Centre de diffusion de revues académiques mathématiques


Journées équations aux dérivées partielles

Table des matières de ce volume | Article précédent | Article suivant
Pierre Germain
Space-time resonances
Journées équations aux dérivées partielles (2010), Exp. No. 8, 10 p., doi: 10.5802/jedp.65
Article PDF

Résumé - Abstract

This article is a short exposition of the space-time resonances method. It was introduced by Masmoudi, Shatah, and the author, in order to understand global existence for nonlinear dispersive equations, set in the whole space, and with small data. The idea is to combine the classical concept of resonances, with the feature of dispersive equations: wave packets propagate at a group velocity which depends on their frequency localization. The analytical method which follows from this idea turns out to be a very general tool.


[1] Bernicot, Frédéric; Germain, Pierre, Bilinear oscillatory integrals and boundedness for new bilinear multipliers. Accepted by Advances in Mathematics.  MR 2431352 |  Zbl pre05796746
[2] Bourgain, Jean, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I: Schrödinger equations, II: The KdV equation, Geom. Funct. Anal. 3 (1993).  MR 1209299 |  Zbl 0787.35098
[3] Bourgain, Jean, Global solutions of nonlinear Schrödinger equations. American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI, 1999.  MR 1691575 |  Zbl 0933.35178
[4] Coifman, Ronald; Meyer, Yves, Au delà des opérateurs pseudo-différentiels. Astérisque, 57. Société Mathématique de France, Paris, 1978  MR 518170 |  Zbl 0483.35082
[5] Germain, Pierre, Global existence for coupled Klein-Gordon equations with different speeds. arxiv 1005.5238. arXiv
[6] Germain, Pierre; Masmoudi, Nader; Shatah, Jalal, Global solutions for 3D quadratic Schrödinger equations, IMRN 2009, 414-432.  MR 2482120 |  Zbl 1156.35087
[7] Germain, Pierre; Masmoudi, Nader; Shatah, Jalal, Global solutions for 2D quadratic Schrödinger equations, arxiv 1001.5158. arXiv
[8] Germain, Pierre; Masmoudi, Nader; Shatah, Jalal, Global solutions for the gravity water waves equation in dimension 3, arxiv 0906.5343.  MR 2542891
[9] Georgiev, Vladimir; Lindblad, Hans; Sogge, Chris, Weighted Strichartz estimates and global existence for semilinear wave equations. Amer. J. Math. 119 (1997), 1291–1319.  MR 1481816 |  Zbl 0893.35075
[10] John, Fritz, Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math. 28 (1979), 235–268.  MR 535704 |  Zbl 0406.35042
[11] Klainerman, Sergiu, The null condition and global existence to nonlinear wave equations. Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), 293–326, Lectures in Appl. Math., 23, Amer. Math. Soc., Providence, RI, 1986  MR 837683 |  Zbl 0599.35105
[12] Klainerman, Sergiu; Machedon, Matei, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993).  Zbl 0803.35095
[13] Lacey, Michael; Thiele, Christoph, $L^p$ estimates on the bilinear Hilbert transform for $2 < p <\infty $, Annals of Math. 146 (1997), 693–724.  MR 1491450 |  Zbl 0914.46034
[14] Shatah, Jalal, Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. Pure Appl. Math. 38 (1985), no. 5, 685–696  MR 803256 |  Zbl 0597.35101
[15] Strauss, Walter, Nonlinear scattering at low energy. J. Funct. Anal. 41 (1981), 110–133 and 43 (1981) 281-293.  Zbl 0466.47006
[16] Tao, Terence, Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.  MR 2233925 |  Zbl 1106.35001
Copyright Cellule MathDoc 2018 | Crédit | Plan du site