Dispersive estimates for linearized water wave type equations in ${\mathbb{R}}^d$

Abstract

We derive a $L_x^1({\mathbb{R}}^d)-L_x^{\infty }({\mathbb{R}}^d)$ decay estimate of order $\mathcal{O}\left(t^{-d/2}\right)$ for the linear propagators $$\exp \left(\pm it\sqrt{|D|\left(1+\beta |D{|}^2\right)\tanh |D|}\right),\beta \in \{0,1\}.D=-i\nabla ,$$ 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 ${\mathbb{R}}^2$.

19 pages; To appear in Annales Henri Poincare