Gravity capillary wave equations on the circle, normal forms and long time existence: a review

Since the beginning of the program, results on long time existence for solutions to gravity/capillary wave equations with small data have been described, in particular in the series of lectures of Daniel Tataru during the Introductory workshop. In this talk, we aim at giving a review of a set of results concerning the possibility of going past the "cubic life span" for solutions to the gravity-capillary system with small data in the periodic setting.

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