The initial value problem for the sequence of generalized Korteweg-de Vries equations
type: Item
instance of: scholarly article
title: The initial value problem for the sequence of generalized Korteweg-de Vries equations
zbMATH Open document ID: 0581.35062
DOI: 10.1016/0001-8708(84)90035-5
author: Martin jun. Schwarz
published in: Advances in Mathematics
publication date: 1984-01-01T00:00:00Z
review text: The initial value problem for the nonlinear equation \[ (1)\quad \frac{\partial u}{\partial t}=K_ m(u),\quad u\quad periodic\quad in\quad x, \] which is the m-th member of the Korteweg-de Vries hierarchy is considered. Let \(H_ n(S^ 1)\) denote the classical Sobolev space with the norm \(\| v\|^ 2_ n=\sum_{\ell =n}\int^{1}_{0}| D^{\ell}v(x)|^ 2 dx\). The main result is the following Theorem. For each m, if \(n\geq 2m-1\) and the initial datum g is an element of \(H_ n(S^ 1)\) then there exists a global solution u(x,t) to (1) from \(H_ n(S^ 1)\). Furthermore, for \(n\geq 3m-2\), u(x,t) is uniquely determined by g. A similar study for the KdV equation which corresponds to \(m=2\) has been carried previously by Temam, Sjöberg and Kato.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35G25
zbMATH DE Number: 3929572
zbMATH Keywords: a priori estimates
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: initial value problem
zbMATH Keywords: Korteweg-de Vries hierarchy
zbMATH Keywords: Sobolev space
zbMATH Keywords: global solution
MaRDI profile type: Publication
full work available at URL: 0001-8708(84)90035-5
OpenAlex ID: W2100465657
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description: scientific article; zbMATH DE number 3929572