대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제62권1호
  • /
  • Pages.217-236
  • /
  • 2025
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

CLASSICAL SOLUTIONS TO A BGK-TYPE MODEL RELAXING TO THE ISENTROPIC GAS DYNAMICS

  • 투고 : 2024.03.10
  • 심사 : 2024.07.10
  • 발행 : 2025.01.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we consider a BGK-type kinetic model relaxing to the isentropic gas dynamics in the hydrodynamic limit. We introduce a linearization of the equation around the global equilibrium. Then we prove the global existence of classical solutions with an exponential convergence rate toward the equilibrium state in the periodic domain when the initial data is a small perturbation of the global equilibrium.

키워드

과제정보

This research was funded by a 2023 Research Grant from Sangmyung University (2023-A000-0284).

참고문헌

  1. J. L. Anderson, H. R. Witting, A relativistic relaxation-time model for the Boltzmann equation, Physica, 74, (1974), 466–488.
  2. P. Andries, K. Aoki, and B. Perthame, A consistent BGK-type model for gas mixtures, J. Statist. Phys. 106 (2002), no. 5-6, 993–1018. https://doi.org/10.1023/A:1014033703134
  3. G.-C. Bae, C. Klingenberg, M. Pirner, and S. Yun, BGK model of the multi-species Uehling-Uhlenbeck equation, Kinet. Relat. Models 14 (2021), no. 1, 25–44. https://doi.org/10.3934/krm.2020047
  4. G.-C. Bae and S.-B. Yun, Quantum BGK model near a global Fermi-Dirac distribution, SIAM J. Math. Anal. 52 (2020), no. 3, 2313–2352. https://doi.org/10.1137/19M1270021
  5. G.-C. Bae and S.-B. Yun, The Shakhov model near a global Maxwellian, Nonlinear Anal. Real World Appl. 70 (2023), Paper No. 103742, 33 pp. https://doi.org/10.1016/j.nonrwa.2022.103742
  6. F. Berthelin and F. Bouchut, Solution with finite energy to a BGK system relaxing to isentropic gas dynamics, Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 4, 605–630.
  7. F. Berthelin and F. Bouchut, Kinetic invariant domains and relaxation limit from a BGK model to isentropic gas dynamics, Asymptot. Anal. 31 (2002), no. 2, 153–176.
  8. F. Berthelin and F. Bouchut, Relaxation to isentropic gas dynamics for a BGK system with single kinetic entropy, Methods Appl. Anal. 9 (2002), no. 2, 313–327. https://doi.org/10.4310/MAA.2002.v9.n2.a7
  9. F. Berthelin and A. F. Vasseur, From kinetic equations to multidimensional isentropic gas dynamics before shocks, SIAM J. Math. Anal. 36 (2005), no. 6, 1807–1835. https://doi.org/10.1137/S0036141003431554
  10. P. L. Bhatnagar, E. P. Gross, and M. L. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94, (1954), 511–525.
  11. A. V. Bobylev, M. Bisi, M. Groppi, G. Spiga, and I. F. Potapenko, A general consistent BGK model for gas mixtures, Kinet. Relat. Models 11 (2018), no. 6, 1377–1393. https://doi.org/10.3934/krm.2018054
  12. F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Statist. Phys. 95 (1999), no. 1-2, 113–170. https://doi.org/10.1023/A:1004525427365
  13. F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models, Numer. Math. 94 (2003), no. 4, 623–672. https://doi.org/10.1007/s00211-002-0426-9
  14. S. Brull, V. Pavan, and J. Schneider, Derivation of a BGK model for mixtures, Eur. J. Mech. B Fluids 33 (2012), 74–86. https://doi.org/10.1016/j.euromechflu.2011.12.
  15. Y.-P. Choi and B.-H. Hwang, From BGK-alignment model to the pressured Euler alignment system with singular communication weights, J. Differential Equations 379 (2024), 363–412. https://doi.org/10.1016/j.jde.2023.10.010
