BGK model of the multi-species Uehling-Uhlenbeck equation

Gi-Chan Bae; Christian Klingenberg; Marlies Pirner; Seok-Bae Yun

doi:10.3934/krm.2020047

Article Contents

2021, Volume 14, Issue 1: 25-44. Doi: 10.3934/krm.2020047

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BGK model of the multi-species Uehling-Uhlenbeck equation

1.
Department of mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
2.
Department of mathematics, Würzburg University, Emil Fischer Str. 40, 97074 Würzburg, Germany
3.
Department of mathematics, Vienna University, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

Received: January 31, 2020

Revised: June 30, 2020

Early access: September 2020

Published: February 2021

Christian Klingenberg acknowledges support by the DFG grant KL-566/20-2. Marlies Pirner is supported by the Austrian Science Fund (FWF) project F65 and the Humboldt foundation. Seok-Bae Yun is supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-02

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Abstract

We propose a BGK model of the quantum Boltzmann equation for gas mixtures. We also provide a sufficient condition that guarantees the existence of equilibrium coefficients so that the model shares the same conservation laws and $ H $-theorem with the quantum Boltzmann equation. Unlike the classical BGK for gas mixtures, the equilibrium coefficients of the local equilibriums for quantum multi-species gases are defined through highly nonlinear relations that are not explicitly solvable. We verify in a unified way that such nonlinear relations uniquely determine the equilibrium coefficients under the condition, leading to the well-definedness of our model.

Erratum: The month information has been corrected from January to February. We apologize for any inconvenience this may cause.

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Mathematics Subject Classification: 82C40, 76P05, 35Q20, 35Q40, 81-10, 80-10.

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