Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D

Kinoshita, Shinya

doi:10.1016/j.anihpc.2020.08.003

Kinoshita, Shinya ¹

¹Universität Bielefeld, Fakultät für Mathematik, Postfach 10 01 31, 33501, Bielefeld, Germany

Annales de l'I.H.P. Analyse non linéaire, mars – avril 2021, Tome 38 (2021) no. 2, pp. 451-505

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Résumé

This paper is concerned with the Cauchy problem of the 2D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space $H^{s} (R^{2})$ for $s > - 1 / 4$ , and these are optimal up to the endpoint. We utilize the nonlinear version of the classical Loomis-Whitney inequality and develop an almost orthogonal decomposition of the set of resonant frequencies. As a corollary, we obtain global well-posedness in $L^{2} (R^{2})$ .

Reçu le : 2019-05-07
Révisé le : 2020-07-31
Accepté le : 2020-08-06

MR Zbl

DOI : 10.1016/j.anihpc.2020.08.003

Classification : 35Q53, 35A01
Keywords: Well-posedness, Cauchy problem, Low regularity, Bilinear estimate, Nonlinear Loomis-Whitney inequality

Affiliations des auteurs :

Kinoshita, Shinya ¹

¹ Universität Bielefeld, Fakultät für Mathematik, Postfach 10 01 31, 33501, Bielefeld, Germany

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     author = {Kinoshita, Shinya},
     title = {Global well-posedness for the {Cauchy} problem of the {Zakharov-Kuznetsov} equation in {2D}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {451--505},
     year = {2021},
     publisher = {Elsevier},
     volume = {38},
     number = {2},
     doi = {10.1016/j.anihpc.2020.08.003},
     mrnumber = {4211993},
     zbl = {1458.35373},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.08.003/}
}

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Kinoshita, Shinya. Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D. Annales de l'I.H.P. Analyse non linéaire, mars – avril 2021, Tome 38 (2021) no. 2, pp. 451-505. doi: 10.1016/j.anihpc.2020.08.003

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