The quartic integrability and long time existence of steep water waves in 2D

TL;DR

This paper demonstrates the partial integrability of 2D steep water waves by constructing energy functionals that ensure long-time existence of solutions without restrictions on initial steepness or velocity.

Contribution

It introduces explicit energy functionals in physical space that prove long-term existence of steep water wave solutions, extending previous partial integrability results.

Findings

Solutions exist for at least O(ε^{-3}) time under certain norms.

Solutions exist for at least O(ε^{-5/2}) time with only scaling invariant norm small.

No restrictions on initial interface steepness or velocity magnitude.

Abstract

It is known since the work of Dyachenko \& Zakharov \cite{zd} that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the non-trivial resonant manifold. In this paper we study this partial integrability from a different point of view. We construct, directly in the physical space, a sequence of energy functionals $\mathfrak E_j(t)$ which are explicit in the Riemann mapping variable and involve material derivatives of order $j$ of the solutions for the 2d water wave equation, so that $\frac d{dt} \mathfrak E_j(t)$ is quintic or higher order. We show that if some scaling invariant norm, and a norm involving one spatial derivative above the scaling of the initial data are of size no more than $\varepsilon$, then the lifespan of the solution for the 2d water wave equation is at least of order $O(\varepsilon^{-3})$, and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size $\varepsilon$, then the lifespan of the solution is at least of order $O(\varepsilon^{-5/2})$. Our long time existence results do not impose size restrictions on the slope of the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses and initial velocities to have arbitrary large magnitudes.