Sur le caractère bien posé des équations de Schrödinger non linéaires

Gérard, Patrick

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 16, 17 p.

[1] Banica, V., On the nonlinear Schrödinger dynamics on $S^2$. 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 $2$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 $S^d$. 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 $H^s$. 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 ${ℝ}^{1+4}$. 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., $L^2$-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.