Linear dispersive decay estimates for the 3+1 dimensional water wave equation with surface tension

Daniel Spirn, J. Douglas Wright

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Abstract

We consider the linearization of the three-dimensional water waves equation with surface tension about a flat interface. Using oscillatory integral methods, we prove that solutions of this equation demonstrate dispersive decay at the somewhat surprising rate of t -5/6. This rate is due to competition between surface tension and gravitation at O(1) wave numbers and is connected to the fact that, in the presence of surface tension, there is a so-called "slowest wave". Additionally, we combine our dispersive estimates with L 2 type energy bounds to prove a family of Strichartz estimates.

Original languageEnglish (US)
JournalCanadian Mathematical Bulletin
Volume55
Issue number1
DOIs
StatePublished - 2012

Keywords

  • Oscillatory integrals
  • Strichartz estimates
  • Surface tension
  • Water waves

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