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The Vlasov–Poisson–Boltzmann System for a Disparate Mass Binary Mixture

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Abstract

The Vlasov–Poisson–Boltzmann system is often used to govern the motion of plasmas consisting of electrons and ions with disparate masses when collisions of charged particles are described by the two-component Boltzmann collision operator. The perturbation theory of the system around global Maxwellians recently has been well established in Guo (Commun Pure Appl Math 55:1104–1135, 2002). It should be more interesting to further study the existence and stability of nontrivial large time asymptotic profiles for the system even with slab symmetry in space, particularly understanding the effect of the self-consistent potential on the non-trivial long-term dynamics of the binary system. In this paper, we consider the problem in the setting of rarefaction waves. The analytical tool is based on the macro–micro decomposition introduced in Liu et al. (Physica D 188(3–4):178–192, 2004) that we have been able to develop for the case of the two-component Boltzmann equations around local bi-Maxwellians. Our focus is to explore how the disparate masses and charges of particles play a role in the analysis of the approach of the complex coupling system time-asymptotically toward a non-constant equilibrium state whose macroscopic quantities satisfy the quasineutral nonisentropic Euler system.

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References

  1. Andries, P., Aoki, K., Perthame, B.: A consistent BGK-type model for gas mixtures. J. Stat. Phys. 106(5–6), 993–1018 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aoki, K., Bardos, C., Takata, K.: Knudsen layer for gas mixtures. J. Stat. Phys. 112(3–4), 629–655 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arsénio, D., Saint-Raymond, L.: Solutions of the Vlasov–Maxwell–Boltzmann system with long-range interactions. C. R. Math. Acad. Sci. Paris 351(9–10), 357–360 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bernis, L., Desvillettes, L.: Propagation of singularities for classical solutions of the Vlasov–Poisson–Boltzmann equation. Discrete Contin. Dyn. Syst. 24(1), 13–33 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bianca, C., Dogbe, C.: On the Boltzmann gas mixture equation: Linking the kinetic and fluid regimes. Commun. Nonlinear Sci. Number Simul. 29, 240–256 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bianchini, S., Bressan, A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. 161(1), 223–342 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bisi, M., Desvillettes, L.: Formal passage from kinetic theory to incompressible Navier-Stokes equations for a mixture of gases. ESAIM Math. Model. Numer. Anal. 48(4), 1171–1197 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bobylev, A.V., Gamba, I.M.: Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails. J. Stat. Phys. 124(2–4), 497–516 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Bostan, M., Gamba, I.M., Goudon, T., Vasseur, A.: Boundary value problems for the stationary Vlasov–Boltzmann–Poisson equation. Indiana Univ. Math. J. 59(5), 1629–1660 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Boudin, L., Grec, B., Pavic, M., Salvarani, F.: Diffusion asymptotics of a kinetic model for gaseous mixtures. Kinet. Relat. Models 6(1), 137–157 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Briant, M., Daus, E.: The Boltzmann equation for a multi-species mixture close to global equilibrium. Arch. Ration. Mech. Anal. 222(3), 1367–1443 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brull, S.: The stationary Boltzmann equation for a two-component gas in the slab with different molecular masses. Adv. Differ. Equ. 15(11–12), 1103–1124 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Caflisch, R.E., Nicolaenko, B.: Shock profile solutions of the Boltzmann equation. Commun. Math. Phys. 86, 161–194 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Carleman, T.: Sur La Théorie de l’Équation Intégrodifférentielle de Boltzmann. Acta Math. 60, 91–142 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  16. Chapman, S., Colwing, T.G.: The Mathematical Theory of Non-uniform Gases, 3rd edn. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1990)

    Google Scholar 

  17. Charles, F., Després, B., Perthame, B., Sentis, R.: Nonlinear stability of a Vlasov equation for magnetic plasmas. Kinet. Relat. Models 6(2), 269–290 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, F.: Introduction to Plasma Physics and Controlled Fusion, 2nd edn. Plenum Press, New York (1984)

