Bilinear virial identities and applications
[Identités virielles bilinéaires et applications]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 2, pp. 261-290.

On démontre des identités de type viriel bilinéaire pour l'équation de Schrödinger nonlinéaire, qui peuvent être vues comme des extensions des inégalités d'interaction de Morawetz. Ceci permet de retrouver et d'étendre des raffinements bilinéaires des inégalités de Strichartz, et nous donnons également des applications à plusieurs problèmes non-linéaires, notamment sur les domaines à bord.

We prove bilinear virial identities for the nonlinear Schrödinger equation, which are extensions of the Morawetz interaction inequalities. We recover and extend known bilinear improvements to Strichartz inequalities and provide applications to various nonlinear problems, most notably on domains with boundaries.

DOI : 10.24033/asens.2096
Classification : 35Q55
Keywords: nonlinear Schrödinger equation, Virial identity, exterior domain
Mot clés : Équation de Schrödinger non linéaire, identité du Viriel, domaine extérieur
@article{ASENS_2009_4_42_2_261_0,
     author = {Planchon, Fabrice and Vega, Luis},
     title = {Bilinear virial identities and applications},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {261--290},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {2},
     year = {2009},
     doi = {10.24033/asens.2096},
     mrnumber = {2518079},
     zbl = {1192.35166},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2096/}
}
TY  - JOUR
AU  - Planchon, Fabrice
AU  - Vega, Luis
TI  - Bilinear virial identities and applications
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2009
SP  - 261
EP  - 290
VL  - 42
IS  - 2
PB  - Société mathématique de France
UR  - http://archive.numdam.org/articles/10.24033/asens.2096/
DO  - 10.24033/asens.2096
LA  - en
ID  - ASENS_2009_4_42_2_261_0
ER  - 
%0 Journal Article
%A Planchon, Fabrice
%A Vega, Luis
%T Bilinear virial identities and applications
%J Annales scientifiques de l'École Normale Supérieure
%D 2009
%P 261-290
%V 42
%N 2
%I Société mathématique de France
%U http://archive.numdam.org/articles/10.24033/asens.2096/
%R 10.24033/asens.2096
%G en
%F ASENS_2009_4_42_2_261_0
Planchon, Fabrice; Vega, Luis. Bilinear virial identities and applications. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 2, pp. 261-290. doi : 10.24033/asens.2096. http://archive.numdam.org/articles/10.24033/asens.2096/

[1] R. Anton, Global existence for defocusing cubic NLS and Gross-Pitaevskii equations in three dimensional exterior domains, J. Math. Pures Appl. 89 (2008), 335-354. | MR | Zbl

[2] M. D. Blair, H. F. Smith & C. D. Sogge, On Strichartz estimates for Schrödinger operators in compact manifolds with boundary (electronic), Proc. Amer. Math. Soc. 136 (2008), 247-256. | MR | Zbl

[3] J. Bourgain, Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Res. Not. 1998 (1998), 253-283. | MR | Zbl

[4] N. Burq, Smoothing effect for Schrödinger boundary value problems, Duke Math. J. 123 (2004), 403-427. | MR | Zbl

[5] N. Burq, P. Gérard & N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 295-318. | Numdam | MR | Zbl

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

[7] N. Burq & F. Planchon, Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications, J. Funct. Anal. 236 (2006), 265-298. | MR

[8] M. Christ & A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), 409-425. | MR | Zbl

[9] J. Colliander, M. Grillakis & N. Tzirakis, The interaction Morawetz estimate for 2 . (Workshop « Nonlinear waves and dispersive equations »), Oberwolfach Report 44 (2007).

[10] J. Colliander, M. Grillakis & N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, to appear in Comm. Pure Appl. Math. | MR | Zbl

[11] J. Colliander, J. Holmer, M. Visan & X. Zhang, Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on , Commun. Pure Appl. Anal. 7 (2008), 467-489. | MR | Zbl

[12] J. Colliander, M. Keel, G. Staffilani, H. Takaoka & T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on 3 , Comm. Pure Appl. Math. 57 (2004), 987-1014. | MR | Zbl

[13] J. Colliander, M. Keel, G. Staffilani, H. Takaoka & T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in 3 , Ann. of Math. 167 (2008), 767-865. | MR | Zbl

[14] P. Constantin & J.-C. Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J. 38 (1989), 791-810. | MR | Zbl

[15] J. Ginibre & G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. 64 (1985), 363-401. | MR | Zbl

[16] J. Ginibre & G. Velo, Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schrödinger and Hartree equations, to appear in Quart. Appl. Math. | MR | Zbl

[17] A. Hassell, T. Tao & J. Wunsch, A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds, Comm. Partial Differential Equations 30 (2005), 157-205. | MR | Zbl

[18] O. Ivanovici, Precised smoothing effect in the exterior of balls, Asymptot. Anal. 53 (2007), 189-208. | MR

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

[20] C. E. Kenig, G. Ponce & L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69. | MR | Zbl

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

[22] T. Ozawa & Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11 (1998), 201-222. | MR | Zbl

[23] D. Salort, Dispersion and Strichartz inequalities for the one-dimensional Schrödinger equation with variable coefficients, Int. Math. Res. Not. 2005 (2005), 687-700. | MR | Zbl

[24] P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), 699-715. | Zbl

[25] G. Staffilani & D. Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27 (2002), 1337-1372. | MR | Zbl

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

[27] T. Tao, A. Vargas & L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), 967-1000. | MR | Zbl

[28] H. Triebel, Theory of function spaces, Monographs in Mathematics 78, Birkhäuser, 1983. | MR | Zbl

[29] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874-878. | MR | Zbl

[30] L. Vega & N. Visciglia, On the local smoothing for the Schrödinger equation (electronic), Proc. Amer. Math. Soc. 135 (2007), 119-128. | MR | Zbl

Cité par Sources :