Mathematische Zeitschrift

, Volume 233, Issue 1, pp 75–91

On the blow-up rate and the blow-up set of breaking waves for a shallow water equation

  • Adrian Constantin
  • Joachim Escher
Original article

DOI: 10.1007/PL00004793

Cite this article as:
Constantin, A. & Escher, J. Math Z (2000) 233: 75. doi:10.1007/PL00004793


We consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation. Blow-up can occur only in the form of wave-breaking, i.e. the solution remains bounded but its slope becomes unbounded in finite time. A quite detailed description of the wave-breaking phenomenon is given: there is at least a point (in general depending on time) where the slope becomes infinite exactly at breaking time. The precise blow-up rate is established and for a large class of initial data we also determine the blow-up set.

Mathematics Subject Classification (1991):35L05.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Adrian Constantin
    • 1
  • Joachim Escher
    • 1
  1. 1.Institute for Mathematics, University Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland CH