    Book  Google Scholar 

  19. Cordier, S., Degond, P., Markowich, P., Schmeiser, C.: Travelling wave analysis of an isothermal Euler–Poisson model. Ann. Fac. Sci. Toulouse Math. 5, 599–643 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Wiley-Interscience, New York (1948)

    MATH  Google Scholar 

  21. Degond, P., Lucquin-Desreux, B.: The asymptotics of collision operators for two species of particles of disparate masses. Math. Models Methods Appl. Sci. 6(3), 405–436 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Degond, P., Lucquin-Desreux, B.: Transport coefficients of plasmas and disparate mass binary gases. Transp. Theory Stat. Phys. 25(6), 595–633 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Desvillettes, L., Dolbeault, J.: On long time asymptotics of the Vlasov–Poisson–Boltzmann equation. Commun. PDE 16(2–3), 451–489 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Desvillettes, L., Monaco, R., Salvarani, F.: A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions. Eur. J. Mech. B 24(2), 219–236 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dogbe, C.: Fluid dynamic limits for gas mixture. I. Formal derivations. M3AS 18(9), 1633–1672 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Duan, R.-J., Liu, S.-Q.: The Vlasov–Poisson–Boltzmann system without angular cutoff. Commun. Math. Phys. 324(1), 1–45 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Duan, R.-J., Liu, S.-Q.: Stability of rarefaction waves of the Navier–Stokes–Poisson system. J. Differ. Equ. 258(7), 2495–2530 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Duan, R.-J., Liu, S.-Q.: Stability of the rarefaction wave of the Vlasov–Poisson–Boltzmann system. SIAM J. Math. Anal. 47(5), 3585–3647 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Duan, R.-J., Strain, R.M.: Optimal time decay of the Vlasov–Poisson–Boltzmann system in \(\mathbb{R}^3\). Arch. Ration. Mech. Anal. 199(1), 291–328 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Duan, R.-J., Yang, X.-F.: Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier–Stokes–Poisson equations. Commun. Pure Appl. Anal. 12(2), 985–1014 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Duan, R.-J., Yang, T., Zhao, H.-J.: The Vlasov–Poisson–Boltzmann system in the whole space: the hard potential case. J. Differ. Equ. 252(12), 6356–6386 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Duan, R.-J., Yang, T., Zhao, H.-J.: The Vlasov–Poisson–Boltzmann system for soft potentials. Math. Models Methods Appl. Sci. 23(6), 979–1028 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Duan, R.-J., Lei, Y.-J., Yang, T., Zhao, H.-J.: The Vlasov–Maxwell–Boltzmann system near Maxwellians in the whole space with very soft potentials, arXiv:1411.5150

  34. Duan, R.-J., Liu, S.-Q., Yin, H.-Y., Zhu, C.-J.: Stability of the rarefaction wave for a two-fluid plasma model with diffusion. Sci. China Math. (2015). doi:10.1007/s11425-015-5059-4

  35. Ellis, R., Pinsky, M.A.: The first and second fluid approximations to the linearized Boltzmann equation. J. Math. Pures Appl. 9(54), 125–156 (1975)

    MathSciNet  MATH  Google Scholar 

  36. Esposito, R., Guo, Y., Marra, R.: Phase transition in a Vlasov–Boltzmann binary mixture. Commun. Math. Phys. 296, 1–33 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Esposito, R., Guo, Y., Marra, R.: Stability of a Vlasov–Boltzmann binary mixture at the phase transition on an interval. Kinet. Relat. Models 6(4), 761–787 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Glassey, R., Strauss, W.A.: Perturbation of essential spectra of evolution operators and the Vlasov–Poisson–Boltzmann system. Discrete Contin. Dyn. Syst. 5(3), 457–472 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Goldman, E., Sirovich, L.: Equations for gas mixtures. Phys. Fluids 10, 1928–1940 (1967)

