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 ${\mathbb{R}}^3$ representing a 2d air-water interface immediately fails to be $C^3$ with respect to the initial data if the initial data $(h_0,{\psi }_0)\in H^{s+\frac{1}{2}}\otimes H^s$ for $s<3$. Similar results hold in ${\mathbb{R}}^2$ domains with a 1d interface. Furthermore, we discuss the illposedness threshold for the pure gravity water wave system.