Skip to main content
Log in

On well-posedness of the Cauchy problem and the mixed problem for some class of hyperbolic systems and equations with constant coefficients and variable multiplicity of characteristics

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. M. S. Agranovich, “Boundary value problems for systems of first order pseudodifferential operators,” Russ. Math. Surv., 24, No. 1, 59–126 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  2. F. Brini, “Hyperbolicity region in extended thermodynamics with 14 moments,” Contin. Mech. Thermodyn., 13, 1–8 (2001).

    Article  MATH  Google Scholar 

  3. G.-Q. Chen, C. D. Levermore, and T.-P. Luui, “Hyperbolic conservation laws with stiff relaxation terms and entropy,” Commun. Pure Appl. Math., 47, 787–830 (1994).

    Article  MATH  Google Scholar 

  4. W. Dreyer and H. Struchtrup, “Heat pulse experiments revisted,” Contin. Mech. Thermodyn., 5, 3–50 (1993).

    Article  MathSciNet  Google Scholar 

  5. Yu. V. Egorov, M. A. Shubin, and R. V. Gamkrelidze (eds.), Partial Differential Equations I. Foundations of the Classical Theory, Encyclopaedia of Mathematical Sciences, 30, Springer-Verlag, Berlin (1992).

    MATH  Google Scholar 

  6. S. K. Godunov, Ordinary differential equations with constant coefficients, Translations of Mathematical Monographs, 169, AMS, Providence (1997).

    Google Scholar 

  7. C. Hermite, Oeuvres I, Gauthier-Villars, Paris (1905).

    MATH  Google Scholar 

  8. R. Hersh, “Mixed problems in several variables,” J. Math. Mech., 12, 317–334 (1963).

    MATH  MathSciNet  Google Scholar 

  9. L. Hörmander, Linear Partial Differential Operators. Mathematics and Applications, Springer-Verlag, Berlin-Göttingen-Heidelberg (1963).

    Google Scholar 

  10. J. Leray, Hyperbolic Differential Equations [in Russian], Nauka, Moscow (1984).

    MATH  Google Scholar 

  11. C. D. Levermore, “Moment closure hierarchies for kinetic theories,” J. Statist. Phys., 83, 1021–1065 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  12. I. Müller and T. Ruggeri, Extended Thermodynamics, Springer-Verlag, New York (1993).

    MATH  Google Scholar 

  13. I. G. Petrovskii, “On the Cauchy problem for systems of differential equations with partial derivatives,” Selected Works. Systems of Equations with Partial Derivatives. Algebraic Geometry [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  14. E. V. Radkevich, “Well-posedness of mathematical models in continuum mechanics and thermodynamics,” J. Math. Sci., 124, No. 3, 5335–5363 (2004).

    Article  MathSciNet  Google Scholar 

  15. P. Sakamoto, “Mixed problems for hyperbolic equations I, II,” J. Math. Kyoto Univ., 10, No. 3, 349–373, 403–417 (1970).

    MATH  MathSciNet  Google Scholar 

  16. L. R. Volevich and S. G. Gindikin, Mixed Problems for Partial Differential Equations with Quasihomogeneous Principal Part, Translations of Mathematical Monographs, 147, AMS, Providence (1999).

    Google Scholar 

  17. L. R. Volevich and V. Ya. Ivrij, Comments to I. G. Petrovskii, “On the Cauchy problem for systems of differential equtions with partial derivatives,” Selected Works. Systems of Equations with Partial Derivatives. Algebraic Geometry [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  18. L. R. Volevich and E. V. Radkevich, “Uniform estimates of solutions of the Cauchy problem for hyperbolic equations with a small parameter multiplying higher derivatives,” Differ. Eq., 39, No. 4, 521–535 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  19. L. R. Volevich and E. V. Radkevich, “Stable pencils of hyperbolic polynomials and Cauchy problems for hyperbolic equations with a small parameter multiplying higher derivatives,” Tr. Mosk. Mat. Obshch., 65, 69–113 (2004).

    MathSciNet  Google Scholar 

  20. G. B. Whitham, Linear and Nonlinear Waves. Pure and Applied Mathematics, John Wiley & Sons, New York-London-Sydney (1974).

    MATH  Google Scholar 

  21. P. A. Zakharchenko and E. V. Radkevich, “On hyperbolic pencils of Grad systems of moments arising in nonequilibrium thermodynamics,” Tr. Semin. im. I. G. Petrovskogo, 24, 67–94 (2004).

    MathSciNet  Google Scholar 

  22. P. A. Zakharchenko and E. V. Radkevich, “On properties of dispersion equations of systems of moments for the Fokker-Planck equation,” Differ. Uravn., 42, No. 1, 1–10 (2006).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to E. V. Radkevich.

Additional information

__________

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 16, Differential and Functional Differential Equations. Part 2, 2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Radkevich, E.V. On well-posedness of the Cauchy problem and the mixed problem for some class of hyperbolic systems and equations with constant coefficients and variable multiplicity of characteristics. J Math Sci 149, 1580–1607 (2008). https://doi.org/10.1007/s10958-008-0083-3

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-0083-3

Keywords

Navigation