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

Aynur Bulut

Abstract

We obtain a decay estimate for solutions to the linear dispersive equation $iu_t-(-Δ)^{1/4}u=0$ for $(t,x)\in\mathbb{R}\times\mathbb{R}$. This corresponds to a factorization of the linearized water wave equation $u_{tt}+(-Δ)^{1/2}u=0$. 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 $iu_t-(-Δ)^{α/2}u=0$ for a wider range of $α$.

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

Abstract

We obtain a decay estimate for solutions to the linear dispersive equation for . This corresponds to a factorization of the linearized water wave equation . In particular, by making use of the Littlewood-Paley decomposition and stationary phase estimates, we obtain decay of order for solutions corresponding to data , assuming only bounds on and . As another application of these ideas, we give an extension to equations of the form for a wider range of .

Paper Structure

This paper contains 3 sections, 4 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Proof of the main result
  3. Concluding remarks

Key Result

Theorem \oldthetheorem

Let $\Phi:\mathbb{R}\rightarrow\mathbb{R}$ be given by $\Phi(\xi)=|\xi|^{1/2}$ for $\xi\in\mathbb{R}$. Then there exists $C>0$ such that estimate holds for every $t\in\mathbb{R}$ and $\varphi\in \mathcal{S}(\mathbb{R})$, where the operator $e^{it\Phi(D)}$ is defined by

Theorems & Definitions (8)

  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • proof : Proof of Theorem $\ref{['thm1']}$
  • Proposition \oldthetheorem
  • proof : Sketch of proof