Abstract
We prove global existence and a modified scattering property for the solutions of the 2D gravity water waves system in the infinite depth setting for a class of initial data, which is only required to be small above the level . No assumption is made below this level. Therefore, the nonlinear solution can have infinite energy. As a direct consequence, the momentum condition assumed on the physical velocity in all previous small energy results by Ionescu‐Pusateri, Alazard‐Delort, and Ifrim‐Tataru is removed.© 2017 Wiley Periodicals, Inc.
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