Dirichlet–Neumann Operator and the Good Unknown
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It is well known that it is not possible to directly derive energy estimates with no loss of derivatives for the solutions of the capillary-gravity water waves equations in the original Zakharov variables (η, ψ). The way to overcome such an issue is now well understood, and it is based on the introduction of a paradifferential good unknown. In this chapter we adapt to our framework the Alazard–Métivier construction that permits to rewrite the water waves equations as an hyperbolic quasi-linear paradifferential system that allows energy estimates in terms of a new complex valued variable.