Archive for Rational Mechanics and Analysis

, Volume 229, Issue 2, pp 601–625 | Cite as

Three-Scale Singular Limits of Evolutionary PDEs

  • Bin Cheng
  • Qiangchang Ju
  • Steve SchochetEmail author
Open Access


Singular limits of a class of evolutionary systems of partial differential equations having two small parameters and hence three time scales are considered. Under appropriate conditions solutions are shown to exist and remain uniformly bounded for a fixed time as the two parameters tend to zero at different rates. A simple example shows the necessity of those conditions in order for uniform bounds to hold. Under further conditions the solutions of the original system tend to solutions of a limit equation as the parameters tend to zero.


  1. 1.
    Browning G., Kreiss H.-O.: Problems with different time scales for nonlinear partial differential equations. SIAM J. Appl. Math., 42(4), 704–718 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cheng B.: Improved accuracy of incompressible approximation of compressible Euler equations. SIAM J. Math. Anal., 46(6), 3838–3864 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Friedman, A.: Partial Differential Equations. Krieger, Huntington, NY, 1976Google Scholar
  4. 4.
    Gallagher I.: Applications of Schochet’s methods to parabolic equations. J. Math. Pures Appl., 77(10), 989–1054 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Grenier E.: Pseudo-differential energy estimates of singular perturbations. Commun. Pure Appl. Math., 50(9), 821–865 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kato T.: A Short Introduction to Perturbation Theory for Linear Operators. Springer, New York (1982)CrossRefzbMATHGoogle Scholar
  7. 7.
    Klainerman S., Majda A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math, 34, 481–524 (1981)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Klainerman S., Majda A.: Compressible and incompressible fluids. Commun. Pure Appl. Math., 35, 629–653 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables, Vol. 53 Applied Mathematical Sciences. Springer, New York, 1984Google Scholar
  10. 10.
    Majda, A., Klein, R.: Systematic multiscale models for the tropics. J. Atmos. Sci., 60(393–408), 2003Google Scholar
  11. 11.
    Métivier G., Schochet S.: The incompressible limit of the non-isentropic euler equations. Arch. Ration. Mech. Anal, 158, 61–90 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Schochet S.: Asymptotics for symmetric hyperbolic systems with a large parameter. J. Differ. Equ., 75, 1–27 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Schochet S.: Fast singular limits of hyperbolic PDEs. J. Differ. Equ., 114(2), 476–512 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SurreyGuildfordUK
  2. 2.Institute of Applied Physics and Computational MathematicsBeijingChina
  3. 3.School of Mathematical SciencesTel-Aviv UniversityTel AvivIsrael

Personalised recommendations