In this article, we develop the local Cauchy theory for the gravity water waves system, for rough initial data which do not decay at infinity. We work in the context of -based uniformly local Sobolev spaces introduced by Kato [22]. We prove a classical well-posedness result (without loss of derivatives). Our result implies also a local well-posedness result in Hölder spaces (with loss of derivatives). As an illustration, we solve a question raised by Boussinesq in [9] on the water waves problem in a canal. We take benefit of an elementary observation to show that the strategy suggested in [9] does indeed apply to this setting.
@article{AIHPC_2016__33_2_337_0, author = {Alazard, T. and Burq, N. and Zuily, C.}, title = {Cauchy theory for the gravity water waves system with non-localized initial data}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {337--395}, publisher = {Elsevier}, volume = {33}, number = {2}, year = {2016}, doi = {10.1016/j.anihpc.2014.10.004}, zbl = {1339.35227}, mrnumber = {3465379}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.10.004/} }
TY - JOUR AU - Alazard, T. AU - Burq, N. AU - Zuily, C. TI - Cauchy theory for the gravity water waves system with non-localized initial data JO - Annales de l'I.H.P. Analyse non linéaire PY - 2016 SP - 337 EP - 395 VL - 33 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2014.10.004/ DO - 10.1016/j.anihpc.2014.10.004 LA - en ID - AIHPC_2016__33_2_337_0 ER -
%0 Journal Article %A Alazard, T. %A Burq, N. %A Zuily, C. %T Cauchy theory for the gravity water waves system with non-localized initial data %J Annales de l'I.H.P. Analyse non linéaire %D 2016 %P 337-395 %V 33 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2014.10.004/ %R 10.1016/j.anihpc.2014.10.004 %G en %F AIHPC_2016__33_2_337_0
Alazard, T.; Burq, N.; Zuily, C. Cauchy theory for the gravity water waves system with non-localized initial data. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 2, pp. 337-395. doi : 10.1016/j.anihpc.2014.10.004. http://www.numdam.org/articles/10.1016/j.anihpc.2014.10.004/
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