Local in time solution to ES-BGK model with correct Prandtl number

Sung-Jun Son; Seok-Bae Yun

doi:10.3934/krm.2024024

Article Contents

2025, Volume 18, Issue 4: 499-519. Doi: 10.3934/krm.2024024

This issue Previous Article Next Article Explicit and approximate solutions for the fragmentation equation in the presence of source and efflux terms: A coupled meshfree approach and its convergence analysis

Local in time solution to ES-BGK model with correct Prandtl number

Sung-Jun Son^1, and
Seok-Bae Yun^2, ,

1.
Department of Mathematics, Pohang University of Science and Technology, South Korea
2.
Department of mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

^*Corresponding author: Seok-Bae Yun
^*Corresponding author: Seok-Bae Yun

Received: March 2024

Revised: October 2024

Early access: November 2024

Published: August 2025

Seok-Bae Yun is supported by The National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2023R1A2C1005737). Sung-Jun Son was supported by The National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2023-00219980).

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we study the existence of a unique mild solution for the ellipsoidal BGK model with the critical Prandtl parameter $ \nu = -1/2 $. The key difficulty is the breakdown of the equivalence between the local temperature and the temperature tensor in this critical case. To overcome this, we use the fact that the quadratic polynomial of the temperature tensor can be written as the difference of the directional temperatures, which enables us to control the temperature tensor from below for a finite time.

Keywords:

