# American Institute of Mathematical Sciences

February  2021, 14(1): 25-44. doi: 10.3934/krm.2020047

## 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  February 2020 Revised  July 2020 Published  February 2021 Early access  September 2020

Fund Project: 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

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.

Citation: Gi-Chan Bae, Christian Klingenberg, Marlies Pirner, Seok-Bae Yun. BGK model of the multi-species Uehling-Uhlenbeck equation. Kinetic & Related Models, 2021, 14 (1) : 25-44. doi: 10.3934/krm.2020047
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##### References:
 [1] Marcel Braukhoff. Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation. Kinetic & Related Models, 2020, 13 (1) : 187-210. doi: 10.3934/krm.2020007 [2] Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic & Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009 [3] Michael Herty, Gabriella Puppo, Sebastiano Roncoroni, Giuseppe Visconti. The BGK approximation of kinetic models for traffic. Kinetic & Related Models, 2020, 13 (2) : 279-307. doi: 10.3934/krm.2020010 [4] Bertrand Lods, Clément Mouhot, Giuseppe Toscani. Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models. Kinetic & Related Models, 2008, 1 (2) : 223-248. doi: 10.3934/krm.2008.1.223 [5] Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 219-251. doi: 10.3934/krm.2014.7.219 [6] Raffaele Esposito, Mario Pulvirenti. Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas. Kinetic & Related Models, 2010, 3 (2) : 281-297. doi: 10.3934/krm.2010.3.281 [7] Karsten Matthies, George Stone, Florian Theil. The derivation of the linear Boltzmann equation from a Rayleigh gas particle model. Kinetic & Related Models, 2018, 11 (1) : 137-177. doi: 10.3934/krm.2018008 [8] Alexander Bobylev, Mirela Vinerean, Åsa Windfäll. Discrete velocity models of the Boltzmann equation and conservation laws. Kinetic & Related Models, 2010, 3 (1) : 35-58. doi: 10.3934/krm.2010.3.35 [9] Seiji Ukai. Time-periodic solutions of the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 579-596. doi: 10.3934/dcds.2006.14.579 [10] Lingbing He, Yulong Zhou. High order approximation for the Boltzmann equation without angular cutoff. Kinetic & Related Models, 2018, 11 (3) : 547-596. doi: 10.3934/krm.2018024 [11] Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic & Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019 [12] Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575 [13] Karsten Matthies, George Stone. Derivation of a non-autonomous linear Boltzmann equation from a heterogeneous Rayleigh gas. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3299-3355. doi: 10.3934/dcds.2018143 [14] Arthur Henrique Caixeta, Irena Lasiecka, Valéria Neves Domingos Cavalcanti. On long time behavior of Moore-Gibson-Thompson equation with molecular relaxation. Evolution Equations & Control Theory, 2016, 5 (4) : 661-676. doi: 10.3934/eect.2016024 [15] Ling-Bing He, Jie Ji, Ling-Xuan Shao. Lower bound for the Boltzmann equation whose regularity grows tempered with time. Kinetic & Related Models, 2021, 14 (4) : 705-724. doi: 10.3934/krm.2021020 [16] Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145 [17] Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic & Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014 [18] Zheng-an Yao, Yu-Long Zhou. High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials. Kinetic & Related Models, 2020, 13 (3) : 435-478. doi: 10.3934/krm.2020015 [19] Esther S. Daus, Shi Jin, Liu Liu. Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel. Kinetic & Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034 [20] Alexander V. Bobylev, Marzia Bisi, Maria Groppi, Giampiero Spiga, Irina F. Potapenko. A general consistent BGK model for gas mixtures. Kinetic & Related Models, 2018, 11 (6) : 1377-1393. doi: 10.3934/krm.2018054

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