Complex Formulation of the Equation and Diagonalization of the Matrix Symbol
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In order to prove our main result, the first step is to reduce the capillary-gravity water waves equations to a paradifferential system in a convenient complex unknown, suitable to prove energy estimates. The paralinearization of the water waves equations is obtained in Chap. 7, based on the analysis of Chap. 6. The goal of the present chapter is to present the general form of such a paradifferential system and to state the main Theorem concerning almost global existence of its solutions with a small initial datum. In a preliminary section we present the key algebraic properties of Reality, Parity and Reversibility of such a system that will play a fundamental role for getting energy estimates of the normal form system. The first part of the proof of the main Theorem is presented in the last section of the chapter.