Skip to main content Accessibility help
×
Home

Linear Stability of Hyperbolic Moment Models for Boltzmann Equation

  • Yana Di (a1), Yuwei Fan (a2), Ruo Li (a3) and Lingchao Zheng (a2)

Abstract

Grad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models with the presentation of collision terms. It is proved that the regularized models are linearly stable at the local equilibrium and satisfy Yong's first stability condition with commonly used approximate collision terms, and particularly with Boltzmann's binary collision model.

Copyright

Corresponding author

*Corresponding author. Email addresses: yndi@lsec.cc.ac.cn (Y. Di), ywfan@pku.edu.cn (Y. Fan), rli@math.pku.edu.cn (R. Li), lczheng@pku.edu.cn (L. Zheng)

References

Hide All
[1] Bhatnagar, P. L., Gross, E. P. and Krook, M., A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94(3) (1954), pp. 511525.
[2] Bobylev, A. V., The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Sov. Phys. Dokl., 27(1) (1982), pp. 2931.
[3] Boltzmann, L., Weitere studien über das wärmegleichgewicht unter gas-molekülen, Wiener Berichte, 66 (1872), pp. 275370.
[4] Cai, Z., Fan, Y. and Li, R., Globally hyperbolic regularization of Grad's moment system in one dimensional space, Commun. Math. Sci., 11(2) (2013), pp. 547571.
[5] Cai, Z., Fan, Y. and Li, R., Globally hyperbolic regularization of Grad's moment system, Commun. Pure Appl. Math., 67(3) (2014), pp. 464518.
[6] Cai, Z., Fan, Y. and Li, R., On hyperbolicity of 13-moment system, Kinetic and Related Models, 7(3) (2014), pp. 415432.
[7] Cai, Z., Fan, Y. and Li, R., A framework on moment model reduction for kinetic equation, SIAM J. Appl. Math., 75(5) (2015), pp. 20012023.
[8] Cai, Z., Fan, Y., Li, R. and Qiao, Z., Dimension-reduced hyperbolic moment method for the Boltzmann equation with BGK-type collision, Commun. Comput. Phys., 15(5) (2014), pp. 13681406.
[9] Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, 3rd Edition, Cambridge University Press, 1990.
[10] Fan, Y., Koellermeier, J., Li, J., Li, R. and Torrilhon, M., Model reduction of kinetic equations by operator projection, J. Stat. Phys., 161(4) (2015).
[11] Fan, Y. W., Development and Application of Moment Method in Gas Kinetic Theory (in Chinese), PhD thesis, Peking University, June 2016.
[12] Grad, H., Note on N-dimensional Hermite polynomials, Commun. Pure Appl. Math., 2(4) (1949), pp. 325330.
[13] Grad, H., On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2(4) (1949), pp. 331407.
[14] Holway, L. H., New statistical models for kinetic theory: Methods of construction, Phys. Fluids, 9(1) (1966), pp. 16581673.
[15] Müller, I. and Ruggeri, T., Rational Extended Thermodynamics, 2nd Edition, vol. 37 of Springer Tracts in Natural Philosophy, Springer-Verlag, New York, 1998.
[16] Rosenau, P., Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev. A, 40(12) (1989), pp. 71937196.
[17] Shakhov, E. M., Generalization of the Krook kinetic relaxation equation, Fluid Dyn., 3(5) (1968), pp. 9596.
[18] Struchtrup, H., Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory, Springer, 2005.
[19] Struchtrup, H. and Torrilhon, M., Regularization of Grad's 13 moment equations: Derivation and linear analysis, Phys. Fluids, 15(9) (2003), pp. 26682680.
[20] Torrilhon, M., Convergence study of moment approximations for boundary value problems of the Boltzmann-BGK equation, Commun. Comput. Phys., 18(3) (2015), pp. 529557.
[21] Yong, W. A., Singular Perturbation of First-order Hyperbolic Systems, PhD Thesis, Universität Heidelbery, 1992.
[22] Yong, W. A., Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Diff. Equations, 155(1) (1999), pp. 89132.

Keywords

MSC classification

Linear Stability of Hyperbolic Moment Models for Boltzmann Equation

  • Yana Di (a1), Yuwei Fan (a2), Ruo Li (a3) and Lingchao Zheng (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed