Nonlinear smoothing for dispersive PDE: a unified approach

TL;DR

This paper introduces a unified method using normal form reductions to prove nonlinear smoothing for various dispersive PDEs, improving and extending existing results across multiple equations and regularity ranges.

Contribution

A general theorem for nonlinear smoothing in dispersive PDEs is developed, applicable to several classical equations, and enhances previous results by broadening regularity conditions.

Findings

Nonlinear smoothing holds for five classical dispersive equations.

The method matches or improves existing smoothing results.

The approach extends the range of Sobolev regularities where smoothing occurs.

Abstract

In several cases of nonlinear dispersive PDEs, the difference between the nonlinear and linear evolutions with the same initial data, i.e. the integral term in Duhamel's formula, exhibits improved regularity. This property is usually called nonlinear smoothing. We employ the method of infinite iterations of normal form reductions to obtain a very general theorem yielding existence of nonlinear smoothing for dispersive PDEs, contingent only on establishing two particular bounds. We then apply this theorem to show that the nonlinear smoothing property holds, depending on the regularity of the initial data, for five classical dispersive equations: in $\mathbb{R}$, the cubic nonlinear Schr\"odinger, the Korteweg-de Vries, the modified Korteweg-de Vries and the derivative Schr\"odinger equations; in $\mathbb{R}^2$, the modified Zakharov-Kuznetsov equation. For the aforementioned one-dimensional equations, this unifying methodology matches or improves the existing nonlinear smoothing results, while enlarging the ranges of Sobolev regularities where the property holds.