Bilinear virial identities and applications
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 2, p. 261-290

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.

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.

DOI : https://doi.org/10.24033/asens.2096
Classification:  35Q55
Keywords: nonlinear Schrödinger equation, Virial identity, exterior domain
@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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {2},
     year = {2009},
     pages = {261-290},
     doi = {10.24033/asens.2096},
     zbl = {1192.35166},
     mrnumber = {2518079},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2009_4_42_2_261_0}
}
Planchon, Fabrice; Vega, Luis. Bilinear virial identities and applications. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 2, pp. 261-290. doi : 10.24033/asens.2096. http://www.numdam.org/item/ASENS_2009_4_42_2_261_0/

[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 2401142 | Zbl 1148.35081

[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 2350410 | Zbl 1169.35012

[3] J. Bourgain, Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Res. Not. 1998 (1998), 253-283. | MR 1616917 | Zbl 0917.35126

[4] N. Burq, Smoothing effect for Schrödinger boundary value problems, Duke Math. J. 123 (2004), 403-427. | MR 2066943 | Zbl 1061.35024

[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 2068304 | Zbl 1061.35126

[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 2058384 | Zbl 1067.58027

[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 2227135 | Zbl pre05037260

[8] M. Christ & A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), 409-425. | MR 1809116 | Zbl 0974.47025

[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 2527809 | Zbl 1185.35250

[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 2379437 | Zbl 1157.35103

[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 2053757 | Zbl 1060.35131

[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 2415387 | Zbl 1178.35345

[14] P. Constantin & J.-C. Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J. 38 (1989), 791-810. | MR 1017334 | Zbl 0712.35022

[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 839728 | Zbl 0535.35069

[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 2598884 | Zbl 1186.35201

[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 2131050 | Zbl 1068.35119

[18] O. Ivanovici, Precised smoothing effect in the exterior of balls, Asymptot. Anal. 53 (2007), 189-208. | MR 2350738 | Zbl pre05541559

[19] M. Keel & T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980. | MR 1646048 | Zbl 0922.35028

[20] C. E. Kenig, G. Ponce & L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69. | MR 1101221 | Zbl 0738.35022

[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 515228 | Zbl 0395.35070

[22] T. Ozawa & Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11 (1998), 201-222. | MR 1741843 | Zbl 1008.35070

[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 2146323 | Zbl 1160.35509

[24] P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), 699-715. | Zbl 0631.42010

[25] G. Staffilani & D. Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27 (2002), 1337-1372. | MR 1924470 | Zbl 1010.35015

[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 512086 | Zbl 0372.35001

[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 1625056 | Zbl 0924.42008

[28] H. Triebel, Theory of function spaces, Monographs in Mathematics 78, Birkhäuser, 1983. | MR 781540 | Zbl 0546.46027

[29] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874-878. | MR 934859 | Zbl 0654.42014

[30] L. Vega & N. Visciglia, On the local smoothing for the Schrödinger equation (electronic), Proc. Amer. Math. Soc. 135 (2007), 119-128. | MR 2280200 | Zbl 1173.35107