  16. Y.-P. Choi and B.-H. Hwang, Global existence of weak solutions to a BGK model relaxing to the barotropic Euler equations, Nonlinear Anal. 238 (2024), Paper No. 113414, 42 pp. https://doi.org/10.1016/j.na.2023.113414
  17. C. M. Cuesta, S. Hittmeir, and C. Schmeiser, Weak shocks of a BGK kinetic model for isentropic gas dynamics, Kinet. Relat. Models 3 (2010), no. 2, 255–279. https://doi.org/10.3934/krm.2010.3.255
  18. Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math. 55 (2002), no. 9, 1104–1135. https://doi.org/10.1002/cpa.10040
  19. Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math. 153 (2003), no. 3, 593–630. https://doi.org/10.1007/s00222-003-0301-z
  20. Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J. 53 (2004), no. 4, 1081–1094. https://doi.org/10.1512/iumj.2004.53.2574
  21. L. H. Holway Jr., Kinetic theory of shock structure using an ellipsoidal distribution function, Rarefied Gas Dynamics, Vol. I (Proc. Fourth Internat. Sympos., Univ. Toronto, 1964), 193–215, Academic Press, New York, 1965.
  22. B.-H. Hwang, M.-S. Lee, and S.-B. Yun, Relativistic BGK model for gas mixtures, J. Stat. Phys. 191 (2024), no. 5, Paper No. 59, 27 pp. https://doi.org/10.1007/s10955-024-03271-2
  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), no. 3, 2906–2947. https://doi.org/10.1137/21M1404946
  24. I. M. Khalatnikov, Introduction to the Theory of Superfluidity (Russian), Izdat. 'Nauka', Moscow, 1965.
  25. C. Klingenberg, M. Pirner, and G. Puppo, A consistent kinetic model for a two component mixture with an application to plasma, Kinet. Relat. Models 10 (2017), no. 2, 445–465. https://doi.org/10.3934/krm.2017017
  26. P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), no. 1, 169–191. https://doi.org/10.2307/2152725
  27. P.-L. Lions, B. Perthame, and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems, Comm. Math. Phys. 163 (1994), no. 2, 415–431. http://projecteuclid.org/euclid.cmp/1104270470 104270470
  28. L. Liu and M. Pirner, Hypocoercivity for a BGK model for gas mixtures, J. Differential Equations 267 (2019), no. 1, 119–149. https://doi.org/10.1016/j.jde.2019.01.006
  29. C. Marle, Modele cinétique pour l'établissement des lois de la conduction de la chaleur et de la viscosité en théorie de la relativité, C. R. Acad. Sci. Paris, 260, (1965), 6539–6541.
  30. C.-M. Marle, Sur l'etablissement des équations de l'hydrodynamique des fluides relativistes dissipatifs. II. Méthodes de résolution approchée de l'équation de Boltzmann relativiste, Ann. Inst. H. Poincaré Sect. A (N.S.) 10 (1969), 127–194.
  31. S. Pennisi, T. Ruggeri, A new BGK model for relativistic kinetic theory of monatomic and polyatomic gases, J. Phys. Conf. Ser., 1035, (2018), 012005.
  32. B. Perthame and E. Tadmor, A kinetic equation with kinetic entropy functions for scalar conservation laws, Comm. Math. Phys. 136 (1991), no. 3, 501–517. http://projecteuclid.org/euclid.cmp/1104202434 104202434
  33. E. M. Shakhov, Generalization of the Krook kinetic relaxation equation, Fluid dynamics, 3, (1968), 95–96.
  34. P. Welander, On the temperature jump in a rarefied gas, Ark. Fys. 7 (1954), 507–553.
  35. S.-B. Yun, Cauchy problem for the Boltzmann-BGK model near a global Maxwellian, J. Math. Phys. 51 (2010), no. 12, 123514, 24 pp. https://doi.org/10.1063/1.3516479
  36. S.-B. Yun, Elipsoidal BGK model near a global Macuellian, SIAM J. Math. Anal. 47 (2015), no. 3, 2324-2354. https://doi.org/10.1137/130932399