Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations

Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay

doi:10.1016/j.ansens.2004.11.003

Burq, Nicolas ; Gérard, Patrick ; Tzvetkov, Nikolay

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 38 (2005) no. 2, pp. 255-301

MR Zbl | 12 citations dans Numdam

DOI : 10.1016/j.ansens.2004.11.003

@article{ASENS_2005_4_38_2_255_0,
     author = {Burq, Nicolas and G\'erard, Patrick and Tzvetkov, Nikolay},
     title = {Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear {Schr\"odinger} equations},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {255--301},
     year = {2005},
     publisher = {Elsevier},
     volume = {Ser. 4, 38},
     number = {2},
     doi = {10.1016/j.ansens.2004.11.003},
     mrnumber = {2144988},
     zbl = {02211346},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.ansens.2004.11.003/}
}

TY  - JOUR
AU  - Burq, Nicolas
AU  - Gérard, Patrick
AU  - Tzvetkov, Nikolay
TI  - Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2005
SP  - 255
EP  - 301
VL  - 38
IS  - 2
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.ansens.2004.11.003/
DO  - 10.1016/j.ansens.2004.11.003
LA  - en
ID  - ASENS_2005_4_38_2_255_0
ER  -

%0 Journal Article
%A Burq, Nicolas
%A Gérard, Patrick
%A Tzvetkov, Nikolay
%T Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations
%J Annales scientifiques de l'École Normale Supérieure
%D 2005
%P 255-301
%V 38
%N 2
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.ansens.2004.11.003/
%R 10.1016/j.ansens.2004.11.003
%G en
%F ASENS_2005_4_38_2_255_0

Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay. Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 38 (2005) no. 2, pp. 255-301. doi: 10.1016/j.ansens.2004.11.003

Bibliographie
Cité par

[1] Bourgain J., Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations, Geom. Funct. Anal. 3 (1993) 107-156. | Zbl | MR

[2] Bourgain J., Exponential sums and nonlinear Schrödinger equations, Geom. Funct. Anal. 3 (1993) 157-178. | Zbl | MR

[3] Bourgain J., Eigenfunction bounds for the Laplacian on the n-torus, Internat. Math. Res. Notices 3 (1993) 61-66. | Zbl | MR

[4] Bourgain J., Remarks on Strichartz' inequalities on irrational tori, Personal communication, 2004.

[5] Burq N., Gérard P., Tzvetkov N., Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (3) (2004) 569-605. | Zbl | MR

[6] Burq N., Gérard P., Tzvetkov N., An instability property of the nonlinear Schrödinger equation on $S^{d}$ , Math. Res. Lett. 9 (2-3) (2002) 323-335. | Zbl | MR

[7] Burq N., Gérard P., Tzvetkov N., The Cauchy problem for the nonlinear Schrödinger equation on compact manifold, J. Nonlinear Math. Phys. 10 (2003) 12-27. | MR

[8] Burq N., Gérard P., Tzvetkov N., Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005) 187-223. | Zbl | MR

[9] Burq N., Gérard P., Tzvetkov N., Multilinear estimates for Laplace spectral projectors on compact manifolds, C. R. Acad. Sci. Paris Ser. I 338 (2004) 359-364. | Zbl | MR

[10] Cazenave T., Semi-Linear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, American Mathematical Society, Providence, RI, 2003. | Zbl | MR

[11] Christ M., Colliander J., Tao T., Ill-posedness for nonlinear Schrödinger and wave equations, Preprint, 2003. | MR

[12] Gallot S., Hulin D., Lafontaine J., Riemannian Geometry, Universitext, Springer-Verlag, Berlin, 1990. | Zbl | MR

[13] Ginibre J., Velo G., On a class of nonlinear Schrödinger equations, J. Funct. Anal. 32 (1979) 1-71. | Zbl | MR

[14] 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. | Zbl | MR | Numdam

[15] Hörmander L., The spectral function of an elliptic operator, Acta Math. 121 (1968) 193-218. | Zbl | MR

[16] Hörmander L., Oscillatory integrals and multipliers on $F L^{p}$ , Ark. Math. 11 (1973) 1-11. | Zbl | MR

[17] Kato T., On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré Phys. Théor. 46 (1987) 113-129. | Zbl | MR | Numdam

[18] Klainerman S., Machedon M., Remark on Strichartz-type inequalities, Internat. Math. Res. Notices 5 (1996) 201-220, With appendices by J. Bourgain and D. Tataru. | Zbl | MR

[19] Klainerman S., Machedon M., Finite energy solutions of the Yang-Mills equations in $R^{3 + 1}$ , Ann. of Math. (2) 142 (1) (1995) 39-119. | Zbl | MR

[20] Koch H., Tataru D., Personal communication, 2004.

[21] Lions J.-L., Quelques méthodes de résolution des équations aux dérivées partielles non linéaires, Dunod, Paris, 1969. | Zbl

[22] Sogge C., Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986) 43-65. | Zbl | MR

[23] Sogge C., Concerning the $L^{p}$ norm of spectral clusters for second order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988) 123-138. | Zbl | MR

[24] Sogge C., Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, 1993. | Zbl | MR

[25] Stein E.M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Monographs in Harmonic Analysis, vol. III, Princeton University Press, Princeton, NJ, 1993. | Zbl | MR

[26] Szegö G., Orthogonal Polynomials, Colloq. Publications, American Mathematical Society, Providence, RI, 1974.

[27] Zygmund A., On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974) 189-201. | Zbl | MR

[28] Tao T., Multilinear weighted convolutions of $L^{2}$ functions, and applications to non-linear dispersive equations, Amer. J. Math. 123 (2001) 839-908. | Zbl | MR

Cité par Sources :