Consider a three-dimensional fluid in a rectangular tank, bounded by a flat bottom, vertical walls and a free surface evolving under the influence of gravity. We prove that one can estimate its energy by looking only at the motion of the points of contact between the free surface and the vertical walls. The proof relies on the multiplier technique, the Craig–Sulem–Zakharov formulation of the water-wave problem, a Pohozaev identity for the Dirichlet to Neumann operator, previous results about the Cauchy problem and computations inspired by the analysis done by Benjamin and Olver of the conservation laws for water waves.
@article{AIHPC_2018__35_3_751_0,
author = {Alazard, Thomas},
title = {Boundary observability of gravity water waves},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {751--779},
publisher = {Elsevier},
volume = {35},
number = {3},
year = {2018},
doi = {10.1016/j.anihpc.2017.07.006},
mrnumber = {3778651},
zbl = {1445.76016},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2017.07.006/}
}
TY - JOUR AU - Alazard, Thomas TI - Boundary observability of gravity water waves JO - Annales de l'I.H.P. Analyse non linéaire PY - 2018 SP - 751 EP - 779 VL - 35 IS - 3 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2017.07.006/ DO - 10.1016/j.anihpc.2017.07.006 LA - en ID - AIHPC_2018__35_3_751_0 ER -
%0 Journal Article %A Alazard, Thomas %T Boundary observability of gravity water waves %J Annales de l'I.H.P. Analyse non linéaire %D 2018 %P 751-779 %V 35 %N 3 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2017.07.006/ %R 10.1016/j.anihpc.2017.07.006 %G en %F AIHPC_2018__35_3_751_0
Alazard, Thomas. Boundary observability of gravity water waves. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 751-779. doi : 10.1016/j.anihpc.2017.07.006. http://archive.numdam.org/articles/10.1016/j.anihpc.2017.07.006/
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