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Transactions of the American Mathematical Society

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Sharp Strichartz estimates for water waves systems


Author: Huy Quang Nguyen
Journal: Trans. Amer. Math. Soc. 370 (2018), 8797-8832
MSC (2010): Primary 35Q31; Secondary 35S50
DOI: https://doi.org/10.1090/tran/7419
Published electronically: September 13, 2018
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Abstract: Water waves are well-known to be dispersive at the linearization level. Considering the fully nonlinear systems, we prove for reasonably smooth solutions the optimal Strichartz estimates for pure gravity waves and the semi-classical Strichartz estimates for gravity-capillary waves for both 2D and 3D waves. Here, by optimal we mean the gains of regularity (over the Sobolev embedding from Sobolev spaces to Hölder spaces) obtained for the linearized systems. Our proofs combine the paradifferential reductions of Alazard-Burq-Zuily with a dispersive estimate using a localized wave package type parametrix of Koch-Tataru.


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Additional Information

Huy Quang Nguyen
Affiliation: Program in Applied $&$ Computational Mathematics, Princeton University, Princeton, New Jersey 08544
Email: qn@math.princeton.edu

DOI: https://doi.org/10.1090/tran/7419
Received by editor(s): September 20, 2016
Received by editor(s) in revised form: June 25, 2017, and September 21, 2017
Published electronically: September 13, 2018
Additional Notes: The author was supported in part by Agence Nationale de la Recherche project ANAÉ ANR-13-BS01-0010-03.
This work was partially supported by the labex LMH through grant no. ANR-11-LABX-0056-LMH in the “Programme des Investissements d’Avenir”.
Article copyright: © Copyright 2018 American Mathematical Society

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