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It is known since the work of Dyachenko \& Zakharov \cite{zd} that for theweakly nonlinear 2d infinite depth water waves, there are no 3-waveinteractions and all of the 4-wave interaction coefficients vanish on thenon-trivial resonant manifold. In this paper we study this partialintegrability from a different point of view.  We construct, directly in the physical space, a sequence of energyfunctionals $\mathfrak E_j(t)$ which are explicit in the Riemann mappingvariable and involve material derivatives of order $j$ of the solutions for the2d water wave equation, so that $\frac d{dt} \mathfrak E_j(t)$ is quintic orhigher order. We show that if some scaling invariant norm, and a norm involvingone spatial derivative above the scaling of the initial data are of size nomore than $\varepsilon$, then the lifespan of the solution for the 2d waterwave equation is at least of order $O(\varepsilon^{-3})$, and the solutionremains as regular as the initial data during this time. If only the scalinginvariant norm of the data is of size $\varepsilon$, then the lifespan of thesolution is at least of order $O(\varepsilon^{-5/2})$. Our long time existenceresults do not impose size restrictions on the slope of the initial interfaceand the magnitude of the initial velocity, they allow the interface to havearbitrary large steepnesses and initial velocities to have arbitrary largemagnitudes.

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