Skip to main content
SpringerLink
Log in

Stability of Nonlinear Wave Patterns to the Bipolar Vlasov–Poisson–Boltzmann System

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction waves for the bipolar Vlasov–Poisson–Boltzmann (VPB) system. To this end, motivated by the micro–macro decomposition to the Boltzmann equation in Liu and Yu (Commun Math Phys 246:133–179, 2004) and Liu et al. (Physica D 188:178–192, 2004), we first set up a new micro–macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction waves are proved for the 1D bipolar VPB system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of the rarefaction wave fan to compressible Euler equations is proved for the 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in Li et al. (Indiana Univ Math J 65(2):665–725, 2016) sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. 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 

  3. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases, 3rd edn. Cambridge University Press, Cambridge, 1990

  4. Duan R.J., Liu S.Q.: Global 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Duan R.J., Yang T.: Stability of the one-species Vlasov–Poisson–Boltzmann system. SIAM J. Math. Anal. 41, 2353–2387 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Duan R.J., Yang T., Zhu C.J.: Boltzmann equation with external force and Vlasov–Poisson–Boltzmann system in infinite vacuum. Discrete Contin. Dyn. Syst. 16, 253–277 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Grad, H.: Asymptotic theory of the Boltzmann equation II. In: Rarefied Gas Dynamics, Vol. 1 (Eds. Laurmann J.A.) Academic Press, New York, 26–59, 1963

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

    Article  MATH  Google Scholar 

  11. Huang F.M., Li J., Matsumura A.: Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier–Stokes system. Arch. Ration. Mech. Anal. 197, 89–116 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huang F.M., Matsumura A.: Stability of a composite wave of two viscous shock waves for the full compressible Navier–Stokes equation. Commun. Math. Phys. 289, 841–861 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Huang F.M., Matsumura A., Xin Z.P.: Stability of contact discontinuities for the 1-D compressible Navier–Stokes equations. Arch. Ration. Mech. Anal. 179, 55–77 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huang F.M., Xin Z.P., Yang T.: Contact discontinuities with general perturbation for gas motion. Adv. Math. 219, 1246–1297 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang F.M., Yang T.: Stability of contact discontinuity for the Boltzmann equation. J. Differ. Equ. 229, 698–742 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Lax P.: Hyperbolic systems of conservation laws, II. Commun. Pure Appl. Math. 10, 537–566 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, H.L., Yang, T., Zhong, M.Y.: Spectrum analysis for the Vlasov–Poisson–Boltzmann system, Preprint, 2014

  18. Li H.L., Yang T., Zhong M.Y.: Spectrum analysis and optimal decay rates of the bipolar Vlasov–Poisson–Boltzmann equations. Indiana Univ. Math. J. 65(2), 665–725 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu T.P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Am. Math. Soc. 56, 1–108 (1985)

    ADS  MathSciNet  Google Scholar 

  20. Liu T.P., Xin Z.P.: Pointwise decay to contact discontinuities for systems of viscous conservation laws. it Asian J. Math. 1, 34–84 (1997)

    MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. 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, 333–371 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, T.P., Zeng, Y.N.: Shock waves in conservation laws with physical viscosity. Mem. Am. Math. Soc. 234(1105), 1–168 (2015)

  26. Markowich, A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations. Springer, Vienna, x+248, 1990

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

    Article  MATH  Google Scholar 

  28. Matsumura A., Nishihara K.: Asymptotics toward the rarefaction wave 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 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Nishihara K., Yang T., Zhao H.J.: Nonlinear stability of strong rarefaction wave for compressible Navier–Stokes equations. SIAM J. Math. Anal. 35, 1561–1597 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Smoller, J.: Shock Waves and Reaction–Diffusion Equations. Springer, New York, 1994

  32. 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 

  33. Szepessy A., Xin Z.P.: Nonlinear stability of viscous shock waves. Arch. Ration. Mech. Anal. 122, 53–103 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, L.S., Xiao, Q.H., Xiong, L.J., Zhao, H.J., The Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions. Acta. Math. Sci. 36, 1049–1097 (2016).

  35. Wang T., Wang Y.: Stability of superposition of two viscous shock waves for the Boltzmann equation. SIAM J. Math. Anal. 47, 1070–1120 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  37. Xin, Z.P.: On Nonlinear Stability of Contact Discontinuities. Hyperbolic Problems: Theory, Numerics, Applications. Stony Brook, NY, 1994. World Scientific Publishing, River Edge, 249–257, 1996

  38. Yang T., Yu H.J.: Optimal convergence rates of classica l solutions for Vlasov–Poisson–Boltzmann system. Commun. Math. Phys. 301, 319–355 (2011)

    Article  ADS  MATH  Google Scholar 

  39. Yang T., Yu H.J., Zhao H.J.: Cauchy problem for the Vlasov–Poisson–Boltzmann system. Arch. Ration. Mech. Anal. 182, 415–470 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yang T., Zhao H.J.: A half-space problem for the Boltzmann equation with specular reflection boundary condition. Commun. Math. Phys. 255, 683–726 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Yang T., Zhao H.J.: Global existence of classical solutions to the Vlasov–Poisson–Boltzmann system. Commun. Math. Phys. 268, 569–605 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhong M.Y.: Optimal time-decay rates of the Boltzmann equation. Sci. China Math. 57(4), 807–822 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zumbrun, K.: Stability of large-amplitude shock waves of compressible Navier–Stokes equations. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, Vol. III. North-Holland, Amsterdam, 311–533, 2004

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Hailiang Li

  2. Beijing Center of Mathematics and Information Sciences, Beijing, 100048, China

    Hailiang Li & Yi Wang

  3. CEMS, HCMS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yi Wang

  4. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yi Wang

  5. Department of Mathematics, City University of Hong Kong, Hong Kong, China

    Tong Yang

  6. Department of Mathematics, Guangxi University, Nanning, China

    Mingying Zhong

Authors
  1. Hailiang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mingying Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi Wang.

Additional information

Communicated by T.-P. Liu

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, H., Wang, Y., Yang, T. et al. Stability of Nonlinear Wave Patterns to the Bipolar Vlasov–Poisson–Boltzmann System. Arch Rational Mech Anal 228, 39–127 (2018). https://doi.org/10.1007/s00205-017-1185-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-017-1185-1

Search

Navigation