@article{SEDP_2005-2006____A16_0, author = {G\'erard, Patrick}, title = {Sur le caract\`ere bien pos\'e des \'equations de {Schr\"odinger} non lin\'eaires}, journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"}, note = {talk:16}, pages = {1--17}, publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2005-2006}, mrnumber = {2276081}, language = {fr}, url = {http://www.numdam.org/item/SEDP_2005-2006____A16_0/} }
TY - JOUR AU - Gérard, Patrick TI - Sur le caractère bien posé des équations de Schrödinger non linéaires JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:16 PY - 2005-2006 SP - 1 EP - 17 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/item/SEDP_2005-2006____A16_0/ LA - fr ID - SEDP_2005-2006____A16_0 ER -
%0 Journal Article %A Gérard, Patrick %T Sur le caractère bien posé des équations de Schrödinger non linéaires %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:16 %D 2005-2006 %P 1-17 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/item/SEDP_2005-2006____A16_0/ %G fr %F SEDP_2005-2006____A16_0
Gérard, Patrick. Sur le caractère bien posé des équations de Schrödinger non linéaires. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 16, 17 p. http://www.numdam.org/item/SEDP_2005-2006____A16_0/
[1] Banica, V., On the nonlinear Schrödinger dynamics on . J. Math. Pures Appl. , 83 (2004), 77–98. | MR | Zbl
[2] Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations. Geom. and Funct. Anal., 3 (1993), 107–156. | MR | Zbl
[3] Bourgain, J., Exponential sums and nonlinear Schrödinger equations. Geom. and Funct. Anal., 3 (1993) 157–178. | MR | Zbl
[4] Bourgain, J., Global Solutions of Nonlinear Schrödinger equations. Colloq. Publications, American Math. Soc., 1999. | MR | Zbl
[5] Bourgain, J., Remarks on Strichartz’ inequalities on irrational tori. Prépublication, 2004, à paraître dans Mathematical Aspects of nonlinear PDE, Annals Math. Studies, Princeton.
[6] Bourgain, J., Refinements of Strichartz’ inequality and applications to D-NLS with critical nonlinearity, IMRN, 5 (1998), 253-283. | Zbl
[7] Burq, N., Gérard, P., Tzvetkov, N., Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer. J. Math., 126 (2004), 569–605. | MR | Zbl
[8] Burq, N., Gérard, P., Tzvetkov, N., An instability property of the nonlinear Schrödinger equation on . Math. Res. Lett., 9 (2002), 323–335. | MR | Zbl
[9] Burq, N., Gérard, P., Tzvetkov, N., Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces. Invent. math. 159 (2005), 187–223. | MR | Zbl
[10] Burq, N., Gérard, P., Tzvetkov, N., Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations. Ann. Scient. Éc. Norm. Sup. 38 (2005), 255–301. | Numdam | MR | Zbl
[11] Burq, N., Gérard, P., Tzvetkov, N., Global solutions for the nonlinear Schrödinger equation on three-dimensional compact manifolds. To appear in Mathematical Aspects of nonlinear PDE, Annals Math. Studies, Princeton. | MR
[12] Burq, N., Zworski, M., Instability for the semiclassical non-linear Schrödinger equation. Comm. Math. Phys. 260 (2005), 45–58. | MR | Zbl
[13] Cazenave, T., Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, 10. New York University, American Mathematical Society, Providence, RI, 2003. | MR | Zbl
[14] Cazenave, T., Weissler, F., The Cauchy problem for the critical nonlinear Schrödinger equation in . Nonlinear Analysis, Theory, Methods and Applications, 14 (1990), 807–836. | MR | Zbl
[15] Christ, M., Colliander, J., Tao, T., Ill-posedness for nonlinear Schrödinger and wave equations. Prépublication, math.AP/0311048, à paraître à Ann. I. H. Poincaré-AN.
[16] Gérard, P., Nonlinear Schrödinger equations on compact manifolds. In European Congress of Mathematics, Stokholm, June 27-July 2, 2004 (ed. by Ari Laptev). European Mathematical Society, Zürich, 2005, 121–139. | MR | Zbl
[17] Gérard, P., Nonlinear Schrödinger Equations in Inhomogeneous Media : Wellposedness and Illposedness of the Cauchy Problem. In Proceedings of the International Congress of Mathematics, Madrid, August 2006, à paraître. | MR | Zbl
[18] Ginibre, J., Velo, G., On a class of nonlinear Schrödinger equations. J. Funct. Anal., 32 (1979) 1-71. | MR | Zbl
[19] Ginibre, J., Velo, G., The global Cauchy problem for the nonlinear Schrödinger equation. Ann. I. H. Poincaré-AN, 2 (1985) 309-327. | Numdam | MR | Zbl
[20] Ginibre, J., Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace (d’après Bourgain). Séminaire Bourbaki, Exp. 796, Astérisque 237 (1996), 163–187. | Numdam | Zbl
[21] Kato, T., On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Physique théorique 46 (1987), 113–129. | Numdam | MR | Zbl
[22] Keel, M., Tao, T., Endpoint Strichartz estimates. Amer. J. Math., 120 (1998), 955–980. | MR | Zbl
[23] Ryckman, E., Visan, M., Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrodinger equation in . Prépublication, math.AP/0501462, à paraître à Amer. J. Math. | MR | Zbl
[24] Strichartz, R., Restrictions of Fourier transforms to quadratic surfaces and decay of soltions of wave equations. Duke Math. J. 44 (1977), 705-714. | MR | Zbl
[25] Tsutsumi, Y., -solutions for nonlinear Schrödinger equations and nonlinear groups. Funkcial. Ekvac. 30 (1987), 115–125. | MR | Zbl
[26] Tzvetkov, N., Illposedness issues for nonlinear dispersive equations. Prépublication, September 2004.
[27] Yajima, K., Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), 415-426. | MR | Zbl
[28] Zakharov, V.E., Collapse of Langmuir waves. Sov. Phys. JETP 35 (1972), 980-914.