Rôle des oscillations quadratiques dans des équations de Schrödinger non linéaire

Rémi Carles[1]; Clotilde Fermanian–Kammerer[2]; Isabelle Gallagher[3]

  • [1] MAB, UMR CNRS 5466, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex, France
  • [2] Université de Cergy-Pontoise, Mathématiques, 2 avenue Adolphe Chauvin, BP 222, Pontoise, 95302 Cergy-Pontoise cedex, France
  • [3] Centre de Mathématiques de l’École polytechnique, UMR CNRS 7640, 91128 Palaiseau Cedex, France

Séminaire Équations aux dérivées partielles (2002-2003)

  • Volume: 2002-2003, page 1-12

How to cite

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Carles, Rémi, Fermanian–Kammerer, Clotilde, and Gallagher, Isabelle. "Rôle des oscillations quadratiques dans des équations de Schrödinger non linéaire." Séminaire Équations aux dérivées partielles 2002-2003 (2002-2003): 1-12. <http://eudml.org/doc/11075>.

@article{Carles2002-2003,
affiliation = {MAB, UMR CNRS 5466, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex, France; Université de Cergy-Pontoise, Mathématiques, 2 avenue Adolphe Chauvin, BP 222, Pontoise, 95302 Cergy-Pontoise cedex, France; Centre de Mathématiques de l’École polytechnique, UMR CNRS 7640, 91128 Palaiseau Cedex, France},
author = {Carles, Rémi, Fermanian–Kammerer, Clotilde, Gallagher, Isabelle},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {nonlinear Schrödinger equation; initial data; nonlinear oscillatory behavior; Strichartz estimates},
language = {eng},
pages = {1-12},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Rôle des oscillations quadratiques dans des équations de Schrödinger non linéaire},
url = {http://eudml.org/doc/11075},
volume = {2002-2003},
year = {2002-2003},
}

TY - JOUR
AU - Carles, Rémi
AU - Fermanian–Kammerer, Clotilde
AU - Gallagher, Isabelle
TI - Rôle des oscillations quadratiques dans des équations de Schrödinger non linéaire
JO - Séminaire Équations aux dérivées partielles
PY - 2002-2003
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2002-2003
SP - 1
EP - 12
LA - eng
KW - nonlinear Schrödinger equation; initial data; nonlinear oscillatory behavior; Strichartz estimates
UR - http://eudml.org/doc/11075
ER -

References

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  1. H. Bahouri et P. Gérard: High frequency approximation of solutions to critical nonlinear wave equations, American Journal of Mathematics, 121, pages 131–175, 1999. Zbl0919.35089MR1705001
  2. H. Bahouri et P. Gérard: Concentration effects in critical nonlinear wave equations and scattering theory, Geometrical Optics and Related Topics (F. Colombini and N. Lerner eds), Progress in Nonlinear Differential Equation and Applications, vol. 32, Birkhäuser, Boston, 17–30, 1997. Zbl0926.35090MR2033489
  3. R. Carles: Geometric optics with caustic crossing for some nonlinear Schrödinger equations, Indiana Univ. Math. J.49, pages 475–551, 2000. Zbl0970.35143MR1793681
  4. R. Carles, C. Fermanian et I. Gallagher: On the role of quadratic oscillations in nonlinear Schrödinger equations, arXiv : math . AP / 0212171 . Zbl1059.35134MR2003356
  5. T. Cazenave: An introduction to nonlinear Schrödinger equations, Text. Met. Mat., vol. 26, Univ. Fed. Rio de Jan., 1993. 
  6. T. Cazenave et F. Weissler: Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147, 75–100, 1992. Zbl0763.35085MR1171761
  7. J. Duistermaat: Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. Pure Appl. Math., 27, pages 207–281, 1974. Zbl0285.35010MR405513
  8. I. Gallagher: Profile decomposition for the Navier–Stokes equations, Bulletin de la Société Mathématique de France, 129, pages 285–316, 2001. Zbl0987.35120MR1871299
  9. P. Gérard: Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal. 141, pages 60–98, 1996. Zbl0868.35075MR1414374
  10. P. Gérard: Description du défaut de compacité de l’injection de Sobolev, ESAIM Contrôle Optimal et Calcul des Variations, 3, pages 213–233, 1998 (version électronique: http://www.emath.fr/cocv/). 
  11. J.-L. Joly, G. Metivier et J. Rauch: Caustics for dissipative semilinear oscillations, Mem. Amer. Math. Soc. 144, 2000. Zbl0963.35114MR1682244
  12. S. Keraani: On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175, no. 2, 353–392, 2001. Zbl1038.35119MR1855973
  13. F. Merle: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J., 69, no. 2, 427–454, 1993. Zbl0808.35141MR1203233
  14. F. Merle et L. Vega: Compactness at blow-up time for L 2 solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, no. 8, 399–425, 1998. Zbl0913.35126MR1628235
  15. J. Rauch: Partial Differential Equations, Graduate Texts in Math., vol. 128, Springer-Verlag, New York, 1991. Zbl0742.35001MR1223093

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