Reduction to a Constant Coefficients Operator and Proof of the Main Theorem
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In the fourth chapter of the book, the capillary-gravity water waves equations have been reduced, up to a smoothing remainder, to a first order diagonal paradifferential system. In this chapter, we reduce this operator to a constant coefficients one, still up to smoothing operators. If the latter operator were self-adjoint, and if we could neglect the smoothing remainders, then we would conclude that the solutions of the evolution problem have Sobolev norms uniformly bounded. Nevertheless, the constant coefficient symbol we find has non zero imaginary part. We thus need to perform a normal forms procedure in order to eliminate such non self-adjoint part, as well as the smoothing contributions, up to remainders that do not make grow significantly the energy over time intervals of size O(𝜖−N). This provides the proof of the almost-global existence result.
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