Existence globale et diffusion pour l’équation de Schrödinger non linéaire répulsive cubique sur mathbbR 3 en dessous l’espace d’énergie
Journées équations aux dérivées partielles (2002), article no. 10, 15 p.

We sketch a proof of global existence and scattering for the defocusing cubic nonlinear Schrödinger equation in H s (ℝ 3 ) for s>4 5. The proof uses a new estimate of Morawetz type.

Nous profilons une démonstration de l’existence globale et diffusion pour l’équation de Schrödinger nonlinéaire répulsive cubique avec données à H s (ℝ 3 ) pour s>4 5. Le raisonnement utilise une estimation nouvelle de type de Morawetz. Nous détaillerons la démonstration ailleurs.

@incollection{JEDP_2002____A10_0,
     author = {J. Colliander and M. Keel and G. Staffilani and H. Takaoka and T. Tao},
     title = {Existence globale et diffusion pour l{\textquoteright}\'equation de {Schr\"odinger} non lin\'eaire r\'epulsive cubique sur $mathbb{R}^3$ en dessous l{\textquoteright}espace d{\textquoteright}\'energie},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {10},
     pages = {1--15},
     year = {2002},
     publisher = {Universit\'e de Nantes},
     doi = {10.5802/jedp.608},
     mrnumber = {1968206},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.608/}
}
TY  - JOUR
AU  - J. Colliander
AU  - M. Keel
AU  - G. Staffilani
AU  - H. Takaoka
AU  - T. Tao
TI  - Existence globale et diffusion pour l’équation de Schrödinger non linéaire répulsive cubique sur $mathbb{R}^3$ en dessous l’espace d’énergie
JO  - Journées équations aux dérivées partielles
PY  - 2002
SP  - 1
EP  - 15
PB  - Université de Nantes
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.608/
DO  - 10.5802/jedp.608
LA  - en
ID  - JEDP_2002____A10_0
ER  - 
%0 Journal Article
%A J. Colliander
%A M. Keel
%A G. Staffilani
%A H. Takaoka
%A T. Tao
%T Existence globale et diffusion pour l’équation de Schrödinger non linéaire répulsive cubique sur $mathbb{R}^3$ en dessous l’espace d’énergie
%J Journées équations aux dérivées partielles
%D 2002
%P 1-15
%I Université de Nantes
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.608/
%R 10.5802/jedp.608
%G en
%F JEDP_2002____A10_0
J. Colliander; M. Keel; G. Staffilani; H. Takaoka; T. Tao. Existence globale et diffusion pour l’équation de Schrödinger non linéaire répulsive cubique sur $mathbb{R}^3$ en dessous l’espace d’énergie. Journées équations aux dérivées partielles (2002), article  no. 10, 15 p.. doi: 10.5802/jedp.608

[1] M. Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations Ann. Math. 118, (1983), 187-214. | Zbl | MR

[2] J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity Intern. Mat. Res. Notices, 5, (1998), 253-283. | Zbl | MR

[3] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12, (1999), 145-171. | Zbl | MR

[4] J. Bourgain. Scattering in the energy space and below for 3D NLS, Jour. D'Anal. Math., 75:267-297, 1998. | Zbl | MR

[5] J. Bourgain, Global solutions of nonlinear Schrödinger equations American Math. Society, Providence, R.I., 1999. | Zbl | MR

[6] J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, GAFA, 3 (1993), 107-156. | Zbl | MR

[7] T. Cazenave, F. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in H 1. Manuscripta Math. 61 (1988), 477-494. | Zbl | MR

[8] J. Colliander, G. Staffilani, H. Takaoka, Global wellposedness for KdV below L 2, Math. Res. Lett. 6 (1999), no. 5-6, 755-778. | Zbl | MR

[9] J. Colliander, J. Delort, C. Kenig, G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. Amer. Math. Soc., 353 (2001), no. 8, 3307-3325. | Zbl | MR

[10] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Global well-posedness for KdV in Sobolev spaces of negative index, Electronic Jour. Diff. Eq. 2001 (2001), No 26, 1-7. | Zbl | MR

[11] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Sharp Global WellPosedness of KdV and Modified KdV on the R and T, submitted to Jour. Amer. Math. Soc. | Zbl | MR

[12] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Multilinear estimates for periodic KdV equations, and applications, preprint. | MR

[13] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Global well-posedness for the Schrödinger equations with derivative, Siam Jour. Math. Anal., 33 (2001), 649-669. | Zbl | MR

[14] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T.Tao, A refined global wellposedness result for Schrödinger equations with derivative, to appear in Siam Jour. of Math. Anal.. | Zbl | MR

[15] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao Almost conservation laws and global rough solutions to a Nonlinear Schrödinger Equation, to appear in Math. Res. Letters. X-13 | Zbl | MR

[16] G. Fonseca, F. Linares, and G. Ponce. Global well-posedness of the modified Korteweg-de Vries equation, Comm. Partial Differential Equations, 24 (1999), 683-705. | Zbl | MR

[17] J. Ginibre and G. Velo. Scattering theory in the energy space for a class of nonlinear SchrŽödinger equations, J. Math. Pures Appl., 9, (1985), 363-401. | Zbl | MR

[18] J. Ginibre, G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 123 (1989), 535-573. | Zbl | MR

[19] M. Keel, and T. Tao, Global well-posedness for large data for the MaxwellKlein-Gordon equation below the energy norm, preprint.

[20] M. Keel, T. Tao, Local and global well-posedness of wave maps on R 1+1 for rough data, Intl. Math. Res. Notices 21 (1998), 1117-1156. | Zbl | MR

[21] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980. | Zbl | MR

[22] C. Kenig, G. Ponce, L. Vega; Global well-posedness for semi-linear wave equations. Comm. Partial Diff. Eq. 25 (2000), 1741-1752. | Zbl | MR

[23] J. Lin, W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, Journ. Funct. Anal. 30, (1978), 245-263. | Zbl | MR

[24] C. Morawetz, Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. A 306 (1968), 291-29 | Zbl | MR

[25] C. Morawetz, W. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math. 25 (1972), 1-31. | Zbl | MR

[26] I.E. Segal, Space-time decay for solutions of wave equations, Adv. Math. 22 (1976), 304-311. | Zbl | MR

[27] R.S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705-774. | Zbl

[28] H. Takaoka, Global well-posedness for the Schrödinger equations with derivative in a nonlinear term and data in low order Sobolev space, Electronic Jour. Diff. Eq., 42 (2001), 1-23. | Zbl | MR

[29] H. Takaoka, N. Tzvetkov, Global low regularity solutions for KadomtsevPetviashvili equation, Internat. Math. Res. Notices, 2001, No. 2, 77-114. | Zbl

[30] T. Tao, Global regularity of wave maps. II. Small energy in two dimensions, Comm. Math. Phys. 224 (2001), 443-544. | Zbl | MR

[31] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987) | Zbl | MR

Cité par Sources :