Illposedness of the gravity-capillary equations

Details Illposedness of the gravity-capillary equations Illposedness of the gravity-capillary equations

2011-11-22

arxiv.org/abs/1111.5361v1

Mathematics

Analysis of PDEs

30 pages, no figures

Scientific paper

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 $(h_0, \psi_0) \in H^{s+\frac12} \otimes H^{s}$ 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.

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