Space-time resonances

Pierre Germain[1]

  • [1] Courant Institute of Mathematical Sciences, New York University, New York, NY 10012-1185 USA

Journées Équations aux dérivées partielles (2010)

  • page 1-10
  • ISSN: 0752-0360

Abstract

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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.

How to cite

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Germain, Pierre. "Space-time resonances." Journées Équations aux dérivées partielles (2010): 1-10. <http://eudml.org/doc/116389>.

@article{Germain2010,
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.},
affiliation = {Courant Institute of Mathematical Sciences, New York University, New York, NY 10012-1185 USA},
author = {Germain, Pierre},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
month = {6},
pages = {1-10},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Space-time resonances},
url = {http://eudml.org/doc/116389},
year = {2010},
}

TY - JOUR
AU - Germain, Pierre
TI - Space-time resonances
JO - Journées Équations aux dérivées partielles
DA - 2010/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 10
AB - 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.
LA - eng
UR - http://eudml.org/doc/116389
ER -

References

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  1. Bernicot, Frédéric; Germain, Pierre, Bilinear oscillatory integrals and boundedness for new bilinear multipliers. Accepted by Advances in Mathematics. Zbl1204.42018MR2431352
  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). Zbl0787.35098MR1209299
  3. Bourgain, Jean, Global solutions of nonlinear Schrödinger equations. American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI, 1999. Zbl0933.35178MR1691575
  4. Coifman, Ronald; Meyer, Yves, Au delà des opérateurs pseudo-différentiels. Astérisque, 57. Société Mathématique de France, Paris, 1978 Zbl0483.35082MR518170
  5. Germain, Pierre, Global existence for coupled Klein-Gordon equations with different speeds. arxiv 1005.5238. Zbl1255.35162
  6. Germain, Pierre; Masmoudi, Nader; Shatah, Jalal, Global solutions for 3D quadratic Schrödinger equations, IMRN 2009, 414-432. Zbl1156.35087MR2482120
  7. Germain, Pierre; Masmoudi, Nader; Shatah, Jalal, Global solutions for 2D quadratic Schrödinger equations, arxiv 1001.5158. Zbl1156.35087
  8. Germain, Pierre; Masmoudi, Nader; Shatah, Jalal, Global solutions for the gravity water waves equation in dimension 3, arxiv 0906.5343. Zbl1241.35003MR2542891
  9. Georgiev, Vladimir; Lindblad, Hans; Sogge, Chris, Weighted Strichartz estimates and global existence for semilinear wave equations. Amer. J. Math. 119 (1997), 1291–1319. Zbl0893.35075MR1481816
  10. John, Fritz, Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math. 28 (1979), 235–268. Zbl0406.35042MR535704
  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 Zbl0599.35105MR837683
  12. Klainerman, Sergiu; Machedon, Matei, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993). Zbl0803.35095
  13. Lacey, Michael; Thiele, Christoph, L p estimates on the bilinear Hilbert transform for 2 &lt; p &lt; ∞ , Annals of Math. 146 (1997), 693–724. Zbl0914.46034MR1491450
  14. Shatah, Jalal, Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. Pure Appl. Math. 38 (1985), no. 5, 685–696 Zbl0597.35101MR803256
  15. Strauss, Walter, Nonlinear scattering at low energy. J. Funct. Anal. 41 (1981), 110–133 and 43 (1981) 281-293. Zbl0466.47006
  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. Zbl1106.35001MR2233925

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