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 . 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  on the water waves problem in a canal. We take benefit of an elementary observation to show that the strategy suggested in  does indeed apply to this setting.
Cite this article
T. Alazard, N. Burq, C. Zuily, Cauchy theory for the gravity water waves system with non-localized initial data. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 2, pp. 337–395DOI 10.1016/J.ANIHPC.2014.10.004