no. 8
Partial differential equations
Remark on the semilinear ill-posedness for a periodic higher-order KP-I equation
[Remarque sur le caractère semi-linéairement mal posé pour une équation KP-I périodique d'ordre supérieur]

Présenté par : Haïm Brézis

Robert, Tristan1
1 Université de Cergy-Pontoise, Laboratoire AGM, 2, av. Adolphe-Chauvin, 95302 Cergy-Pontoise cedex, France
Comptes Rendus. Mathématique, Tome 356 (2018) no. 8, pp. 891-898.

On montre que, pour un tore irrationnel bien choisi, le flot pour l'équation KP-I d'ordre 5 périodique n'est pas localement uniformément continu sur l'espace d'énergie, même sur les hyperplans de données initiales à moyenne en x fixée.

We prove that, for some irrational torus, the flow map of the periodic fifth-order KP-I equation is not locally uniformly continuous on the energy space, even on the hyperplanes of fixed x-mean value.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.06.002
Robert, Tristan 1

1 Université de Cergy-Pontoise, Laboratoire AGM, 2, av. Adolphe-Chauvin, 95302 Cergy-Pontoise cedex, France 
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Robert, Tristan. Remark on the semilinear ill-posedness for a periodic higher-order KP-I equation. Comptes Rendus. Mathématique, Tome 356 (2018) no. 8, pp. 891-898. doi : 10.1016/j.crma.2018.06.002. http://www.numdam.org/articles/10.1016/j.crma.2018.06.002/

[1] Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal., Volume 3 (1993), pp. 209-262

[2] Bourgain, J. On the Cauchy problem for the Kadomtsev–Petviashvili equation, Geom. Funct. Anal., Volume 3 (1993), pp. 315-341

[3] Bourgain, J. Periodic Korteweg de Vries equation with measures as initial data, Sel. Math., Volume 3 (1997), pp. 115-159

[4] Hadac, M.; Herr, S.; Koch, H. Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009), pp. 917-941

[5] Herr, S. Well-Posedness Results for Dispersive Equations with Derivative Nonlinearities, 2006 (PhD thesis)

[6] Ionescu, A.; Kenig, C. Global well-posedness of the Benjamin–Ono equation in low-regularity spaces, J. Amer. Math. Soc., Volume 20 (2007), pp. 753-798

[7] Ionescu, A.; Kenig, C.; Tataru, D. Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., Volume 173 (2008), pp. 265-304

[8] Iório, R.J.; Nunes, W.V.L. On equations of KP-type, Proc. R. Soc. Edinb., Sect. A, Math., Volume 128 (1998), pp. 725-743

[9] Koch, H.; Tzvetkov, N. On the local well-posedness of the Benjamin–Ono equation in Hs(R), Int. Math. Res. Not., Volume 2003 (2003), pp. 1449-1464

[10] Koch, H.; Tzvetkov, N. On finite energy solutions of the KP-I equation, Math. Z., Volume 258 (2008), pp. 55-68

[11] Molinet, L. Global well-posedness in L2 for the periodic Benjamin–Ono equation, Amer. J. Math., Volume 130 (2008), pp. 635-683

[12] Molinet, L.; Saut, J.-C.; Tzvetkov, N. Ill-posedness issues for the Benjamin–Ono and related equations, SIAM J. Math. Anal., Volume 33 (2001), pp. 982-988

[13] Molinet, L.; Saut, J.-C.; Tzvetkov, N. Well-posedness and ill-posedness results for the Kadomtsev–Petviashvili-I equation, Duke Math. J., Volume 115 (2002), pp. 353-384

[14] T. Robert, On the Cauchy problem for the periodic fifth-order KP-I equation, arXiv e-prints, 2017.

[15] Robert, T. Global well-posedness of partially periodic KP-I equation in the energy space and application, Ann. Inst. Henri Poincaré, Anal. Non Linéaire (2018) (ISSN: 0294-1449) | DOI

[16] Saut, J.-C.; Tzvetkov, N. On periodic KP-I type equations, Commun. Math. Phys., Volume 221 (2001), pp. 451-476

[17] Takaoka, H.; Tzvetkov, N. On the local regularity of the Kadomtsev–Petviashvili-II equation, Int. Math. Res. Not., Volume 2001 (2001), pp. 77-114

[18] Tzvetkov, N. Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris, Ser. I, Volume 329 (1999), pp. 1043-1047

[19] N. Tzvetkov, Ill-posedness issues for nonlinear dispersive equations, arXiv mathematics e-prints, 2004.

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