Linear Dispersive Decay Estimates for the 3+1 Dimensional Water Wave Equation with Surface Tension

http://dx.doi.org/10.4153/CMB-2011-057-3
Canad. Math. Bull. 55(2012), 176-187

  Published:2011-03-31
 Printed: Mar 2012

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.