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Dirichlet–Neumann Operator and the Good Unknown

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

Abstract

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.

References

  1. 4.
    Alazard, T., Métivier, G.: Paralinearization of the Dirichlet to Neumann operator, and regularity of three-dimensional water waves. Commun. Partial Differ. Equ. 34(10–12), 1632–1704 (2009). https://doi.org/10.1080/03605300903296736 MathSciNetCrossRefGoogle Scholar
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    Alazard, T., Burq, N., Zuily, C.: On the water-wave equations with surface tension. Duke Math. J. 158(3), 413–499 (2011). https://doi.org/10.1215/00127094-1345653 MathSciNetCrossRefGoogle Scholar
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    Alazard, T., Burq, N., Zuily, C.: On the Cauchy problem for gravity water waves. Invent. Math. 198(1), 71–163 (2014). https://doi.org/10.1007/s00222-014-0498-z MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

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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