no. 3
Subcritical well-posedness results for the Zakharov–Kuznetsov equation in dimension three and higher
[Résultats sous-critiques sur le caractère bien posé de l’équation de Zakharov–Kuznetsov en dimension supérieure ou égale à trois]
Herr, Sebastian1 ; Kinoshita, Shinya2
1 Universität Bielefeld Fakultät für Mathematik Postfach 10 01 31 33501 Bielefeld (Germany)
2 Department of Mathematics Graduate School of Science and Engineering Saitama University Saitama 338-8570 (Japan)
Annales de l'Institut Fourier, Tome 73 (2023) no. 3, pp. 1203-1267.

On considère l’équation de Zakharov–Kuznetsov en dimension d≥3. On établit que le problème de Cauchy est localement bien posé dans H s pour tout exposant sous-critique s>(d-4)/2. Ceci est optimal jusqu’au cas limite. Comme corollaire, il s’ensuit que l’équation est globalement bien posée dans L 2 (ℝ 3 ) et, sous une hypothèse de petitesse, dans H 1 (ℝ 4 ).

The Zakharov–Kuznetsov equation in space dimension d≥3 is considered. It is proved that the Cauchy problem is locally well-posed in H s (ℝ d ) in the full subcritical range s>(d-4)/2, which is optimal up to the endpoint. As a corollary, global well-posedness in L 2 (ℝ 3 ) and, under a smallness condition, in H 1 (ℝ 4 ), follow.

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DOI : 10.5802/aif.3547
Classification : 35Q55
Keywords: Well-posedness, Cauchy problem, Zakharov–Kuznetsov equation, bilinear estimate, nonlinear Loomis–Whitney inequality.
Mot clés : Caractère bien posé, problème de Cauchy, équation de Zakharov–Kuznetsov, estimation bilinéaire, inégalité de Loomis–Whitney non-linéaire
Herr, Sebastian 1 ; Kinoshita, Shinya 2

1 Universität Bielefeld Fakultät für Mathematik Postfach 10 01 31 33501 Bielefeld (Germany)
2 Department of Mathematics Graduate School of Science and Engineering Saitama University Saitama 338-8570 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Herr, Sebastian; Kinoshita, Shinya. Subcritical well-posedness results for the Zakharov–Kuznetsov equation in dimension three and higher. Annales de l'Institut Fourier, Tome 73 (2023) no. 3, pp. 1203-1267. doi : 10.5802/aif.3547. https://aif.centre-mersenne.org/articles/10.5802/aif.3547/

[1] Bejenaru, I.; Herr, S.; Holmer, J.; Tataru, D. On the 2D Zakharov system with L 2 -Schrödinger data, Nonlinearity, Volume 22 (2009) no. 5, pp. 1063-1089 | DOI | MR | Zbl

[2] Bejenaru, Ioan; Herr, Sebastian Convolutions of singular measures and applications to the Zakharov system, J. Funct. Anal., Volume 261 (2011) no. 2, pp. 478-506 | DOI | MR | Zbl

[3] Bejenaru, Ioan; Herr, Sebastian; Tataru, Daniel A convolution estimate for two-dimensional hypersurfaces, Rev. Mat. Iberoam., Volume 26 (2010) no. 2, pp. 707-728 | DOI | MR | Zbl

[4] Bellan, Paul M. Fundamentals of Plasma Physics, Cambridge University Press, Cambridge, 2006 | DOI

[5] Bennett, Jonathan; Carbery, Anthony; Wright, James A non-linear generalisation of the Loomis–Whitney inequality and applications, Math. Res. Lett., Volume 12 (2005) no. 4, pp. 443-457 | DOI | MR | Zbl

[6] Bhattacharya, Debdeep; Farah, Luiz Gustavo; Roudenko, Svetlana Global well-posedness for low regularity data in the 2d modified Zakharov–Kuznetsov equation (https://arxiv.org/abs/1906.05822)

[7] Biagioni, H. A.; Linares, F. Well-posedness results for the modified Zakharov–Kuznetsov equation, Nonlinear equations: methods, models and applications (Bergamo, 2001) (Progr. Nonlinear Differential Equations Appl.), Volume 54, Birkhäuser, Basel, 2003, pp. 181-189 | MR | Zbl

[8] Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., Volume 3 (1993) no. 3, pp. 209-262 | DOI | MR | Zbl

[9] Candy, Timothy; Herr, Sebastian Transference of bilinear restriction estimates to quadratic variation norms and the Dirac-Klein-Gordon system, Anal. PDE, Volume 11 (2018) no. 5, pp. 1171-1240 | DOI | MR | Zbl

