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

2023/02/07 by Simão Correia, Correia, Simão, Filipe Oliveira +3 · 2 citations
Mathematics · Physics and Astronomy · #35A01 #35B65 #35Q53 #35Q55 #42B37 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2302.03575

openalex publication_date 2023/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Schrödinger equation and the quartic Korteweg-de Vries equation.

Cited by

• Nonlinear smoothing implies improved lower bounds on the radius of spatial analyticity for nonlinear dispersive equations
• Exponential mixing for Korteweg-de Vries equation with localized noise

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