Geodesic
Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D
Kinoshita, Shinya1
1 Universität 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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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 Hs(R2) 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 L2(R2).

Reçu le :
Révisé le :
Accepté le :
DOI : 10.1016/j.anihpc.2020.08.003
Classification : 35Q53, 35A01
Keywords: Well-posedness, Cauchy problem, Low regularity, Bilinear estimate, Nonlinear Loomis-Whitney inequality

Kinoshita, Shinya 1

1 Universität Bielefeld, Fakultät für Mathematik, Postfach 10 01 31, 33501, Bielefeld, Germany 
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     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},
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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. http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.08.003/

[1] Bejenaru, I.; Herr, S.; Holmer, J.; Tataru, D. On the 2D Zakharov system with L2 Schrödinger data, Nonlinearity, Volume 22 (2009), pp. 1063-1089 | MR | Zbl | DOI

[2] Bejenaru, I.; Herr, S. Convolutions of singular measures and applications to the Zakharov system, J. Funct. Anal., Volume 261 (2011), pp. 478-506 | MR | Zbl | DOI

[3] Bejenaru, I.; Herr, S.; Tataru, D. A convolution estimate for two-dimensional hypersurfaces, Rev. Mat. Iberoam., Volume 26 (2010), pp. 707-728 | MR | Zbl | DOI

[4] Bennett, J.; Carbery, A.; Wright, J. A non-linear generalization of the Loomis-Whitney inequality and applications, Math. Res. Lett., Volume 12 (2005), pp. 443-457 | MR | Zbl | DOI

[5] Biagioni, H.A.; Linares, F. Well-posedness results for the modified Zakharov-Kuznetsov equation, Nonlinear Equations: Methods, Models and Applications, Progr. Nonlinear Differential Equations Appl., vol. 54, Birkhäuser, 2003, pp. 181-189 | MR | Zbl

[6] Bourgain, J. Periodic Korteweg de Vries equation with measures as initial data, Sel. Math. New Ser., Volume 3 (1997) no. 2, pp. 115-159 | MR | Zbl | DOI

[7] Faminskii, A.V. The Cauchy problem for the Zakharov-Kuznetsov equation, Differ. Uravn., Volume 31 (1995), pp. 1070-1081 (in Russian), translation in Differ. Equ., 31, 1995, pp. 1002-1012 | Zbl | MR

[8] Fatah, L.G.; Linares, F.; Pastor, A. A note on the 2D generalized Zakharov–Kuznetsov equation: local, global, and scattering results, J. Differ. Equ., Volume 253 (2012), pp. 2558-2571 | MR | Zbl | DOI

[9] Ginibre, J.; Tsutsumi, Y.; Velo, G. On the Cauchy problem for the Zakharov system, J. Funct. Anal., Volume 151 (1997) no. 2, pp. 384-436 | MR | Zbl | DOI

[10] Grünrock, A. A remark on the modified Zakharov–Kuznetsov equation in three space dimensions, Math. Res. Lett., Volume 21 (2014), pp. 127-131 | MR | Zbl | DOI

[11] Grünrock, A. On the generalized Zakharov–Kuznetsov equation at critical regularity, 2015 | arXiv

[12] Grünrock, A.; Herr, S. The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., Volume 34 (2014), pp. 2061-2068 | MR | Zbl | DOI

[13] Holmer, J. Local ill-posedness of the 1D Zakharov system, Electron. J. Differ. Equ., Volume 24 (2007) (22 pp) | MR | Zbl

[14] Kenig, C.E.; Ponce, G.; Vega, L. Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., Volume 40 (1991), pp. 33-69 | MR | Zbl | DOI

[15] Kenig, C.; Ponce, G.; Vega, L. A bilinear estimate with applications to the KdV equation, J. Am. Soc., Volume 9 (1996), pp. 573-603 | MR | Zbl

[16] Laedke, E.W.; Spatschek, K.-H. Nonlinear ion-acoustic waves in weak magnetic fields, Phys. Fluids, Volume 25 (1982), pp. 985-989 | MR | Zbl | DOI

[17] Lannes, D.; Linares, F.; Saut, J.-C. The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Prog. Nonlinear Differ. Equ. Appl., Volume 84 (2013), pp. 181-213 | MR | Zbl

[18] Linares, F.; Pastor, A. Well-posedness for the two-dimensional modified Zakharov–Kuznetsov equation, SIAM J. Math. Anal., Volume 41 (2009), pp. 1323-1339 | MR | Zbl | DOI

[19] Linares, F.; Pastor, A. Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation, J. Funct. Anal., Volume 260 (2011), pp. 1060-1085 | MR | Zbl | DOI

[20] Linares, F.; Saut, J.-C. The Cauchy problem for the 3D Zakharov–Kuznetsov equation, Discrete Contin. Dyn. Syst., Volume 24 (2009), pp. 547-565 | MR | Zbl | DOI

[21] Loomis, L.; Whitney, H. An inequality related to the isoperimetric inequality, Bull. Am. Math. Soc., Volume 55 (1949), pp. 961-962 | MR | Zbl | DOI

[22] Molinet, L.; Pilod, D. Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 32 (2015), pp. 347-371 | MR | Zbl | mathdoc-id | DOI

[23] Ribaud, F.; Vento, S. A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris, Volume 350 (2012), pp. 499-503 | MR | Zbl | DOI

[24] Ribaud, F.; Vento, S. Well-posedness results for the three-dimensional Zakharov–Kuznetsov equation, SIAM J. Math. Anal., Volume 44 (2012), pp. 2289-2304 | MR | Zbl | DOI

[25] Tao, T. Nonlinear Dispersive Equations: Local and Global Analysis, AMS, 2006 | MR | Zbl

[26] Zakharov, V.E.; Kuznetsov, E.A. Three-dimensional solitons, Sov. Phys. JETP, Volume 39 (1974), pp. 285-286

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