On the regularity of the flow map for the gravity-capillary equations

We prove via explicitly constructed initial data that solutions to the gravity-capillary wave system in R3 representing a 2d air–water interface immediately fail to be C3 with respect to the initial data if the initial data for s<3. Similar results hold in R2 domains with a 1d interface. Furthermore, we discuss the related threshold for the pure gravity water wave system.