Sharp Local Well-Posedness and Nonlinear Smoothing for Dispersive Equations through Frequency-Restricted Estimates

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Mathematical Analysis Pub Date : 2024-08-07 DOI:10.1137/23m156923x

Simão Correia, Filipe Oliveira, Jorge Drumond Silva

引用次数: 0

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5604-5633, August 2024.
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.

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通过频率限制估算实现离散方程的锐利局部拟合和非线性平滑

SIAM 数学分析期刊》，第 56 卷第 4 期，第 5604-5633 页，2024 年 8 月。摘要。我们考虑的问题是将非线性平滑作为非线性色散方程的一般特征，即相对于初始数据和线性演化的相应正则性，Duhamel 公式中积分项的正则性得到改善，以及这一特性与局部好求解性的关系。首先，我们展示了该问题一般如何简化为特定频率限制估计值的推导，即仅在空间频率上的乘数估计值。然后，我们使用精确的方法，证明了修正的扎哈罗夫-库兹涅佐夫方程、三次方和五次方非线性薛定谔方程以及四次方 Korteweg-de Vries 方程特定情况下的这些估计值。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Mathematical Analysis 数学-应用数学

CiteScore

3.30

自引率

5.00%

发文量

175

审稿时长

12 months

期刊介绍： SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.