Weak Continuity of Dynamical Systems for the KdV and mKdV Equations

2009/09/04 by Shangbin Cui, Carlos E. Kenig, Cui, Shangbin +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.0909.0794

Abstract

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.

Citations

• On the ill-posedness of some canonical dispersive equations

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