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Bilinear Dispersive Estimates Via Space Time Resonances, Dimensions Two and Three

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Abstract

Consider a bilinear interaction between two linear dispersive waves with a generic resonant structure (roughly speaking, space and time resonant sets intersect transversally). We derive an asymptotic equivalent of the solution for data in the Schwartz class, and bilinear dispersive estimates for data in weighted Lebesgue spaces. An application to water waves with infinite depth, gravity and surface tension is also presented.

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Authors and Affiliations

  1. Laboratoire Jean Leray, CNRS-Université de Nantes, 2, rue de la Houssinière, 44322, Nantes Cedex 3, France

    Frédéric Bernicot

  2. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012-1185, USA

    Pierre Germain

Authors
  1. Frédéric Bernicot
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  2. Pierre Germain
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Corresponding author

Correspondence to Frédéric Bernicot.

Additional information

Communicated by L. Saint-Raymond

P. Germain is partially supported by NSF grant DMS-1101269, a start-up grant from the Courant Institute, and a Sloan fellowship.

F. Bernicot is partially supported by the ANR under the project AFoMEN no. 2011-JS01-001-01.

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Bernicot, F., Germain, P. Bilinear Dispersive Estimates Via Space Time Resonances, Dimensions Two and Three. Arch Rational Mech Anal 214, 617–669 (2014). https://doi.org/10.1007/s00205-014-0764-7

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  • DOI: https://doi.org/10.1007/s00205-014-0764-7

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