A dispersive estimate for the linearized Water-Waves equations in finite depth

TL;DR

This paper establishes a dispersive decay estimate of order 1/3 for solutions to the linearized Water-Waves equations in one dimension with finite depth, aiding the analysis of the full nonlinear system.

Contribution

It provides the first dispersive estimate for the linearized Water-Waves equations in finite depth and applies it to prove existence results for the nonlinear system.

Findings

Decay rate of 1/3 for solutions with weighted Sobolev initial data

Variants with different decay rates for practical applications

Existence results for the full Water-Waves equations in weighted spaces

Abstract

We prove a dispersive estimate for the solutions of the linearized Water-Waves equations in dimension 1 in presence of a flat bottom. We prove a decay with respect to time t of order 1/3 for solutions with initial data in weighted Sobolev spaces. We also give variants to this result with different decays for a more convenient use of the dispersive estimate. We then give an existence result for the full Water-Waves equations in weighted spaces for practical uses of the proven dispersive estimates.