The Schrödinger equation on a compact manifold : Strichartz estimates and applications
Journées équations aux dérivées partielles (2001), article no. 5, 18 p.

Nous établissons des estimations de Strichartz avec perte de dérivée fractionnaire pour l'équation de Schrödinger sur toute variété riemannienne compacte. Nous en déduisons des théorèmes d'existence globale pour le problème de Cauchy d'équations de Schrödinger non-linéaires sur les surfaces dans le cas de non-linéarités polynomiales défocalisantes, et sur les variétés de dimension trois dans le cas de non-linéarités quadratiques. Nous discutons également l'optimalité de ces estimées de Strichartz sur les sphères.

We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any riemannian compact manifold. As a consequence we infer global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three-manifolds in the case of quadratic nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres.

@article{JEDP_2001____A5_0,
     author = {Burq, Nicolas and G\'erard, Patrick and Tzvetkov, Nikolay},
     title = {The {Schr\"odinger} equation on a compact manifold : {Strichartz} estimates and applications},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {5},
     pages = {1--18},
     publisher = {Universit\'e de Nantes},
     year = {2001},
     doi = {10.5802/jedp.589},
     mrnumber = {1843406},
     zbl = {01808681},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.589/}
}
TY  - JOUR
AU  - Burq, Nicolas
AU  - Gérard, Patrick
AU  - Tzvetkov, Nikolay
TI  - The Schrödinger equation on a compact manifold : Strichartz estimates and applications
JO  - Journées équations aux dérivées partielles
PY  - 2001
SP  - 1
EP  - 18
PB  - Université de Nantes
UR  - http://www.numdam.org/articles/10.5802/jedp.589/
DO  - 10.5802/jedp.589
LA  - en
ID  - JEDP_2001____A5_0
ER  - 
%0 Journal Article
%A Burq, Nicolas
%A Gérard, Patrick
%A Tzvetkov, Nikolay
%T The Schrödinger equation on a compact manifold : Strichartz estimates and applications
%J Journées équations aux dérivées partielles
%D 2001
%P 1-18
%I Université de Nantes
%U http://www.numdam.org/articles/10.5802/jedp.589/
%R 10.5802/jedp.589
%G en
%F JEDP_2001____A5_0
Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay. The Schrödinger equation on a compact manifold : Strichartz estimates and applications. Journées équations aux dérivées partielles (2001), article  no. 5, 18 p. doi : 10.5802/jedp.589. http://www.numdam.org/articles/10.5802/jedp.589/

[Besse] A. Besse Manifolds all of whose geodesics are closed Springer-Verlag, Berlin-New York, 1978. | MR | Zbl

[Bo1] J. Bourgain 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

[Bo2] J. Bourgain Exponential sums and nonlinear Schrödinger equations, Geom. and Funct. Anal. 3 1993, 157-178. | MR | Zbl

[Bo3] J. Bourgain Global solutions of nonlinear Schrödinger equations, Colloq. Publications, American Math. Soc., 1999. | MR | Zbl

[Bo4] J. Bourgain Global wellposedness of defocusing critical nonlinear Schrödinger equations in the radial case, J. Amer. Math. Soc. 12 1999, 145-171. | MR | Zbl

[BG] H. Brézis, T. Gallouët Nonlinear Schrödinger evolution equations, Nonlinear Analysis, Theory, Methods and Applications, 4 1980, 677-681. | MR | Zbl

[BGT] N. Burq, P. Gérard, N. Tzvetkov Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, preprint 2001.

[C] T. Cazenave An introduction to nonlinear Schrödinger equations, Text. Met. Mat. 22, Inst. Mat., Rio de Janeiro, 1989.

[CW] T. Cazenave, F. Weissler The Cauchy problem for the critical nonlinear Schrödinger equation in H s , Nonlinear Analysis, Theory, Methods and Applications, 1990, 807-836 | MR | Zbl

[CdV] Y. Colin De Verdière Le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques, Comment. Math. Helvetici 54 1979, 508-522. | MR | Zbl

[D] E. B. Davies Spectral theory and differential operators, Cambridge University Press 1995. | MR | Zbl

[GV1] J. Ginibre, G. Velo The global Cauchy problem for the nonlinear Schrödinger equation, Ann. I.H.P. (Anal. non lin.) 2 1985, 309-327. | Numdam | MR | Zbl

[GV2] J. Ginibre and G. Velo Smoothing properties and retarded estimates for some dispersive evolution equations, Commun. Math. Phys. 144 1992, 163-188. | MR | Zbl

[Gr] E. Grosswald Representations of Integers as Sums of Squares, Springer-Verlag, 1985. | MR | Zbl

[Gu] V. Guillemin Lectures on spectral theory of elliptic operators, Duke Math. J. 44 1977, 129-137. | MR | Zbl

[HS] B. Helffer, J. Sjöstrand Equation de Schrödinger avec champ magnetique et équation de Harper, Lecture notes in Physics, 345 1989, 118-197. | MR | Zbl

[Ka] L. Kapitanskii Some generalizations of the Strichartz-B-Brenner inequality, Leningrad Math. J. 1 1990, 693-726. | MR | Zbl

[K] T. Kato On nonlinear Schrödinger equations, Ann. I.H.P. (Phys. Théor.) 46 1987, 113-129. | Numdam | MR | Zbl

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

[Leb] G. Lebeau Contrôle de l'équation de Schrödinger, J. Math. Pures Appl. 71 1992, 267-291. | MR | Zbl

[ReSi] M. Reed, B. Simon Methods of Modern Mathematical Physics, vol.2, Academic Press, 1975. | MR | Zbl

[Ro] D. Robert Autour de l'approximation semi-classique Progress in Mathematics, vol. 68, Birkhäuser, 1987. | MR | Zbl

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

[Sogge2] C. D. Sogge Concerning the L p norm of spectral clusters for second order elliptic operators on compact manifolds, J. Funct. Anal. 77 1988, 123-138. | MR | Zbl

[Sogge3] C. D. Sogge Fourier integrals in classical analysis, Cambridge tracts in Mathematics, 1993. | MR | Zbl

[SZ] C. D. Sogge, S. Zelditch Riemannian manifolds with maximal eigenfunction growth, Preprint 2001. | MR

[StTa] G. Staffilani, D. Tataru Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Preprint 2000. | MR

[S] R. Strichartz Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 1977, 705-714. | MR | Zbl

[To] P. A. Tomas A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 1975, 477-478. | MR | Zbl

[Y] K. Yajima Existence of solutions for Schrödinger evolution equations, Commun. Math. Phys. 110 1987, 415-426. | MR | Zbl

Cité par Sources :