Global well-posedness and scattering for small data for the 2D and 3D KP-II Cauchy problem
Journées équations aux dérivées partielles (2015), article no. 4, 9 p.

We discuss global well-posedness for the Kadomtsev-Petviashvili II in two and three space dimensions with small data. The crucial points are new bilinear estimates and the definition of the function spaces. As by-product we obtain that all solutions to small initial data scatter as $t\to ±\infty$.

DOI: 10.5802/jedp.633
Keywords: Kadomtsev-Petviashvili, Galilean transform, Bilinear estimate
Koch, Herbert 1

1 Mathematisches Institut Universität Bonn Endenicher Allee 60 53115 Bonn Germany
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Koch, Herbert. Global well-posedness and scattering for small data for the 2D and 3D KP-II Cauchy problem. Journées équations aux dérivées partielles (2015), article  no. 4, 9 p. doi : 10.5802/jedp.633. http://www.numdam.org/articles/10.5802/jedp.633/

[1] Bourgain, Jean On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., Volume 3 (1993) no. 4, pp. 315-341 | DOI | MR | Zbl

[2] Hadac, Martin On the local well-posedness of the Kadomtsev-Petviashvili II equation., Universität Dortmund (2007) (Ph. D. Thesis) | Zbl

[3] Hadac, Martin Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations, Trans. Amer. Math. Soc., Volume 360 (2008) no. 12, pp. 6555-6572 | DOI | MR | Zbl

[4] Hadac, Martin; Herr, Sebastian; Koch, Herbert Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 26 (2009) no. 3, pp. 917-941 | DOI | Numdam | MR | Zbl

[5] Hadac, Martin; Herr, Sebastian; Koch, Herbert Erratum to “Well-posedness and scattering for the KP-II equation in a critical space”, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 27 (2010) no. 3, pp. 971-972 | DOI | Numdam | MR | Zbl

[6] Isaza, Pedro; López, Juan; Mejía, Jorge The Cauchy problem for the Kadomtsev-Petviashvili (KPII) equation in three space dimensions, Comm. Partial Differential Equations, Volume 32 (2007) no. 4-6, pp. 611-641 | DOI | MR | Zbl

[7] Isaza, Pedro; Mejía, Jorge Local and global Cauchy problems for the Kadomtsev-Petviashvili (KP-II) equation in Sobolev spaces of negative indices, Comm. Partial Differential Equations, Volume 26 (2001) no. 5-6, pp. 1027-1054 | DOI | MR | Zbl

[8] Keel, Markus; Tao, Terence Endpoint Strichartz estimates, Amer. J. Math., Volume 120 (1998) no. 5, pp. 955-980 http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5keel.pdf | MR | Zbl

[9] Klein, Christian; Saut, Jean-Claude Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations, J. Nonlinear Sci., Volume 22 (2012) no. 5, pp. 763-811 | DOI | MR | Zbl

[10] Koch, Herbert; Tataru, Daniel Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., Volume 58 (2005) no. 2, pp. 217-284 | DOI | MR | Zbl

[11] Koch, Herbert; Tataru, Daniel; Vişan, Monica Dispersive Equations and Nonlinear Waves (2014)

[12] Lépingle, D. La variation d’ordre $p$ des semi-martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Volume 36 (1976) no. 4, pp. 295-316 | MR | Zbl

[13] Lyons, Terry J. Differential equations driven by rough signals, Rev. Mat. Iberoamericana, Volume 14 (1998) no. 2, pp. 215-310 | DOI | MR | Zbl

[14] Takaoka, H.; Tzvetkov, N. On the local regularity of the Kadomtsev-Petviashvili-II equation, Internat. Math. Res. Notices (2001) no. 2, pp. 77-114 | DOI | MR | Zbl

[15] Takaoka, Hideo Well-posedness for the Kadomtsev-Petviashvili II equation, Adv. Differential Equations, Volume 5 (2000) no. 10-12, pp. 1421-1443 | MR | Zbl

[16] Tzvetkov, Nickolay On the Cauchy problem for Kadomtsev-Petviashvili equation, Comm. Partial Differential Equations, Volume 24 (1999) no. 7-8, pp. 1367-1397 | DOI | MR | Zbl

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