Skip to main content
SpringerLink
Log in

Global existence of smooth solutions and stability of the constant state for dissipative hyperbolic systems with applications to extended thermodynamics

  • Conference paper
Trends and Applications of Mathematics to Mechanics

Abstract

The entropy principle plays an important role in hyperbolic systems of balance laws: symmetrization, principal subsystems and nesting theories, equilibrium manifold. After a brief survey on these questions we present recent results concerning the local and global well-posedness of the Cauchy problem for smooth solutions with particular attention to the genuine coupling Kawashima condition. These results are applied to the case of extended thermodynamics and we prove that the K-condition is satisfied in the case of the 13-moment Grad theory with the consequence that there exist global smooth solutions for small initial data and the solutions converge to constant equilibrium states.

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

Access this chapter

Log in via an institution
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 PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Boillat, G. (1974): Sur l’existence et la recherche d’équations de conservation supplémentaires pour les systémes hyperboliques. C.R. Acad. Sci. Paris Sér. A 278, 909–912 Boillat, G. (1996): Nonlinear hyperbolic fields and waves. In: Ruggeri, T. (ed.): Recent mathematical methods in nonlinear wave propagation. (Lecture Notes in Mathematics, vol. 1640). Springer, Berlin, pp. 1–47

    MathSciNet  MATH  Google Scholar 

  2. Ruggeri, T., Strumia, A. (1981): Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics. Ann. Inst. H. Poincaré Sect. A (N.S.) 34, 65–84

    MathSciNet  Google Scholar 

  3. Boillat, G., Ruggeri, T. (1997): Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Rational Mech. Anal. (London) 137, 305–320

    Article  MathSciNet  MATH  Google Scholar 

  4. Friedrichs, K.O., Lax, P.D. (1971): Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. USA 68, 1686–1688

    Article  MathSciNet  MATH  Google Scholar 

  5. Godunov, S.K. (1961): An interesting class of quasi-linear systems (Russian). Dokl. Akad. Nauk SSSR 139, 521–523; translated as: Sov. Math. Dokl. 2, 947-949

    MathSciNet  Google Scholar 

  6. Boillat, G., Ruggeri, T. (1998): On the shock structure problem for hyperbolic system of balance laws and convex entropy. Contin. Mech. Thermodyn. 10, 285–292

    Article  MathSciNet  MATH  Google Scholar 

  7. Ruggeri, T. (2000): Maximum of entropy density in equilibrium and minimax principle for an hyperbolic system of balance laws. In: Albers, B. (ed.): Contributions to continuum theories. Anniversary volume for KrzysztoffWilmanski. (WIAS-Report No. 18). Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, pp. 207–214

    Google Scholar 

  8. Ruggeri, T., Serre, D. (2004): Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. Quart. Appl. Math. 62, 163–179

    MathSciNet  MATH  Google Scholar 

  9. Kawashima, S. (1987): Large-time behavior of solutions to hyperbolic-parabolic systems of conservation laws and applications. Proc. Roy. Soc. Edinburgh Sect. A 106, 169–194

    MathSciNet  Google Scholar 

  10. Fischer, A.E., Marsden, J.E. (1972): The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. I. Comm. Math. Phys. 28, 1–38

    Article  MathSciNet  MATH  Google Scholar 

  11. Majda, A. (1984): Compressible fluid flow and systems of conservation laws in several space variables. Springer, New York

    MATH  Google Scholar 

  12. Dafermos, C. (2000): Hyperbolic conservation laws in continuum physics. Springer, Berlin

    MATH  Google Scholar 

  13. Zeng, Y. (1999): Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Ration. Mech. Anal. 150, 225–279

    Article  MathSciNet  MATH  Google Scholar 

  14. Hanouzet, B., Natalini, R. (2003): Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Rat. Mech. Anal. 169, 89

    Article  MathSciNet  MATH  Google Scholar 

  15. Müller, I., Ruggeri, T. (1998): Rational extended thermodynamics. 2nd edition. (Springer Tracts in Natural Philosophy, vol. 37). Springer, New York

    Google Scholar 

  16. Grad, H. (1949): On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2,331–407

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Center of Applied Mathematics, C.I.R.A.M., University of Bologna, Bologna, Italy

    Tommaso Ruggeri

Authors
  1. Tommaso Ruggeri
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. of Mathematics Renato Caccioppoli, University of Naples Federico II, Naples, Italy

    Salvatore Rionero

  2. Dept. of Scienza delle Costruzioni, University of Naples Federico II, Naples, Italy

    Giovanni Romano

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Italia

About this paper

Cite this paper

Ruggeri, T. (2005). Global existence of smooth solutions and stability of the constant state for dissipative hyperbolic systems with applications to extended thermodynamics. In: Rionero, S., Romano, G. (eds) Trends and Applications of Mathematics to Mechanics. Springer, Milano . https://doi.org/10.1007/88-470-0354-7_17

Download citation

  • DOI: https://doi.org/10.1007/88-470-0354-7_17

  • Publisher Name: Springer, Milano

  • Print ISBN: 978-88-470-0269-2

  • Online ISBN: 978-88-470-0354-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation