Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estimates

TL;DR

This paper investigates nonlinear smoothing in dispersive equations, demonstrating how frequency-restricted estimates can establish local well-posedness and improved regularity for specific equations like NLS, ZK, and KdV.

Contribution

It introduces a methodology to derive frequency-restricted estimates, linking nonlinear smoothing to local well-posedness for various dispersive PDEs.

Findings

Proved frequency-restricted estimates for NLS, ZK, and KdV equations.

Established nonlinear smoothing as a feature of these dispersive equations.

Connected smoothing properties to local well-posedness through these estimates.

Abstract

We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e. the improved regularity of the integral term in Duhamel's formula, with respect to the initial data and the corresponding regularity of the linear evolution, and how this property relates to local well-posedness. In a first step, we show how the problem generally reduces to the derivation of specific frequency-restricted estimates, which are multiplier estimates in the spatial frequency alone. Then, using a precise methodology, we prove these estimates for the specific cases of the modified Zakharov-Kuznetsov equation, the cubic and quintic nonlinear Schr\"odinger equation and the quartic Korteweg-de Vries equation.