On the ill-posedness of some canonical dispersive equations.
type: Item
instance of: scholarly article
title: On the ill-posedness of some canonical dispersive equations.
zbMATH Open document ID: 1034.35145
DOI: 10.1215/S0012-7094-01-10638-8
author: C. E. Kenig
author: G. Ponce
author: L. Vega
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: This paper is concerned with the local well-posedness of the initial value problem for some classical nonlinear dispersive equations on the real line, namely \[ i\partial_tu+\partial_x^2 u\pm| u|^2 u= 0 \] (cubic nonlinear Schrödinger), \[ \partial_tu+ \partial^3_x u+ | u|^2\partial_xu=0 \] (complex modified Korteweg-de Vries), \[ \partial_t u+\partial_x^3 u+u^k\partial_xu=0,\;k\in \mathbb{Z}^+ \] (\(k\)-generalized Korteweg-de Vries). The authors establish the minimum regularity property required in the data which guarantees the local well-posedness of the problem. Measuring this regularity in the classical Sobolev spaces they show ill-posedness results for Sobolev index above the value suggested by the scaling argument.
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 1820790
zbMATH Keywords: generalized Korteweg-de Vries
zbMATH Keywords: local well-posedness
zbMATH Keywords: nonlinear Schrödinger
zbMATH Keywords: modified Korteweg-de Vries
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: ill-posedness
zbMATH Keywords: Sobolev index
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
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full work available at URL: s0012-7094-01-10638-8
OpenAlex ID: W1982778586
description: scientific article; zbMATH DE number 1820790