## Differential and Integral Equations

### Weak continuity of dynamical systems for the KdV and mKdV equations

#### Abstract

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

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

Dates
First available in Project Euclid: 20 December 2012

https://projecteuclid.org/euclid.die/1356019070

Mathematical Reviews number (MathSciNet)
MR2742475

Zentralblatt MATH identifier
1240.35448

Subjects