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

TL;DR

This paper demonstrates that solutions to the gravity-capillary wave system in 3D are not smoothly dependent on initial data in certain Sobolev spaces, revealing ill-posedness at low regularity levels.

Contribution

It constructs explicit initial data showing immediate failure of $C^3$ regularity for solutions, and discusses ill-posedness thresholds for pure gravity water waves.

Findings

Solutions are not $C^3$ with respect to initial data for $s<3$ in 3D.

Similar ill-posedness results hold in 2D domains.

Discusses the ill-posedness threshold for the gravity water wave system.

Abstract

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.