    Article  ADS  Google Scholar 

  40. Goldman, E., Sirovich, L.: Equations for gas mixtures II. Phys. Fluids 12, 245–247 (1969)

    Article  ADS  Google Scholar 

  41. Golse, F., Perthame, B., Sulem, C.: On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Ration. Mech. Anal. 103, 81–96 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  42. Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Ration. Mech. Anal. 95(4), 325–344 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  43. Grad, H.: Asymptotic theory of the Boltzmann equation. Phys. Fluids 6(2), 147–181 (1963)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Guo, Y.: The Vlasov-Poisson-Boltzmann system near vacuum. Commun. Math. Phys. 218(2), 293–313 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Guo, Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Commun. Pure Appl. Math. 55, 1104–1135 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Guo, Y.: The Vlasov–Maxwell–Boltzmann system near Maxwellians. Invent. Math. 153(3), 593–630 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Guo, Y.: The Vlasov–Poisson–Landau system in a periodic box. J. Am. Math. Soc. 25, 759–812 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. Guo, Y., Jang, J.: Global Hilbert expansion for the Vlasov–Poisson–Boltzmann system. Commun. Math. Phys. 299(2), 469–501 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Guo, Y., Pausader, B.: Global smooth ion dynamics in the Euler–Poisson system. Commun. Math. Phys. 303, 89–125 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Guo, Y., Pu, X.: KdV limit of the Euler–Poisson system. Arch. Ration. Mech. Anal. 211(2), 673–710 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. Guo, Y., Ionescu, A., Pausader, B.: Global solutions of the Euler–Maxwell two-fluid system in 3D. Ann. Math. 183(2), 377–498 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  52. Han-Kwan, D.: Quasineutral limit of the Vlasov–Poisson system with massless electrons. Commun. Partial Differ. Equ. 36(8), 1385–1425 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  53. Han-Kwan, D.: From Vlasov–Poisson to Kortewegde Vries and Zakharov–Kuznetsov. Commun. Math. Phys. 324(3), 961–993 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Huang, Y.-T.: Spectrum structure and behavior of the Vlasov-Maxwell-Boltzmann system without angular cutoff. J. Differ. Equ. (2015). doi:10.1016/j.jde.2015.10.015

  55. Huang, F.-M., Xin, Z.-P., Yang, T.: Contact discontinuity with general perturbations for gas motions. Adv. Math. 219(4), 1246–1297 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  56. Huang, F.-M., Wang, Y., Wang, Y., Yang, T.: The limit of the Boltzmann equation to the Euler equations for Riemann problems. SIAM J. Math. Anal. 45(3), 1741–1811 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  57. Jang, J., Masmoudi, N.: Derivation of Ohm’s law from the kinetic equations. SIAM J. Math. Anal. 44(5), 3649–3669 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  58. Kosuge, S., Aoki, K., Takata, S.: Shock-wave structure for a binary gas mixture: finite-difference analysis of the Boltzmann equation for hard-sphere molecules. Eur. J. Mech. B 20, 87–126 (2001)

