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 $i{u}_t-(-\Delta {)}^{1/4}u=0$ 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 $u(0)=\phi$, assuming only bounds on $\Vert \phi {\Vert }_{H_x^1(\mathbb{R})}$ and $\Vert x{\partial }_x\phi {\Vert }_{L_x^2(\mathbb{R})}$. As another application of these ideas, we give an extension to equations of the form $i{u}_t-(-\Delta {)}^{\alpha /2}u=0$ for a wider range of $\alpha$.

New result added (see Section 3). To appear in Proc. Amer. Math. Soc

References in corpus (3)

• Two dimensional water waves in holomorphic coordinates
• Global analysis of a model for capillary water waves in 2D
• Nonstandard estimates for a class of 1D dispersive equations and applications to linearized water waves