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 − ( − Δ ) 1 / 4 u = 0 iu_t-(-\Delta )^{1/4}u=0 for ( t , x ) ∈ R × R (t,x)\in \mathbb {R}\times \mathbb {R} . This corresponds to a factorization of the linearized water wave equation u t t + ( − Δ ) 1 / 2 u = 0 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 |t|^{-1/2} for solutions corresponding to data u ( 0 ) = φ u(0)=\varphi , assuming only bounds on ‖ φ ‖ H x 1 ( R ) \lVert \varphi \rVert _{H_x^1(\mathbb {R})} and ‖ x ∂ x φ ‖ L x 2 ( R ) \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 i u t − ( − Δ ) α / 2 u = 0 iu_t-(-\Delta )^{\alpha /2}u=0 for a wider range of α \alpha .