Dispersive Estimates for Linearized Water Wave-Type Equations in \(\mathbb {R}^d\)

Abstract

We derive a \(L^1_x (\mathbb {R}^d)-L^{\infty }_x (\mathbb {R}^d)\) decay estimate of order \(\mathcal O \left( t^{-d/2}\right) \) for the linear propagators

$$\begin{aligned} \exp \left( {\pm it \sqrt{ |D|\left( 1+ \beta |D|^2\right) \tanh |D | } }\right) , \qquad \beta \in \{0, 1\}. \quad D= -i\nabla , \end{aligned}$$

with a loss of 3d/4 or d/4–derivatives in the case \(\beta =0\) or \(\beta =1\), respectively. These linear propagators are known to be associated with the linearized water wave equations, where the parameter \(\beta \) measures surface tension effects. As an application, we prove low regularity well-posedness for a Whitham–Boussinesq-type system in \(\mathbb {R}^d\), \(d\ge 2\). This generalizes a recent result by Dinvay, Selberg and the third author where they proved low regularity well-posedness in \(\mathbb {R}\) and \(\mathbb {R}^2\).