Strictly Hyperbolic Cauchy Problems with Coefficients Low-Regular in Time and Space

  • Daniel LorenzEmail author
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We consider the strictly hyperbolic Cauchy problem
$$\displaystyle \begin {cases} D_t^m u - \sum \limits _{j = 0}^{m-1} \sum \limits _{|\gamma |+j = m} a_{m-j,\,\gamma }(t,\,x) D_x^\gamma D_t^j u = 0,\\ D_t^{k-1}u(0,\,x) = g_k(x),\,k = 1,\,\ldots ,\,m, \end {cases} $$
for \((t,\,x) \in [0,\,T]\times \mathbb {R}^n\) with coefficients belonging to the Zygmund class \(C_\ast ^s\) in x and having a modulus of continuity below Lipschitz in t. Imposing additional conditions to control oscillations, we obtain a global (on [0, T]) L2 energy estimate without loss of derivatives for \(s \geq \max \{1+\varepsilon ,\,\frac {2m_0}{2-m_0}\}\), where m0 is linked to the modulus of continuity of the coefficients in time.


Cauchy problem Modulus of continuity Zygmund space Low-regular Strictly hyperbolic Higher order 



The author wants to express his gratitude to Michael Reissig for many fruitful discussions and suggestions. Furthermore, he wants to thank Daniele Del Santo for his hospitality and the suggested improvements during the authors stay at Trieste University.


  1. 1.
    Abels, H.: Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients. Comm. Partial Differential Equations 30.10 (2005), 1463–1503, doi:
  2. 2.
    Agliardi, R., Cicognani, M.: Operators of p-evolution with nonregular coefficients in the time variable. J. Differential Equations 202.1 (2004), 143–157, doi:
  3. 3.
    Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations., Springer, Berlin and Heidelberg (2011).CrossRefGoogle Scholar
  4. 4.
    Cicognani, M., Colombini, F.: Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem. J. Differential Equations 221.1 (2006), 143–157, doi:
  5. 5.
    Cicognani, M., Lorenz, D.: Strictly hyperbolic equations with coefficients low-regular in time and smooth in space. J. Pseudo-Differ. Oper. Appl. (2017), online first, doi:
  6. 6.
    Colombini, F., De Giorgi E., Spagnolo, S.: Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Scuola. Norm.-Sci. 6.3 (1979), 511–559.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Colombini, F., Del Santo, D., Kinoshita, T.: Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients. Ann. Sc. Norm. Super. Pisa Cl. Sci. 1.2 (2002), 327–358.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Colombini, F., Del Santo, D., Reissig, M.: On the optimal regularity of coefficients in hyperbolic Cauchy problems. Bull. Sci. Math. 127.4 (2003), 328–347, doi:
  9. 9.
    Colombini, F., Lerner, N.: Hyperbolic operators with non-Lipschitz coefficients. Duke Math. J. 77.3 (1995), 657–698, doi:
  10. 10.
    Hirosawa, F.: On the Cauchy problem for second order strictly hyperbolic equations with non-regular coefficients. Math. Nachr. 256 (2003), 29–47, doi:
  11. 11.
    Hirosawa, F., Reissig, M.: Well-Posedness in Sobolev Spaces for Second-Order Strictly Hyperbolic Equations with Nondifferentiable Oscillating Coefficients. Ann. Global Anal. Geom. 25.2 (2004), 99–119, doi:
  12. 12.
    Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin, Heidelberg (1963).CrossRefGoogle Scholar
  13. 13.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators., Reprint of the 1994 ed., Springer, Berlin (2007).Google Scholar
  14. 14.
    Hurd, A. E., Sattinger, D. H.: Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients. Trans. Amer. Math. Soc. 132 (1968), 159–174, doi:
  15. 15.
    Jachmann, K., Wirth, J.: Diagonalisation schemes and applications. Ann. Mat. Pura Appl (4) 189.4 (2010), 571–590.Google Scholar
  16. 16.
    Kinoshita, T., Reissig, M.: About the loss of derivatives for strictly hyperbolic equations with non-Lipschitz coefficients. Adv. differential Equations 10.2 (2005), 191–222.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kumano-go, H.: Pseudo-differential operators., English-language ed. MIT Press, Cambridge, Massachusetts 1982.zbMATHGoogle Scholar
  18. 18.
    Marschall, J., Löfström, J., Marschall, J.: Pseudo-differential operators with nonregular symbols of the class Sm p p. Comm. Partial Differential Equations 12.8 (1987), 921–965, doi:
  19. 19.
    Mizohata, S.: The theory of partial differential equations., Cambridge University Press, New York 1973.zbMATHGoogle Scholar
  20. 20.
    Reissig, M.: Hyperbolic equations with non-Lipschitz coefficients. Rend. Sem. Mat. Univ. Politec. Torino 61.2 (2003), 135–182.MathSciNetzbMATHGoogle Scholar
  21. 21.
    Tataru, D.: On the Fefferman-Phong inequality and related problems. Comm. Partial Differential Equations 27.11-12 (2002), 2101–2138, doi:
  22. 22.
    Taylor, M.: Pseudodifferential Operators and Nonlinear PDE., Vol. 100. Progress in Mathematics. Birkäuser, Boston 1973.Google Scholar
  23. 23.
    Triebel, H.: Theory of function spaces., Modern Birkhäuser Classics. Springer, Basel 2010.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.TU Bergakademie Freiberg, Faculty of Mathematics and Computer ScienceInstitute of Applied AnalysisFreibergGermany

Personalised recommendations