An optimal decay estimate for the linearized water wave equation in 2D

TL;DR

This paper establishes decay estimates of order |t|^{-1/2} for solutions to the linearized water wave equation in 2D, using Littlewood-Paley and stationary phase techniques, with implications for a broader class of dispersive equations.

Contribution

It provides the first decay estimate for the linearized water wave equation in 2D using harmonic analysis methods, extending results to equations with fractional Laplacians.

Findings

Decay of order |t|^{-1/2} for solutions with initial data in certain Sobolev spaces

Extension of decay estimates to equations with general fractional powers of Laplacian

Use of Littlewood-Paley decomposition and stationary phase in dispersive PDE analysis

Abstract

We obtain a decay estimate for solutions to the linear dispersive equation $iu_t-(-\Delta)^{1/4}u=0$ for $(t,x)\in\mathbb{R}\times\mathbb{R}$. This corresponds to a factorization of the linearized water wave equation $u_{tt}+(-\Delta)^{1/2}u=0$. In particular, by making use of the Littlewood-Paley decomposition and stationary phase estimates, we obtain decay of order $|t|^{-1/2}$ for solutions corresponding to data $u(0)=\varphi$, assuming only bounds on $\lVert \varphi\rVert_{H_x^1(\mathbb{R})}$ and $\lVert x\partial_x\varphi\rVert_{L_x^2(\mathbb{R})}$. As another application of these ideas, we give an extension to equations of the form $iu_t-(-\Delta)^{\alpha/2}u=0$ for a wider range of $\alpha$.