Time-decay estimates for the linearized water wave type equations

Abstract

Recently, A. Bulut showed that the free waves $S_\alpha (t) f:=\exp \left( it |\nabla |^{\alpha }\right) f$ in 1D for $\alpha \in (1/3, 1/2]$, which are known to be associated with the linearized gravity water wave equations, decay at time scale of order $|t|^{-1/2}$ for large t, provided that the $H^1_x(\mathbb {R})$-norm of f and the $L^2_x(\mathbb {R})$-norm of $x\partial _x f$ are bounded. In this note we derive a decay estimate of order $(1-\alpha )^{-1/2} (\alpha |t|)^{-d/2}$ on $S_\alpha (t)f$ for all $\alpha \in (0, 1)$ and $d\ge 1$, assuming a bound only on the $\dot{B}_{1, 1}^{d(1-\alpha /2)} (\mathbb {R}^d)$-norm of f. Our estimate extends to any dimension, a wider range of $\alpha$ and describes well the behaviour of the decay near $\alpha =0$ and $\alpha =1$, without requiring a spatial-decay assumption on f or its derivative.