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

Published:2011-03-31
Printed: Mar 2012
• Daniel Spirn,
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
• J. Douglas Wright,
Department of Mathematics, Drexel University, Philadelphia, PA 19104, U.S.A.
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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.
 Keywords: oscillatory integrals, water waves, surface tension, Strichartz estimates
 MSC Classifications: 76B07 - Free-surface potential flows 76B15 - Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30] 76B45 - Capillarity (surface tension) [See also 76D45]