Global solutions for the gravity water waves equation in dimension 3

Abstract

We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of $L^2$ related norms, with dispersive estimates, which give decay in $L^\infty$. To obtain these dispersive estimates, we use an analysis in Fourier space; the study of space and time resonances is then the crucial point.

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      DOI = {10.2977/prims/1195184016},
      ZBLNUMBER = {0493.76018},
      }
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    @article{zz06,
      author={Zhang, P. and Zhang, Z.},
      TITLE = {On the free boundary problem of three-dimensional incompressible {E}uler equations},
      JOURNAL = {Comm. Pure Appl. Math.},
      VOLUME = {61},
      YEAR = {2008},
      MRNUMBER = {2410409},
      ZBLNUMBER = {1158.35107},
      DOI = {10.1002/cpa.20226},
      }

Authors

Pierre Germain

Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY 10012

Nader Masmoudi

Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY 10012

Jalal Shatah

Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY 10012