JournalsaihpcVol. 33, No. 2pp. 337–395

Cauchy theory for the gravity water waves system with non-localized initial data

  • T. Alazard

    Département de Mathématiques et Applications, UMR 8553 du CNRS, École Normale Supérieure, 45, rue d'Ulm, 75005 Paris Cedex, France
  • N. Burq

    Laboratoire de Mathématiques d'Orsay, UMR 8628 du CNRS, Université Paris-Sud, 91405 Orsay Cedex, France, Département de Mathématiques et Applications, UMR 8553 du CNRS, École Normale Supérieure, 45, rue d'Ulm, 75005 Paris Cedex, France
  • C. Zuily

    Laboratoire de Mathématiques d'Orsay, UMR 8628 du CNRS, Université Paris-Sud, 91405 Orsay Cedex, France
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Abstract

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 L2L^{2}-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/2d/ 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.

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. 337–395

DOI 10.1016/J.ANIHPC.2014.10.004