We consider the problem of global in time existence and uniqueness of solutions of the 3-D infinite depth full water wave problem. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial interface that is sufficiently small in its steepness and velocity, we show that there exists a unique smooth solution of the full water wave problem for all time, and the solution decays at the rate $1/t$.
Global well-posedness and scattering for the defocusing $H^{\frac12}$-subcritical Hartree equation in $\mathbb{R}^d$
Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case
Global well-posedness of the short-pulse and sine-Gordon equations in energy space
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