Geodesic
Cauchy theory for the gravity water waves system with non-localized initial data
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 337-395.

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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 L2-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 d/2 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.

DOI : 10.1016/j.anihpc.2014.10.004
Keywords: Water-waves, Cauchy problem, Uniformly local Sobolev spaces, Paradifferential calculus 
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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, Tome 33 (2016) no. 2, pp. 337-395. doi : 10.1016/j.anihpc.2014.10.004. http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.10.004/

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