Advertisement

First-Order Hyperbolic Systems

  • Carles BonaEmail author
  • Carles Bona-Casas
  • Carlos Palenzuela-Luque
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 783)

From the mathematical point of view, the mixed-type systems (first order in time, second order in space) that we have considered in the previous chapter are associated with the parabolic type of equations. The prototype could be the Navier–Stokes equation of fluid dynamics, where second-order space derivatives appear in the viscosity terms. Parabolic equations are not the ones usually associated with causal propagation phenomena, where a finite propagation speed can be derived in a natural way from the governing equations.

Keywords

Hyperbolic System Energy Estimate Principal Part Characteristic Matrix Characteristic Speed 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    T. De Donder, La Gravifique Einstenienne, Gauthier-Villars, Paris (1921).Google Scholar
  2. 2.
    T. De Donder, The Mathematical Theory of Relativity, Massachusetts Institute of Technology, Cambridge (1927).zbMATHGoogle Scholar
  3. 3.
    K. Lanczos, Ann. Phys. 13, 621 (1922).Google Scholar
  4. 4.
    K. Lanczos, Z. Phys. 23, 537 (1923).Google Scholar
  5. 5.
    V. A. Fock, The Theory of Space, Time and Gravitation, Pergamon, London (1959).zbMATHGoogle Scholar
  6. 6.
    L. E. Kidder, M. A. Scheel and S. A. Teukolsky, Phys. Rev. D64, 064017 (2001).ADSMathSciNetGoogle Scholar
  7. 7.
    O. Sarbach and M. Tiglio, Phys. Rev. D66, 064023 (2002).ADSMathSciNetGoogle Scholar
  8. 8.
    H. O. Kreiss and J. Lorentz, Initial-Boundary Problems and the Navier-Stokes Equations, Academic Press, New York (1989).zbMATHGoogle Scholar
  9. 9.
    Y. Choquet-Bruhat and T. Ruggeri, Commun. Math. Phys. 89, 269 (1983).zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    R. Arnowit, S. Deser and C. W. Misner. Gravitation: An Introduction to Current Research, ed. by L. Witten, Wiley, New York (1962). gr-qc/0405109.Google Scholar
  11. 11.
    C. Bona and J. Massó, Phys. Rev. Lett. 68, 1097 (1992).zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    C. Bona, J. Massó, E. Seidel and J. Stela, Phys. Rev. Lett. 75, 600 (1995).CrossRefADSGoogle Scholar
  13. 13.
    K. Alvi, Class. Quantum Grav, 19, 5153 (2002).zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    M. Holst, et al., Phys. Rev. D70, 084017 (2004).ADSMathSciNetGoogle Scholar
  15. 15.
    O. Brodbeck, S. Frittelli, P. Huebner and O. A. Reula, J. Math. Phys. 40, 909 (1999).zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    S. Frittelli and O. A. Reula, Commun. Math. Phys. 166, 221 (1994).zbMATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    S. Frittelli and O. A. Reula, Phys. Rev. Lett. 76, 4667 (1996).CrossRefADSGoogle Scholar
  18. 18.
    A. Anderson and J. W. York, Jr., Phys. Rev. Lett. 82, 4384 (1999).zbMATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    S. D. Hern, Numerical Relativity and Inhomogeneous Cosmologies. Ph. D. Thesis, gr-qc/0004036 (2000).Google Scholar
  20. 20.
    A. Abrahams, A. Anderson, Y. Choquet-Bruhat and J. W. York, Phys. Rev. Lett. 75, 3377 (1995).zbMATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    H. Friedrich, Class. Quantum. Grav. 13, 1451 (1996).zbMATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    M. H. P. van Putten and D. M. Eardley, Phys. Rev. D53, 3056 (1996).ADSGoogle Scholar
  23. 23.
    M. H. P. van Putten, Phys. Rev. D55, 4705 (1997).ADSGoogle Scholar
  24. 24.
    G. Nagy, O. Ortiz and O. Reula, Phys. Rev. D70, 044012 (2004).ADSMathSciNetGoogle Scholar
  25. 25.
    C. Bona, T. Ledvinka, C. Palenzuela and M. Žáček, Phys. Rev. D69, 064036 (2004).ADSGoogle Scholar
  26. 26.
    C. Bona, T. Ledvinka, C. Palenzuela and M. Žáček, Phys. Rev. D67, 104005 (2003).ADSGoogle Scholar
  27. 27.
    F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005).CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    L. Lindblom et al., Class. Quantum Grav, 23, S447 (2006).zbMATHCrossRefADSMathSciNetGoogle Scholar
  29. 29.
    L. Lindblom and M. A. Scheel, Phys. Rev. D67, 124005 (2003).ADSMathSciNetGoogle Scholar
  30. 30.
    C. Bona and C. Palenzuela, Phys. Rev. D69, 104003 (2004).ADSGoogle Scholar
  31. 31.
    D. Alic, C. Bona and C. Bona-Casas, Phys. Rev. D (2009). ArXiv:0811.1691Google Scholar
  32. 32.
    M. Alcubierre and B. Brügmann, Phys. Rev. D63, 104006 (2001).ADSGoogle Scholar
  33. 33.
    M. Alcubierre et al., Phys. Rev. D67, 084023 (2003).ADSMathSciNetGoogle Scholar
  34. 34.
    P. Olsson, Math. Comput. 64, 1035 (1995).zbMATHCrossRefADSMathSciNetGoogle Scholar
  35. 35.
    W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Veterling, Numerical Recipes, Cambridge University Press, Cambridge (1989).zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Carles Bona
    • 1
    Email author
  • Carles Bona-Casas
    • 1
  • Carlos Palenzuela-Luque
    • 2
  1. 1.Departament de FísicaUniversitat de les Illes BalearsPalma de MallorcaSpain
  2. 2.Max-Planck-Institut für Gravitationsphysik (Albert Einstein Institut)GolmGermany

Personalised recommendations