Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains

Bulletin de la Societe Mathematique de France

Volumn 136, Issue 1, 2008, Pages 27-65

  Anton, Ramona ^a  

^a NONE

Author keywords

Dispersive equations; Lipschitz metric; Nonlinear schr dinger

Indexed keywords

EID: 57049135607     PISSN: 00379484     EISSN: None     Source Type: Journal    
DOI: 10.24033/bsmf.2548     Document Type: Article

References (28)

1. Alinhac, Serge and Gérard, Patrick. Opérateurs pseudo-différentiels et théorème de Nash-Moser. InterÉditions, 1991.
2. Bahouri, Hajer and Chemin, Jean-Yves. Équations d'ondes quasilinéaires et estimations de Strichartz. Amer. J. Math., 121 (1999) 1337-1377. Math Reviews 1719798.
3. Banica, Valeria. Dispersion and Strichartz inequalities for Schrödinger equations with singular coefficients. SIAM J. Math. Anal., 35 (2003) 868-883. Math Reviews 2049025, Zentralblatt 1058.35048.
4. Bourgain, J. Global solutions of nonlinear Schrödinger equations. American Mathematical Society, 1999. Math Reviews 1691575.
5. Brézis, H. and Gallouet, T. Nonlinear Schrödinger evolution equations. Nonlinear Anal., 4 (1980) 677-681. Math Reviews 582536.
6. Burq, Nicolas and Gérard, Patrick and Tzvetkov, N. Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer. J. Math., 126 (2004) 569-605. Math Reviews 2058384.
7. Burq, Nicolas and Planchon, Fabrice. Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications. J. Funct. Anal., 236 (2006) 265-298. Math Reviews 2227135.
8. Castro, C. and Zuazua, E. Concentration and lack of observability of waves in highly heterogeneous media. Arch. Ration. Mech. Anal., 164 (2002) 39-72. Math Reviews 1921162.
9. Cazenave, Thierry. Semilinear Schrödinger equations. New York University Courant Institute of Mathematical Sciences, 2003. Zentralblatt 1055.35003.
10. 8. Nonlinear Anal., 14 (1990) 807-836. Math Reviews 1055532.
11. Gilbarg, David and Trudinger, Neil S. Elliptic partial differential equations of second order. Springer, 2001, Reprint of the 1998 edition. Math Reviews 1814364.
12. Ginibre, J. and Velo, G. The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985) 309-327. Math Reviews 801582.
13. Ginibre, J. and Velo, G. Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pures Appl. (9), 64 (1985) 363-401. Math Reviews 839728.
14. Hörmander, Lars. The analysis of linear partial differential operators. I. Springer, 2003. Math Reviews 1996773, Zentralblatt 1028.35001.
15. Kato, Tosio. On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor., 46 (1987) 113-129. Math Reviews 877998, Zentralblatt 632.35038.
16. Keel, Markus and Tao, Terence. Endpoint Strichartz estimates. Amer. J. Math., 120 (1998) 955-980. Math Reviews 1646048.
17. Ogawa, Takayoshi and Ozawa, Tohru. Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem. J. Math. Anal. Appl., 155 (1991) 531-540. Math Reviews 1097298.
18. Reed, Michael and Simon, Barry. Methods of modern mathematical physics. I. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1980, Functional analysis. Math Reviews 751959.
19. Salort, Delphine. Dispersion and Strichartz inequalities for the one-dimensional Schrödinger equation with variable coefficients. Int. Math. Res. Not., (2005) 687-700. Math Reviews 2146323, Zentralblatt pre02188798.
20. q bounds for Spectral Clusters, (2004).
21. Spivak, Michael. A comprehensive introduction to differential geometry. Vol. One. Published by M. Spivak, Brandeis Univ., Waltham, Mass., 1970.
22. Staffilani, Gigliola and Tataru, Daniel. Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. Comm. Partial Differential Equations, 27 (2002) 1337-1372. Math Reviews 1924470.
23. Strichartz, Robert S. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J., 44 (1977) 705-714. Math Reviews 512086, Zentralblatt 372.35001.
24. Sulem, Catherine and Sulem, Pierre-Louis. The nonlinear Schrödinger equation. Springer, 1999, Self-focusing and wave collapse. Math Reviews 1696311.
25. Tataru, Daniel. Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation. Amer. J. Math., 122 (2000) 349-376. Math Reviews 1749052, Zentralblatt 959.35125.
26. Tsutsumi, Masayoshi. On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in two space dimensions. Nonlinear Anal., 13 (1989) 1051-1056. Math Reviews 1013309, Zentralblatt 693.35133.
27. Vladimirov, M.V. On the solvability of mixed problem for a nonlinear equation of Schrödinger type. Sov. Math. Dokl., 29 (1984) 281-284.
28. Yajima, Kenji. Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys., 110 (1987) 415-426. Math Reviews 891945, Zentralblatt 638.35036.