Dirichlet–Neumann Operator and the Good Unknown

  • Massimiliano Berti
  • Jean-Marc Delort
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 24)


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.


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Authors and Affiliations

  • Massimiliano Berti
    • 1
  • Jean-Marc Delort
    • 2
  1. 1.Department of MathematicsInternational School for Advanced Studies SISSATriesteItaly
  2. 2.LAGASorbonne Paris-Cité/University Paris 13VilletaneuseFrance

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