SpringerLink
Log in

Kinetic Theory and Gas Dynamics, Some Historical Perspectives

  • Conference paper
  • First Online:
Trends in Contemporary Mathematics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 8))

  • 1330 Accesses

Abstract

The purpose of the present paper is to present, through historical perspectives, some of the recent developments on the Boltzmann equation in the kinetic theory and multi-dimensional shock waves in gas dynamics.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Bianchini, A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. 161, 223–242 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. G.-Q. Chen, M. Feldman, Global solutions of shock reflection by large-angle wedges for potential flow. Ann. Math. 171(2), 1067–1182 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  4. D. Coutandd, S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum. Arch. Ration. Mech. Anal. 206, 515–616 (2012)

    Article  MathSciNet  Google Scholar 

  5. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (Springer, Berlin/New York, 1999)

    Google Scholar 

  6. L. Desvillettes, C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invet. Math. 159, 245–316 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. V. Elling, T.-P. Liu, The ellipticity principle for self-similar potential flows. J. Hyperbolic Differ. Equ. 2(4), 909–917 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. V. Elling, T.-P. Liu, Supersonic flow onto a solid wedge. Commun. Pure Appl. math. 61(10), 1347–1448 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 41, 697–715 (1965)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. H. Grad, Principle of the kinetic theory of gases, in Thermodynamics of Gases, ed. by S. Flügge (Springer, Berlin/Heidelberg, 1958)

    Google Scholar 

  13. O. Lanford, The evolution of large classical system, in Dynamical Systems, Theory and Applications, ed. by J. Moser. Lecture Notes in Physics (Springer, Heidleberg, 1975), pp. 1–111

    Google Scholar 

  14. H.W. Liepmann, A. Roshko, Elements of Gasdynamics (Wiley, New York, 1957)

    MATH  Google Scholar 

  15. T.-P. Liu, Transonic gas flow in a duct of varying area. Arch. Ration. Mech. Anal. 80, 1–18 (1982)

    Article  MATH  Google Scholar 

  16. T.-P. Liu, Nonlinear stability and instability of transonic flows through a nozzle. Commun. Math. Phys. 83, 243–260 (1982)

    Article  MATH  Google Scholar 

  17. T.-P. Liu, Z. **n, T. Yang, Vacuum states for compressible flow. Discret. Contin. Dyn. Syst. 4, 1–32 (1998)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. T.-P. Liu, Multi-diemnstional gas flow: some historical perspectives. Bull. Inst. Math. Acad. Sin. (N.S.) 6, 269–291 (2011)

    Google Scholar 

  20. T.-P. Liu, S.-H. Yu, Solving Boltzmann equation, Part I: Green’s function. Bull. Inst. Math. Acad. Sin. (N.S.) 6(1), 115–243 (2011)

    Google Scholar 

  21. T.-P. Liu, S.-H. Yu, Invariant manifolds for steady Boltzmann flows and applications. Arch. Ration. Mech. Anal. 207, 869–997 (2013)

    Article  Google Scholar 

  22. T.-P. Liu, Y. Zeng, Shock Waves in Conservation laws with physical viscosity, memoirs. Am. Math. Soc. (to appear)

    Google Scholar 

  23. Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications (Birkhauser, Boston, 2007)

    Book  Google Scholar 

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

    Article  MATH  Google Scholar 

  25. S.-H. Yu, Hydrodynamics limits with shock waves of the Boltzmann equaiton. Commun. Pure Appl. Math. 58(3), 409–443 (2005)

    Article  MATH  Google Scholar 

  26. S.-H. Yu, Initial and shock layers for Boltzmann equation. Arch. Ration. Mech. Anal. 211, 1–60 (2014)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

Part of the article was presented in INdAm Day Lecture on June 7, 2012 in Genova. The author would like to thank Professor Tommaso Ruggeri for the opportunity. The research is supported in part by Investigator Award of Academia Sinica and National Science Council Grant 96-2628-M-001-011.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Academia Sinica, Taipei, Taiwan

    Tai-** Liu

  2. Stanford University, Stanford, CA, USA

    Tai-** Liu

Authors
  1. Tai-** Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tai-** Liu .

Editor information

Editors and Affiliations

  1. Istituto Nazionale di Alta Matematica "Francesco Severi", Roma, Italy

    Vincenzo Ancona

  2. Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Roma, Italy

    Elisabetta Strickland

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Liu, TP. (2014). Kinetic Theory and Gas Dynamics, Some Historical Perspectives. In: Ancona, V., Strickland, E. (eds) Trends in Contemporary Mathematics. Springer INdAM Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-05254-0_19

Download citation

Publish with us

Policies and ethics

Search

Navigation