Advertisement

Smoothing Effect in Quasilinear Wave Equations

  • Hajer BahouriEmail author
  • Jean-Yves Chemin
  • Raphaël Danchin
Chapter
  • 4.4k Downloads
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 343)

Abstract

Chapter 9 is devoted to the study of a class of quasilinear wave equations which can be seen as a toy model for the Einstein equations. First, by taking advantage of energy methods in the spirit of those of Chapter 4, we establish local well-posedness for “smooth” initial data (i.e., for data in Sobolev spaces embedded in the set of Lipschitz functions). Next, we weaken our regularity assumptions by taking advantage of the dispersive nature of the wave equation. The key to that improvement is a quasilinear Strichartz estimate and a refinement of the paradifferential calculus. To prove the quasilinear Strichartz estimate, we use a microlocal decomposition of the time interval (i.e., a decomposition in some interval, the length of which depends on the size of the frequency) and geometrical optics.

Keywords

Wave Equation Cauchy Sequence Besov Space Geometrical Optic Eikonal Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Hajer Bahouri
    • 1
    Email author
  • Jean-Yves Chemin
    • 2
  • Raphaël Danchin
    • 3
  1. 1.Départment de Mathématiques, Faculté des Sciences de Tunis, Campus UniversitaireUniversité de Tunis El ManarTunisTunisia
  2. 2.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie CurieParis Cedex 05France
  3. 3.Centre de Mathématiques, Faculté de Sciences et TechnologieUniversité Paris XII-Val de MarneCréteil CedexFrance

Personalised recommendations