Advertisement

Cauchy Problem for Some 2 × 2 Hyperbolic Systems of Pseudo-differential Equations with Nondiagonalisable Principal Part

  • Todor GramchevEmail author
  • Michael Ruzhansky
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 84)

Abstract

We investigate the well-posedness of the Cauchy problem for some first-order linear hyperbolic systems having nondiagonalisable principal parts.

Key words

Cauchy problem Hyperbolic systems Pseudo-differential equations 

Notes

Acknowledgements

This work was completed with the support of the research impulse grant of Imperial College London 2011. The first author was also supported by a research grant of the Università di Cagliari (regional funds 2010). The second author was also supported by the EPSRC Leadership Fellowship.

References

  1. 1.
    Arnold, V.: Matrices depending on parameters. Uspekhi Nauk U.S.S.R. 26, 101–114 (1971)Google Scholar
  2. 2.
    Colombini, F., Nishitani, T.: Two by two strongly hyperbolic systems and Gevrey classes. In: Workshop on Partial Differential Equations (Ferrara, 1999). Ann. Univ. Ferrara Sez. VII (N.S.), vol. 45 (1999), suppl., pp. 79–108 (2000)Google Scholar
  3. 3.
    Cordes, H.O.: On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators. J. Funct. Anal. 18, 115–131 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Craig, W.: Nonstrictly hyperbolic nonlinear systems. Math. Ann. 277, 213–232 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    D’Abbicco, M., Taglialatela, G.: Some results on the well-posedness for systems with time dependent coefficients. Ann. Fac. Sci. Toulouse Math., (6) 18, 247–284 (2009)Google Scholar
  6. 6.
    D’Ancona, P., Kinoshita, T., Spagnolo, S.: On the 2 by 2 weakly hyperbolic systems. Osaka J. Math. 45, 921–939 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Duistermaat, J.J.: Fourier integral operators. Progress in Mathematics, vol. 130. Birkhäuser Boston, Inc., Boston (1996)Google Scholar
  8. 8.
    Gramchev, T.: Perturbation of systems with Nilpotent real part. In: Gaeta, G. (ed.) Editor–in–Chief R. A. Meyers, Section Perturbation Theory. Encyclopedia of Complexity and Systems Science, pp. 6649–6667. Springer, Berlin (2009)Google Scholar
  9. 9.
    Gramchev, T., Orrù, N.: Cauchy problem for a class of nondiagonalisable hyperbolic systems. Discrete and Continuous Dynamical Systems. In: Dynamical Systems, Differential Equations and Applications Proceedings of the 8th AIMS Conference at Dresden, Germany, May 25–28, 2010, vol. Supplement 2011, pp. 533–542. American Institute of Mathematical Sciences, Springfield (MO)Google Scholar
  10. 10.
    Kajitani, K.: Cauchy problem for nonstrictly hyperbolic systems in Gevrey classes. J. Math. Kyoto Univ. 23, 599–616 (1983)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kamotski, I., Ruzhansky, M.: Representation of solutions and regularity properties for weakly hyperbolic systems. In: Pseudo-Differential Operators and Related Topics. Operator Theory: Advances and Applications, vol. 164, pp. 53–63. Birkhäuser, Basel (2006)Google Scholar
  12. 12.
    Kamotski, I., Ruzhansky, M.: Regularity properties, representation of solutions, and spectral asymptotics of systems with multiplicities. Comm. Part. Differ. Equat. 32, 1–35 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kumano-go, H.: Pseudodifferential Operators. MIT, Cambridge (1981)Google Scholar
  14. 14.
    Mizohata, S.: On the Cauchy problem. Notes and Reports in Mathematics in Science and Engineering, 3. Academic Press, Inc., Orlando, FL; vi+177 Science Press, Beijing, (1985)Google Scholar
  15. 15.
    Petkov, V.: Microlocal forms for hyperbolic systems. Math. Nachr. 93, 117–131 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Rozenblum, G.: Spectral asymptotic behavior of elliptic systems. (Russian) Zap. LOMI 96, 255–271 (1980)zbMATHGoogle Scholar
  17. 17.
    Ruzhansky, M., Sugimoto, M.: Global L 2-boundedness theorems for a class of Fourier integral operators. Comm. Part. Differ. Equat. 31, 547–569 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ruzhansky, M., Sugimoto, M.: Weighted Sobolev L2 estimates for a class of Fourier integral operators. Math. Nachr. 284, 1715–1738 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Taglialatela, G., Vaillant, J.: Remarks on the Levi conditions for differential systems. Hokkaido Math. J. 37, 463–492 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Taylor, M.: Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28, 457–478 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Taylor, M.: In: Pseudodifferential Operators. Princeton Mathematical Series, vol. 34. Princeton University Press, Princeton (1981)Google Scholar
  22. 22.
    Vaillant, J.: Invariants des systèmes d’opérateurs différentiels et sommes formelles asymptotiques. Jpn. J. Math. New Ser. 25, 1–153 (1999)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Yamahara, H.: Cauchy problem for hyperbolic systems in Gevrey class. A note on Gevrey indices. Ann. Fac. Sci. Toulouse Math., (6) 9, 147–160 (2000)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di CagliariCagliariItaly
  2. 2.Mathematics DepartmentImperial College LondonLondonUK

Personalised recommendations