp. 77-104 ◽  
Author(s):  
HO LEE ◽  
ALAN D. RENDALL
Keyword(s):  
Cauchy Problem ◽  
General Relativity ◽  
Boltzmann Equation ◽  
Cross Section ◽  
Collision Cross Section ◽  
Curved Spacetime ◽  
The Boltzmann Equation ◽  
The Cauchy Problem

The Einstein–Boltzmann (EB) system is studied, with particular attention to the non-negativity of the solution of the Boltzmann equation. A new parametrization of post-collisional momenta in general relativity is introduced and then used to simplify the conditions on the collision cross-section given by Bancel and Choquet-Bruhat. The non-negativity of solutions of the Boltzmann equation on a given curved spacetime has been studied by Bichteler and Tadmon. By examining to what extent the results of these authors apply in the framework of Bancel and Choquet-Bruhat, the non-negativity problem for the EB system is resolved for a certain class of scattering kernels. It is emphasized that it is a challenge to extend the existing theory of the Cauchy problem for the EB system so as to include scattering kernels which are physically well-motivated.

scholarly journals Convergence of a Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

2012 ◽  
Vol 50 (3) ◽  
pp. 1111-1135 ◽  
Author(s):  
Giovanni Russo ◽  
Pietro Santagati ◽  
Seok-Bae Yun
Keyword(s):  
Boltzmann Equation ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Lagrangian Scheme

The study of the Boltzmann equation of solid-gas two-phase flow with three-dimensional BGK model

2018 ◽  
Author(s):  
Chang-jiang Liu ◽  
Song Pang ◽  
Qiang Xu ◽  
Ling He ◽  
Shao-peng Yang ◽  
...  
Keyword(s):  
Boltzmann Equation ◽  
Two Phase Flow ◽  
Three Dimensional ◽  
Phase Flow ◽  
Bgk Model ◽  
Two Phase ◽  
The Boltzmann Equation

"ETERNAL" SOLUTIONS FOR A MODEL OF THE BOLTZMANN EQUATION

1999 ◽  
Vol 09 (01) ◽  
pp. 127-137 ◽  
Author(s):  
HENRI CABANNES
Keyword(s):  
Cauchy Problem ◽  
Boltzmann Equation ◽  
Plane Model ◽  
The Boltzmann Equation ◽  
Eternal Solutions ◽  
The Cauchy Problem

This paper deals with the analysis of the so-called "eternal" solutions to the Cauchy problem for a semidiscrete plane model of the Boltzmann equation. By eternal solutions we mean solutions existing globally for both positive and negative values of time.

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