Abstract
AbstractWe 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 L2 type energy bounds to prove a family of Strichartz estimates.
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