    Article  MATH  Google Scholar 

  59. Krall, N.A., Trivelpiece, A.W.: Principles of Plasma Physics. McGraw-Hill, New York (1973)

    Google Scholar 

  60. Li, H.-L. , Yang, T., Zhong, M.Y.: Spectrum analysis for the Vlasov–Poisson–Boltzmann system. arXiv:1402.3633

  61. Liu, T.-P., Xin, Z.-P.: Nonlinear stability of rarefaction waves for compressible Navier–Stokes equations. Commun. Math. Phys. 118, 451–465 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. Liu, T.-P., Yu, S.-H.: Boltzmann equation: micro-macro decompositions and positivity of shock profiles. Commun. Math. Phys. 246(1), 133–179 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  63. Liu, T.-P., Yu, S.-H.: Invariant manifolds for steady Boltzmann flows and applications. Arch. Ration. Mech. Anal. 209(3), 869–997 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  64. Liu, T.-P., Yang, T., Yu, S.-H.: Energy method for the Boltzmann equation. Physica D 188(3–4), 178–192 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  65. Liu, T.-P., Yang, T., Yu, S.-H., Zhao, H.-J.: Nonlinear stability of rarefaction waves for the Boltzmann equation. Arch. Ration. Mech. Anal. 181(2), 333–371 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  66. Matsumura, A., Nishihara, K.: On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Jpn. J. Appl. Math. 2(1), 17–25 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  67. Matsumura, A., Nishihara, K.: Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas. Jpn. J. Appl. Math. 3, 1–13 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  68. Matsumura, A., Nishihara, K.: Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas. Commun. Math. Phys. 144(2), 325–335 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. Mischler, S.: On the initial boundary value problem for the Vlasov–Poisson–Boltzmann system. Commun. Math. Phys. 210(2), 447–466 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  70. Saint-Raymond, L.: Hydrodynamic Limits of the Boltzmann Equation. Lecture Notes in Mathematics, vol. 1971. Springer, Berlin (2009)

  71. Sone, Y.: Molecular Gas Dynamics, Theory, Techniques, and Applications. Birkhauser, Boston (2006)

    MATH  Google Scholar 

  72. Sotirov, A., Yu, S.-H.: On the solution of a Boltzmann system for gas mixtures. Arch. Ration. Mech. Anal. 195(2), 675–700 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  73. Strain, R.M.: The Vlasov–Maxwell–Boltzmann system in the whole space. Commun. Math. Phys. 268(2), 543–567 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  74. Strain, R.M., Zhu, K.: The Vlasov–Poisson–Landau system in \(\mathbb{R}^3\). Arch. Ration. Mech. Anal. 210(2), 615–671 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  75. Suzuki, M.: Asymptotic stability of stationary solutions to the Euler–Poisson equations arising in plasma physics. Kinet. Relat. Models 4(2), 569–588 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  76. Takata, S.: Diffusion slip for a binary mixture of hard-sphere molecular gases: numerical analysis based on the linearized Boltzmann equation. In: Bartel, T.J., Gallis, M.A. (eds.) Rarefied Gas Dynamics, pp. 22–29. AIP, Melville (2001)

    Google Scholar 

  77. Takata, S., Aoki, K.: The ghost effect in the continuum limit for a vapor–gas mixture around condensed phases: asymptotic analysis of the Boltzmann equation, Transport Theory Statist. Phys. 30: 205–237 (2001); Erratum. Transp. Theory Stat. Phys. 31(2002), 289–290 (2001)

    Google Scholar 

  78. Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Jpn. Acad. 50, 179–184 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  79. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.) Handbook of athematical fluid dynamics, pp. 71–305. North Holland, Amsterdam (2002)

    Chapter  Google Scholar 

  80. Wang, Y.-J.: Decay of the two-species Vlasov–Poisson–Boltzmann system. J. Differ. Equ. 254(5), 2304–2340 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  81. Xin, Z.-P., Zeng, H.-H.: Convergence to the rarefaction waves for the nonlinear Boltzmann equation and compressible Navier–Stokes equations. J. Differ. Equ. 249, 827–871 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  82. Xin, Z.-P., Yang, T., Yu, H.-J.: The Boltzmann equation with soft potentials near a local Maxwellian. Arch. Ration. Mech. Anal. 206(1), 239–296 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  83. Yu, S.-H.: Nonlinear wave propagations over a Boltzmann shock profile. J. Am. Math. Soc. 23(4), 1041–1118 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Renjun Duan has received the General Research Fund (Project No. 14301515) from RGC of Hong Kong. Shuangqian Liu has received grants from the National Natural Science Foundation of China under contracts 11471142 and 11271160.

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  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s Republic of China

    Renjun Duan

  2. Department of Mathematics, Jinan University, Guangzhou, 510632, People’s Republic of China

    Shuangqian Liu

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  1. Renjun Duan
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Duan, R., Liu, S. The Vlasov–Poisson–Boltzmann System for a Disparate Mass Binary Mixture. J Stat Phys 169, 614–684 (2017). https://doi.org/10.1007/s10955-017-1875-y

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