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Linear Dispersive Decay Estimates for the 3+1 Dimensional Water Wave Equation with Surface Tension
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- Journal:
- Canadian Mathematical Bulletin / Volume 55 / Issue 1 / 01 March 2012
- Published online by Cambridge University Press:
- 20 November 2018, pp. 176-187
- Print publication:
- 01 March 2012
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- Article
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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.
