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Stationary Quantum BGK Model for Bosons and Fermions in a Bounded Interval

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In this paper, we consider the existence problem for a stationary relaxational models of the quantum Boltzmann equation. More precisely, we establish the existence of mild solution to the fermionic or bosonic quantum BGK model in a slab with inflow boundary data. Unlike the classical case, it is necessary to verify that the quantum local equilibrium state is well-defined, and the transition from the non-condensed state to the condensated state (bosons), or from the non-saturated state to the saturated state (fermions) does not arise in our solution space.

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  1. 1.

    Allemand, T.: Existence and conservation laws for the Boltzmann–Fermi–Dirac equation in a general domain. C. R. Math. Acad. Sci. Paris 348(13–14), 763–767 (2010)

  2. 2.

    Arkeryd, L., Cercignani, C., Illner, R.: Measure solutions of the steady Boltzmann equation in a slab. Commun. Math. Phys. 142(2), 285–296 (1991)

  3. 3.

    Arkeryd, L., Nouri, A.: Bose condensates in interaction with excitations: a kinetic model. Commun. Math. Phys. 310(3), 765–788 (2012)

  4. 4.

    Arkeryd, L., Nouri, A.: \(L^1\) solutions to the stationary Boltzmann equation in a slab (6) (English, French summary). Ann. Fac. Sci. Toulouse Math. 9(3), 375–413 (2000)

  5. 5.

    Arkeryd, L., Nouri, A.: On the Cauchy problem with large data for a space-dependent Boltzmann–Nordheim boson equation. Commun. Math. Sci. 15(5), 1247–1264 (2017)

  6. 6.

    Arkeryd, L., Nouri, A.: The stationary Boltzmann equation in the slab with given weighted mass for hard and soft forces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27(34), 533–556 (1998). (1999)

  7. 7.

    Bae, G.-C., Yun, S.-B.: Quantum BGK model near a global Fermi–Dirac distribution (2018). arXiv:1809.07790

  8. 8.

    Bang, J., Yun, S.-B.: Stationary solutions for the ellipsoidal BGK model in a slab. J. Differ. Equ. 261(10), 5803–5828 (2016)

  9. 9.

    Braukhoff, M.: Semiconductor Boltzmann–Dirac–Benny equation with BGK-type collision operator: existence of solutions vs. ill-posedness (2017). arXiv:1711.06015

  10. 10.

    Braukhoff, M.: Global analytic solutions of the semiconductor Boltzmann–Dirac–Benny equation with relaxation time approximation (2018). arXiv:1803.00379

  11. 11.

    Briant, M., Einav, A.: On the Cauchy problem for the homogeneous Boltzmann–Nordheim equation for bosons: local existence, uniqueness and creation of moments. J. Stat. Phys. 163(5), 1108–1156 (2016)

  12. 12.

    Brull, S.: The stationary Boltzmann equation for a two-component gas for soft forces in the slab. Math. Methods Appl. Sci. 31(14), 1653–1666 (2008)

  13. 13.

    Brull, S.: The stationary Boltzmann equation for a two-component gas in the slab. Math. Methods Appl. Sci. 31(2), 153–178 (2008)

  14. 14.

    Brull, S., Yun, S.-B.: Stationary flows of the ES-BGK model with the correct Prandtl number (preparation)

  15. 15.

    Dolbeault, J.: Kinetic models and quantum effects: a modified Boltzmann equation for Fermi–Dirac particles. Arch. Ration. Mech. Anal. 127(2), 101–131 (1994)

  16. 16.

    Escobedo, M., Mischler, S., Valle, M.A.: Entropy maximisation problem for quantum relativistic particles. Bull. Soc. Math. France 133(1), 87–120 (2005)

  17. 17.

    Escobedo, M., Mischler, S., Valle, M. A.: Homogeneous Boltzmann equation in quantum relativistic kinetic theory. Electron. J. Differ. Equ. Monograph, 4. Southwest Texas State University, San Marcos, TX, 85 pp (2003)

  18. 18.

    Escobedo, M., Mischler, S., Velázquez, J.J.L.: On the fundamental solution of a linearized Uehling–Uhlenbeck equation. Arch. Ration. Mech. Anal. 186(2), 309–349 (2007)

  19. 19.

    Escobedo, M., Velázquez, J.J.L.: Finite time blow-up and condensation for the bosonic Nordheim equation. Invent. Math. 200(3), 761–847 (2015)

  20. 20.

    Esposito, R., Guo, Y., Kim, C., Marra, R.: Non-isothermal boundary in the Boltzmann theory and Fourier law. Commun. Math. Phys. 323(1), 177–239 (2013)

  21. 21.

    Esposito, R., Guo, Y., Kim, C., Marra, R.: Stationary solutions to the Boltzmann equation in the hydrodynamic limit. Ann. PDE 4(1), 119 (2018). Art. 1

  22. 22.

    Filbet, F., Hu, J., Jin, S.: A numerical scheme for the quantum Boltzmann equation with stiff collision terms. ESAIM Math. Model. Numer. Anal. 46(2), 443–463 (2012)

  23. 23.