Mathematics Subject Classification: Primary: 80C40, 35Q35; Secondary: 76P05.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	P. Andries, J.-F. Bourgat, P. Le Tallec and B. Perthame, Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases, Comput. Methods Appl. Mech. Engrg., 191 (2002), 3369-3390. doi: 10.1016/S0045-7825(02)00253-0.
[2]	P. Andries, P. Le Tallec, J.-P. Perlat and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, Eur. J. Mech. B Fluids, 19 (2000), 813-830. doi: 10.1016/S0997-7546(00)01103-1.
[3]	G.-C. Bae, C. Klingenberg, M. Pirner and S.-B. Yun, BGK model for two-component gases near a global Maxwellian, SIAM J. Math. Anal., 55 (2023), 1007-1047. doi: 10.1137/22M1469535.
[4]	G.-C. Bae and S.-B. Yun, Quantum BGK model near a global Fermi-Dirac distribution, SIAM J. Math. Anal., 52 (2020), 2313-2352. doi: 10.1137/19M1270021.
[5]	G.-C. Bae and S.-B. Yun, Stationary quantum BGK model for bosons and fermions in a bounded interval, J. Stat. Phys., 178 (2020), 845-868. doi: 10.1007/s10955-019-02466-2.
[6]	J. Bang and S.-B. Yun, Stationary solutions fohe ellipsoidal BGK model in a slab, J. Differential Equations, 261 (2016), 5803-5828. doi: 10.1016/j.jde.2016.08.022.
[7]	A. Bellouquid, Global existence and large-time behavior for BGK model for a gas with non-constant cross section, Transport Theory Statist. Phys., 32 (2003), 157-184.
[8]	P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. Small amplitude process in charge and neutral one-component systems, Physical Revies, 94 (1954), 511-525. doi: 10.1103/PhysRev.94.511.
[9]	S. Boscarino and S.-Y. Cho, On the order reduction of semi-Lagrangian methods for BGK model of Boltzmann equation, Appl. Math. Lett., 123 (2022), Paper No. 107488, 9 pp. doi: 10.1016/j.aml.2021.107488.
[10]	S. Boscarino, S.-Y. Cho and G. Russo, A local velocity grid conservative semi-Lagrangian schemes for BGK model, J. Comput. Phys., 460 (2022), Paper No. 111178, 25 pp. doi: 10.1016/j.jcp.2022.111178.
[11]	S. Boscarino, S.-Y. Cho, G. Russo and S.-B. Yun, Convergence estimates of a semi-Lagrangian scheme for the ellipsoidal BGK model for polyatomic molecules, ESAIM Math. Model. Numer. Anal., 56 (2022), 893-942. doi: 10.1051/m2an/2022022.
[12]	S. Boscarino, S.-Y. Cho, G. Russo and S.-B. Yun, High order conservative semi-Lagrangian scheme for the BGK model of the Boltzmann equation, Commun. Comput. Phys., 29 (2021), 1-56. doi: 10.4208/cicp.OA-2020-0050.
[13]	S. Brull and J. Schneider, A new approach for the ellipsoidal statistical model, Contin. Mech. Thermodyn., 20 (2008), 63-74. doi: 10.1007/s00161-008-0068-y.
[14]	S. Brull and J. Schneider, On the ellipsoidal statistical model for polyatomic gases, Contin. Mech. Thermodyn., 20 (2009), 489-508. doi: 10.1007/s00161-009-0095-3.
[15]	S. Brull and S.-B. Yun, Stationary flows of the ES-BGK model with the correct Prandtl number, SIAM J. Math. Anal., 56 (2024), 6361-6397. doi: 10.1137/23M1599628.
[16]	W. Chan, An Energy Method for the BGK Model, Ph.D thesis, City University of Hong Kong in Hong Kong, 2007.
[17]	Z. Chen, Smooth solution to the BGK equation and the ES-BGK equation with infinite energy, J. Differential Equations, 265 (2018), 389-416. doi: 10.1016/j.jde.2018.02.037.
[18]	Z. Chen and X. Zhang, Global existence and uniqueness to the Cauchy problem of the BGK equation with infinite energy, Math. Methods Appl. Sci., 39 (2016), 3116-3135. doi: 10.1002/mma.3757.
[19]	S.-Y. Cho, S. Boscarino, G. Russo and S.-B. Yun, Conservative semi-Lagrangian schemes for kinetic equations Part Ⅰ. Reconstruction, J. Comput. Phys., 432 (2021), Paper No. 110159, 30 pp. doi: 10.1016/j.jcp.2021.110159.
[20]	L. H. Holway, Kinetic theory of shock structure using and ellipsoidal distribution function, Rarefied Gas Dynamics, Proc. Fourth Internat. Sympos., Univ. Toronto, I (1964), 193-215.
[21]	B.-H. Hwang, Global existence of bounded solutions to the relativistic BGK model, Nonlinear Anal. Real World Appl., 63 (2022), Paper No. 103409, 24 pp. doi: 10.1016/j.nonrwa.2021.103409.
[22]	B.-H. Hwang, H. Lee and S.-B. Yun, Relativistic BGK model for massless particles in the FLRW spacetime, Kinet. Relat. Models, 14 (2021), 949-959. doi: 10.3934/krm.2021031.