[10] Côte, Raphaël; Muñoz, Claudio; Pilod, Didier; Simpson, Gideon Asymptotic stability of high-dimensional Zakharov–Kuznetsov solitons, Arch. Ration. Mech. Anal., Volume 220 (2016) no. 2, pp. 639-710 | DOI | MR | Zbl

[11] Faminskiĭ, A. V. The Cauchy problem for the Zakharov–Kuznetsov equation, Differentsial’ nye Uravneniya, Volume 31 (1995) no. 6, p. 1070-1081, 1103 | MR

[12] Farah, Luiz G.; Linares, Felipe; Pastor, Ademir A note on the 2D generalized Zakharov–Kuznetsov equation: local, global, and scattering results, J. Differential Equations, Volume 253 (2012) no. 8, pp. 2558-2571 | DOI | MR | Zbl

[13] 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 | DOI | MR | Zbl

[14] Grünrock, Axel On the generalized Zakharov-Kuznetsov equation at critical regularity (https://arxiv.org/abs/1509.09146)

[15] Grünrock, Axel A remark on the modified Zakharov–Kuznetsov equation in three space dimensions, Math. Res. Lett., Volume 21 (2014) no. 1, pp. 127-131 | DOI | MR | Zbl

[16] Grünrock, Axel; Herr, Sebastian The Fourier restriction norm method for the Zakharov–Kuznetsov equation, Discrete Contin. Dyn. Syst., Volume 34 (2014) no. 5, pp. 2061-2068 | DOI | MR | Zbl

[17] Grünrock, Axel; Panthee, Mahendra; Drumond Silva, Jorge On KP-II type equations on cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 26 (2009) no. 6, pp. 2335-2358 | DOI | Numdam | MR | Zbl

[18] Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., Volume 40 (1991) no. 1, pp. 33-69 | DOI | MR | Zbl

[19] Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., Volume 9 (1996) no. 2, pp. 573-603 | DOI | MR | Zbl

[20] Kinoshita, Shinya Well-posedness for the Cauchy problem of the modified Zakharov-Kuznetsov equation (https://arxiv.org/abs/1911.13265)

[21] Kinoshita, Shinya Global well-posedness for the Cauchy problem of the Zakharov–Kuznetsov equation in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 38 (2021) no. 2, pp. 451-505 | DOI | MR | Zbl

[22] Koch, Herbert; Steinerberger, Stefan Convolution estimates for singular measures and some global nonlinear Brascamp-Lieb inequalities, Proc. Roy. Soc. Edinburgh Sect. A, Volume 145 (2015) no. 6, pp. 1223-1237 | DOI | MR | Zbl

[23] Laedke, E. W.; Spatschek, K. H. Nonlinear ion-acoustic waves in weak magnetic fields, Phys. Fluids, Volume 25 (1982) no. 6, pp. 985-989 | DOI | MR | Zbl

[24] Lannes, David; Linares, Felipe; Saut, Jean-Claude The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Studies in phase space analysis with applications to PDEs (Progr. Nonlinear Differential Equations Appl.), Volume 84, Birkhäuser/Springer, New York, 2013, pp. 181-213 | DOI | MR | Zbl

[25] Linares, Felipe; Pastor, Ademir Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., Volume 41 (2009) no. 4, pp. 1323-1339 | DOI | MR | Zbl

[26] Linares, Felipe; Pastor, Ademir Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation, J. Funct. Anal., Volume 260 (2011) no. 4, pp. 1060-1085 | DOI | MR | Zbl

[27] Linares, Felipe; Saut, Jean-Claude The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., Volume 24 (2009) no. 2, pp. 547-565 | DOI | MR | Zbl

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

[29] Molinet, Luc; Pilod, Didier Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 32 (2015) no. 2, pp. 347-371 | DOI | Numdam | MR | Zbl

[30] Ribaud, Francis; Vento, Stéphane A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 9-10, pp. 499-503 | DOI | MR | Zbl

[31] Ribaud, Francis; Vento, Stéphane Well-posedness results for the three-dimensional Zakharov–Kuznetsov equation, SIAM J. Math. Anal., Volume 44 (2012) no. 4, pp. 2289-2304 | DOI | MR | Zbl

[32] Sulem, Catherine; Sulem, Pierre-Louis The nonlinear Schrödinger equation, Applied Mathematical Sciences, 139, Springer-Verlag, New York, 1999, xvi+350 pages (Self-focusing and wave collapse) | MR | Zbl

[33] Zakharov, V. E.; Kuznetsov, E. A. Three-dimensional solitons, Sov. Phys. JETP, Volume 39 (1974) no. 2, pp. 285-286

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