Geodesic
A sharp Cauchy theory for the 2D gravity-capillary waves
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1793-1836

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This article is devoted to the Cauchy problem for the 2D gravity-capillary water waves in fluid domains with general bottoms. Local well-posedness for this problem with Lipschitz initial velocity was established by Alazard–Burq–Zuily [1]. We prove that the Cauchy problem in Sobolev spaces is uniquely solvable for initial data 14-derivative less regular than the aforementioned threshold, which corresponds to the gain of Hölder regularity of the semi-classical Strichartz estimate for the fully nonlinear system. In order to obtain this Cauchy theory, we establish global, quantitative results for the paracomposition theory of Alinhac [5].

DOI : 10.1016/j.anihpc.2016.12.007
Keywords: Water waves, Cauchy problem, Semi-classical Strichartz estimate, Paracomposition 
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     author = {Nguyen, Huy Quang},
     title = {A sharp {Cauchy} theory for the {2D} gravity-capillary waves},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1793--1836},
     publisher = {Elsevier},
     volume = {34},
     number = {7},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.12.007},
     zbl = {1451.76028},
     mrnumber = {3724757},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.12.007/}
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Nguyen, Huy Quang. A sharp Cauchy theory for the 2D gravity-capillary waves. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1793-1836. doi: 10.1016/j.anihpc.2016.12.007

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