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 $iu_t-(-\Delta )^{1/4}u=0$ 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 $\alpha$.