Skip to main content

The Diffusive Limit of the Bipolar Vlasov–Poisson–Boltzmann Equations

Abstract

In this work, we first prove the existence of the classic solutions to the dimensionless bipolar Vlasov–Poisson–Boltzmann equations by employing hypocoercive properties of the linear Boltzmann operators. Based on the uniform estimates and employing the Ohm’s law, two fluids Navier–Stokes–Poisson system is derived from the dimensionless Vlasov–Poisson–Boltzmann equations.

This is a preview of subscription content, access via your institution.

Data Availability

No data or code were generated or used during the study. The only model used in this work can be find in the reference.

References

  1. Arsénio, D., Saint-Raymond, L.: From the Vlasov–Maxwell–Boltzmann System to Incompressible Viscous Electro-Magneto-Hydrodynamics. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2019)

  2. Bardos, C., Golse, F., Levermore, C.D.: Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Stat. Phys. 63, 323–344 (1991)

    ADS  MathSciNet  Article  Google Scholar 

  3. Bardos, C., Golse, F., Levermore, C.D.: Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Commun. Pure Appl. Math. 46, 667–753 (1993)

    MathSciNet  Article  Google Scholar 

  4. Bardos, C., Ukai, S.: The classical incompressible Navier–Stokes limit of the Boltzmann equation. Math. Models Methods Appl. Sci. 01, 235–257 (1991)

    MathSciNet  Article  Google Scholar 

  5. Bastea, S., Esposito, R., Lebowitz, J.L., Marra, R.: Binary fluids with long range segregating interaction. I. Derivation of kinetic and hydrodynamic equations. J. Stat. Phys. 101, 1087–1136 (2000)

    MathSciNet  Article  Google Scholar 

  6. Boyer, F., Fabrie, P.: Mathematical Tools for the Study of the incompressible Navier–Stokes Equations and Related Models, Applied Mathematical Sciences, vol. 183. Springer, New York (2013)

    MATH  Google Scholar 

  7. Briant, M.: From the Boltzmann equation to the incompressible Navier–Stokes equations on the torus: a quantitative error estimate. J. Differ. Equ. 259, 6072–6141 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  8. DiPerna, R.J., Lions, P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. (2) 130, 321–366 (1989)

    MathSciNet  Article  Google Scholar 

  9. Golse, F., Saint-Raymond, L.: The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155, 81–161 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  10. Golse, F., Saint-Raymond, L.: Hydrodynamic limits for the Boltzmann equation. Riv. Mat. Univ. Parma (7) 4, 1–144 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Guo, M., Jiang, N., Luo, Y.-L.: From Vlasov–Poisson–Boltzmann system to incompressible Navier–Stokes–Fourier–Poisson system: convergence for classical solutions. arXiv preprintarXiv:2006.16514 (2020)

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

    ADS  MathSciNet  Article  Google Scholar 

  13. Guo, Y.: Boltzmann diffusive limit beyond the Navier–Stokes approximation. Commun. Pure Appl. Math. 59, 626–687 (2006)

    MathSciNet  Article  Google Scholar 

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

    ADS  MathSciNet  Article  Google Scholar 

  15. Jiang, N., Xu, C.-J., Zhao, H.: Incompressible Navier–Stokes–Fourier limit from the Boltzmann equation: classical solutions. Indiana Univ. Math. J. 67, 1817–1855 (2018)

    MathSciNet  Article  Google Scholar 

  16. Jiang, N., Zhang, X.: The Boltzmann equation with incoming boundary condition: global solutions and Navier–Stokes limit. arXiv eprintarXiv:1701.04144 (2017)

  17. Jiang, N., Zhang, X.: Sensitivity analysis and incompressible Navier–Stokes–Poisson limit of Vlasov–Poisson–Boltzmann equations with uncertainty. arXiv preprintarXiv:2007.00879 (2020)

  18. Li, H., Yang, T., Zhong, M.: Spectrum analysis for the Vlasov–Poisson–Boltzmann system. arXiv preprintarXiv:1402.3633 (2014)

  19. Li, H.-L., Yang, T., Zhong, M.: Diffusion limit of the Vlasov–Poisson–Boltzmann system. arXiv preprintarXiv:2007.01461 (2020)

  20. Mischler, S.: Kinetic equations with Maxwell boundary conditions. Ann. Sci. Éc. Norm. Supér. (4) 43, 719–760 (2010)

  21. Mouhot, C.: Explicit coercivity estimates for the linearized Boltzmann and Landau operators. Commun. Partial Differ. Equ. 31, 1321–1348 (2006)

    MathSciNet  Article  Google Scholar 

  22. Mouhot, C.: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Commun. Math. Phys. 261, 629–672 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  23. Mouhot, C., Neumann, L.: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity 19, 969–998 (2006)

    ADS  MathSciNet  Article  Google Scholar 

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

    Book  Google Scholar 

  25. Wang, Y.: The diffusive limit of the Vlasov–Boltzmann system for binary fluids. SIAM J. Math. Anal. 43, 253–301 (2011)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The first author was supported by Chongqing University of Posts and Telecommunications startup fund (Grant No. A2018-128). The second author was supported by the National Natural Science Foundation of China (Grant Nos. 11271305, 11531010). The third author was supported by National Natural Science Foundation of China (Grant No. 11901537).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xu Zhang.

Additional information

Communicated by Clement Mouhot.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tong, L., Tan, Z. & Zhang, X. The Diffusive Limit of the Bipolar Vlasov–Poisson–Boltzmann Equations. J Stat Phys 188, 2 (2022). https://doi.org/10.1007/s10955-022-02928-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10955-022-02928-0