Abtract: In this first talk we present a road map for deriving a priori control in low regularity spaces for solutions to several completely integrable equations. This type of control is an essential requirement for any well-posedness result in such spaces. We present this method in the context of the Korteweg-de Vries equation, and then discuss the case of AKNS systems. This is based on joint work with Rowan Killip and Xiaoyi Zhang.
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