Differential and Integral Equations

Weak continuity of dynamical systems for the KdV and mKdV equations

Shangbin Cui and Carlos E. Kenig

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In this paper we study weak continuity of the dynamical systems for the KdV equation in $H^{-3/4}(\mathbb{R})$ and the modified KdV equation in $H^{1/4}(\mathbb{R})$. This topic should have significant applications in the study of other properties of these equations such as finite time blow-up and asymptotic stability and instability of solitary waves. The spaces considered here are borderline Sobolev spaces for the corresponding equations from the viewpoint of the local well-posedness theory. We first use a variant of the method of [5] to prove weak continuity for the mKdV, and next use a similar result for an mKdV system and the generalized Miura transform to get weak continuity for the KdV equation.

Article information

Differential Integral Equations, Volume 23, Number 11/12 (2010), 1001-1022.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]


Cui, Shangbin; Kenig, Carlos E. Weak continuity of dynamical systems for the KdV and mKdV equations. Differential Integral Equations 23 (2010), no. 11/12, 1001--1022. https://projecteuclid.org/euclid.die/1356019070

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