Weak Continuity of Dynamical Systems for the KdV and mKdV Equations

Details Weak Continuity of Dynamical Systems for the KdV and mKdV Equations Weak Continuity of Dynamical Systems for the KdV and mKdV Equations

2009-09-04

arxiv.org/abs/0909.0794v2

Mathematics

Analysis of PDEs

21 pages, no figures

Scientific paper

In this paper we study weak continuity of the dynamical systems for the KdV equation in H^{-3/4}(R) and the modified KdV equation in H^{1/4}(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 a mKdV system and the generalized Miura transform to get weak continuity for the KdV equation.

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