On Sharpness of the Local Kato Smoothing Property of Dispersive Wave Equations

TL;DR

This paper proves that the local Kato smoothing property for solutions of dispersive wave equations in one dimension is sharp, showing that the regularity gain cannot be improved beyond a certain limit for some initial data.

Contribution

It demonstrates the sharpness of the local Kato smoothing property for 1D dispersive equations, establishing the optimality of the known regularity gain.

Findings

Existence of initial data for which smoothing cannot be improved

Sharpness of the local Kato smoothing property in 1D

Limits of regularity gain for dispersive wave solutions

Abstract

Constantin and Saut showed in 1988 that solutions of the Cauchy problem for general dispersive equations $$ w_t +iP(D)w=0,\quad w(x,0)=q (x), \quad x\in \mathbb{R}^n, \ t\in \mathbb{R} , $$ enjoy the local smoothing property $$ q\in H^s (\R ^n) \implies w\in L^2 \Big (-T,T; H^{s+\frac{m-1}{2}}_{loc} \left (\R^n\right )\Big ) , $$ where $m$ is the order of the pseudo-differential operator $P(D)$. This property, now called local Kato smoothing, was first discovered by Kato for the KdV equation and implicitly shown later by Sj\"olin for the linear Schr\"odinger equation. In this paper, we show that the local Kato smoothing property possessed by solutions general dispersive equations in the 1D case is sharp, meaning that there exist initial data $q\in H^s \left (\R \right )$ such that the corresponding solution $w$ does not belong to the space $ L^2 \Big (-T,T; H^{s+\frac{m-1}{2} +\epsilon}_{loc} \left (\R\right )\Big )$ for any $\epsilon >0$.