Weak Continuity of Dynamical Systems for the KdV and mKdV Equations
نویسندگان
چکیده
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 H1/4(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 the “local-smoothing” estimates for the Airy operator together with the uniqueness of the solution of the mKdV in suitable function spaces 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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