[23]	B.-H. Hwang, T. Ruggeri and S.-B. Yun, On a relativistic BGK model for polyatomic gases near equilibrium, SIAM J. Math. Anal., 54 (2022), 2906-2947. doi: 10.1137/21M1404946.
[24]	B.-H. Hwang and S.-B. Yun, Ellipsoidal BGK model near a global Maxwellian in the whole space, J. Math. Phys., 60 (2019), 071507, 28 pp. doi: 10.1063/1.5017899.
[25]	B.-H. Hwang and S.-B. Yun, Stationary solutions to the boundary value problem for the relativistic BGK model in a slab, Kinet. Relat. Models, 12 (2019), 749-764. doi: 10.3934/krm.2019029.
[26]	D. Issautier, Convergence of a weighted particle method for solving the Boltzmann (B.G.K.) equaiton, Siam Journal on Numerical Analysis, 33 (1996), 2099-2119. doi: 10.1137/S0036142994266856.
[27]	D. Kim, M.-S. Lee and S.-B. Yun, Entropy production estimate for the ES-BGK model with the correct Prandtl number, J. Math. Anal. Appl., 514 (2022), Paper No. 126323, 15 pp.. doi: 10.1016/j.jmaa.2022.126323.
[28]	C. Klingenberg and M. Pirner, Existence uniqueness and positivity of solutions for BGK models for mixtures, J. Differential Equations, 264 (2018), 702-727. doi: 10.1016/j.jde.2017.09.019.
[29]	S. Mischler, Uniqueness for the BGK-equation in $\mathbb{R}^n$ and the rate of convergence for a semi-discrete scheme, Differential Integral Equations, 9 (1996), 1119-1138. doi: 10.57262/die/1367871533.
[30]	J. Oh, S.-Y. Cho, S.-B. Yun, E. Park and Y. Hong, Separable physics-informed neural networks for solving the BGK model of the Boltzmann equation, submitted, 2024. arXiv: 2403.06342.
[31]	S.-J. Park and S.-B. Yun, Cauchy problem for the ellipsoidal-BGK model of the Boltzmann equation, J. Math. Phys., 57 (2016), 081512, 19 pp. doi: 10.1063/1.4960745.
[32]	S.-J. Park and S.-B. Yun, Cauchy problem for the ellipsoidal-BGK model for polyatomic particles, J. Differential Equations, 266 (2019), 7678-7708. doi: 10.1016/j.jde.2018.12.013.
[33]	S.-J. Park and S.-B. Yun, Entropy production estimates for the polyatomic ellipsoidal BGK model, Appl. Math. Lett., 58 (2016), 26-33. doi: 10.1016/j.aml.2016.01.021.
[34]	B. Perthame, Global existence to the BGK model of Boltzmann equation, J. Differential Equations, 82 (1989), 191-205. doi: 10.1016/0022-0396(89)90173-3.
[35]	B. Perthame and M. Pulvirenti, Weighted $L^{\infty}$ bounds and uniqueness for the Boltzmann BGK model, Arch. Rational Mech. Anal., 125 (1993), 289-295. doi: 10.1007/BF00383223.
[36]	G. Russo, P. Santagati and S.-B. Yun, Convergence of a semi-Lagrangian scheme for the BGK model of the Boltzmann equation, SIAM J. Numer. Anal., 50 (2012), 1111-1135. doi: 10.1137/100800348.
[37]	S.-J. Son and S.-B. Yun, Cauchy problem for the ES-BGK Model with correct Prandtl number, Partial Differ. Equ. Appl., 3 (2022), Paper No. 41, 9 pp. doi: 10.1007/s42985-022-00175-2.
[38]	S.-J. Son and S.-B. Yun, The ES-BGK for the polyatomic molecules with infinite energy, J. Stat. Phys., 190 (2023), Paper No. 129, 27 pp. doi: 10.1007/s10955-023-03139-x.
[39]	S. Ukai, Stationary solutions of the BGK model equation on a finite interval with large boundary data, Transport Theory Statist. Phys., 21 (1992), 487-500. doi: 10.1080/00411459208203795.
[40]	J. Wei and X. Zhang, On the BGK equation with some force field in $L^p$-space, Nonlinear. Anal., 85 (2013), 52-65. doi: 10.1016/j.na.2013.02.025.
[41]	S.-B. Yun, Ellipsoidal BGK model for polyatomic particles near a global Maxwellian: A dichotomy in the dissipation estimate, J. Differential Equations, 266 (2019), 5566-5614.
[42]	S.-B. Yun, Ellipsoidal BGK model near a global Maxwellian, SIAM J. Math. Anal., 47 (2015), 2324-2354. doi: 10.1137/130932399.
[43]	S.-B. Yun, Cauchy problem for the Boltzmann-BGK model near a global Maxwellian, J. Math. Phy., 51 (2010), 123514, 24 pp. doi: 10.1063/1.3516479.
[44]	S.-B. Yun, Entropy production for ellipsoidal BGK model of the Boltzmann equation, Kinet. Relat. Models, 9 (2016), 605-619. doi: 10.3934/krm.2016009.
[45]	S.-B. Yun, Classical solutions for the ellipsoidal BGK model with fixed collision frequency, J. Differential Equations, 259 (2015), 6009-6037. doi: 10.1016/j.jde.2015.07.016.
[46]	X. Zhang and S. Hu, $L^p$ solutions to the Cauchy problem of the BGK equation, J. Math. Phys., 48 (2007), 113304, 17 pp.