Large Time Well-posedness of the Three-Dimensional Capillary-Gravity Waves in the Long Wave Regime

In the regime of weakly transverse long waves, given long-wave initial data, we prove that a non-dimensionalized water wave system in an infinite strip under the influence of gravity and surface tension on the upper free interface has a unique solution on 0, T/ e{0, T/ \varepsilon} ] for some e{\varepsilon} independent of constant T. We shall prove in the subsequent paper (Ming et al., The long wave approximation to the three-dimensional capillary gravity waves, 2011) that on the same time interval, these solutions can be accurately approximated by sums of solutions of two decoupled Kadomtsev–Petviashvili (KP) equations.