Multilinear estimates for periodic KdV equations, and applications
type: Item
instance of: scholarly article
title: Multilinear estimates for periodic KdV equations, and applications
zbMATH Open document ID: 1062.35109
DOI: 10.1016/S0022-1236(03)00218-0
author: James Colliander
author: Markus Keel
author: Gigliola Staffilani
author: Hideo Takaoka
author: Terence Tao
published in: Journal of Functional Analysis
publication date: 2004-08-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0110049
review text: The authors deal with the Cauchy problem for periodic generalized Korteweg-de Vries (KdV) equations of the form \[ \begin{cases} \partial_t u+\frac{1}{4\pi^2} \partial^3_x u+ (F(u))_x= 0,\quad & u:\mathbb{T}\times [0,T]\to \mathbb{R},\\ u(x,0)= u_0(x),\quad & x\in\mathbb{T},\end{cases}\tag{1} \] where \(F\) is a polynomial of degree \(k+ 1\), the initial data \(u_0\) is in a Sobolev space \(H^s(\mathbb{T})\), and \(\mathbb{T}= \mathbb{R}/\mathbb{Z}\) is the torus. The main result is a sharp multilinear estimate which allows the authors to prove that (1) is locally well-posed in \(H^s(\mathbb{T})\) for \(s> 1/2\).
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 2083558
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: nonlinear dispersive equations
zbMATH Keywords: multilinear harmonic analysis
zbMATH Keywords: locally well-posed
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
arXiv ID: math/0110049
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OpenAlex ID: W2075677712
description: scientific article