    Ghomeshi, S.: Existence and uniqueness of solutions for the Couette problem. J. Stat. Phys. 118(1–2), 265–300 (2005)

  24. 24.

    Hu, J., Jin, S.: On kinetic flux vector splitting schemes for quantum Euler equations. Kinet. Relat. Models 4(2), 517–530 (2011)

  25. 25.

    Hu, J., Jin, S., Wang, L.: An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: a splitting approach. Kinet. Relat. Models 8(4), 707–723 (2015)

  26. 26.

    Jin, S., Pareschi, L.: Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes. J. Comput. Phys. 161(1), 312–330 (2000)

  27. 27.

    Jüngel, A.: Transport Equations for Semiconductors. Lecture Notes in Physics, vol. 773. Springer, Berlin (2009)

  28. 28.

    Li, W., Lu, X.: Global existence of solutions of the Boltzmann equation for Bose–Einstein particles with anisotropic initial data. J. Funct. Anal. 276(1), 231–283 (2019)

  29. 29.

    Lu, X.: A modified Boltzmann equation for Bose–Einstein particles: isotropic solutions and long-time behavior. J. Stat. Phys. 98(5–6), 1335–1394 (2000)

  30. 30.

    Lu, X.: Long time convergence of the Bose–Einstein condensation. J. Stat. Phys. 162(3), 652–670 (2016)

  31. 31.

    Lu, X.: On spatially homogeneous solutions of a modified Boltzmann equation for Fermi–Dirac particles. J. Stat. Phys. 105(1–2), 353–388 (2001)

  32. 32.

    Lu, X.: The Boltzmann equation for Bose–Einstein particles: condensation in finite time. J. Stat. Phys. 150(6), 1138–1176 (2013)

  33. 33.

    Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations. Springer, Vienna (1990)

  34. 34.

    Muljadi, B.P., Yang, J.Y.: Simulation of shock wave diffraction by a square cylinder in gases of arbitrary statistics using a semiclassical Boltzmann–Bhatnagar–Gross–Krook equation solver. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci 468(2139), 651–670 (2012)

  35. 35.

    Nguyen, T.T., Tran, M.B.: Uniform in time lower bound for solutions to a quantum Boltzmann equation of bosons. Arch. Ration. Mech. Anal. 231(1), 63–89 (2019)

  36. 36.

    Nouri, A.: An existence result for a quantum BGK model. Math. Comput. Model. 47(3–4), 515–529 (2008)

  37. 37.

    Nt, A.R.F.: Relaxation time approximation for the Wigner–Boltzmann transport equation (2015). arXiv:1512.05959

  38. 38.

    Reinhard, P.G., Suraud, E.: A quantum relaxation-time approximation for finite fermion systems. Ann. Phys. 354, 183–202 (2015)

  39. 39.

    Shi, Y.-H., Yang, J.Y.: A gas-kinetic BGK scheme for semiclassical Boltzmann hydrodynamic transport. J. Comput. Phys. 227(22), 9389–9407 (2008)

  40. 40.

    Soffer, A., Tran, M.B.: On the dynamics of finite temperature trapped Bose gases. Adv. Math. 325, 533–607 (2018)

  41. 41.

    Ukai, S.: Stationary solutions of the BGK model equation on a finite interval with large boundary data. Transp. Theory Stat. Phys. 21(4–6), 487–500 (1992)

  42. 42.

    Wu, L., Meng, J., Zhang, Y.: Kinetic modelling of the quantum gases in the normal phase. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci 468(2142), 1799–1823 (2012)

  43. 43.

    Yang, J.-Y., Muljadi, B.P., Chen, S.-Y., Li, Z.-H.: Kinetic numerical methods for solving the semiclassical Boltzmann-BGK equation. Comput. Fluids 85(22), 153–165 (2013)

  44. 44.

    Yang, J.-Y., Hung, L.-H.: Lattice Uehling–Uhlenbeck Boltzmann–Bhatnagar–Gross–Krook hydrodynamics of quantum gases. Phys. Rev. E 79(5), 056708 (2009)

  45. 45.

    Yang, J.-Y., Yan, C.-Y., Diaz, M., Huang, J.-C., Li, Z., Zhang, H.: Numerical solutions of ideal quantum gas dynamical flows governed by semiclassical ellipsoidal-statistical distribution. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470(2161), 20130413, 19 (2014)

  46. 46.

    Zakrevskiy, T.: The Euler limit for kinetic models with Fermi–Dirac statistics. Asymptot. Anal. 95(1–2), 59–77 (2015)

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S.-B. Yun was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03935955)

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Correspondence to Seok-Bae Yun.

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Communicated by Eric A. Carlen.

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Bae, G., Yun, S. Stationary Quantum BGK Model for Bosons and Fermions in a Bounded Interval. J Stat Phys 178, 845–868 (2020).

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  • Quantum BGK model
  • Quantum Boltzmann equation
  • Stationary problems
  • Relaxation time approximation
  • Inflow boundary conditions