Bilinear virial identities and applications

Fabrice Planchon; Luis Vega

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 2, page 261-290
  • ISSN: 0012-9593

Abstract

top
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.

How to cite

top

Planchon, Fabrice, and Vega, Luis. "Bilinear virial identities and applications." Annales scientifiques de l'École Normale Supérieure 42.2 (2009): 261-290. <http://eudml.org/doc/272210>.

@article{Planchon2009,
abstract = {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.},
author = {Planchon, Fabrice, Vega, Luis},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {nonlinear Schrödinger equation; Virial identity; exterior domain},
language = {eng},
number = {2},
pages = {261-290},
publisher = {Société mathématique de France},
title = {Bilinear virial identities and applications},
url = {http://eudml.org/doc/272210},
volume = {42},
year = {2009},
}

TY - JOUR
AU - Planchon, Fabrice
AU - Vega, Luis
TI - Bilinear virial identities and applications
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 2
SP - 261
EP - 290
AB - 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.
LA - eng
KW - nonlinear Schrödinger equation; Virial identity; exterior domain
UR - http://eudml.org/doc/272210
ER -

References

top
  1. [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. Zbl1148.35081MR2401142
  2. [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. Zbl1169.35012MR2350410
  3. [3] J. Bourgain, Refinements of Strichartz’ inequality and applications to 2 D-NLS with critical nonlinearity, Int. Math. Res. Not.1998 (1998), 253–283. Zbl0917.35126MR1616917
  4. [4] N. Burq, Smoothing effect for Schrödinger boundary value problems, Duke Math. J.123 (2004), 403–427. Zbl1061.35024MR2066943
  5. [5] N. Burq, P. Gérard & N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire21 (2004), 295–318. Zbl1061.35126MR2068304
  6. [6] N. Burq, P. Gérard & N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math.126 (2004), 569–605. Zbl1067.58027MR2058384
  7. [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. Zbl1293.35264MR2227135
  8. [8] M. Christ & A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal.179 (2001), 409–425. Zbl0974.47025MR1809116
  9. [9] J. Colliander, M. Grillakis & N. Tzirakis, The interaction Morawetz estimate for 2 . (Workshop « Nonlinear waves and dispersive equations »), Oberwolfach Report 44 (2007). Zbl1142.35085
  10. [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. Zbl1185.35250MR2527809
  11. [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. Zbl1157.35103MR2379437
  12. [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. Zbl1060.35131MR2053757
  13. [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. Zbl1178.35345MR2415387
  14. [14] P. Constantin & J.-C. Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J.38 (1989), 791–810. Zbl0712.35022MR1017334
  15. [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. Zbl0535.35069MR839728
  16. [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. Zbl1186.35201MR2598884
  17. [17] A. Hassell, T. Tao & J. Wunsch, A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds, Comm. Partial Differential Equations30 (2005), 157–205. Zbl1068.35119MR2131050
  18. [18] O. Ivanovici, Precised smoothing effect in the exterior of balls, Asymptot. Anal.53 (2007), 189–208. Zbl05541559MR2350738
  19. [19] M. Keel & T. Tao, Endpoint Strichartz estimates, Amer. J. Math.120 (1998), 955–980. Zbl0922.35028MR1646048
  20. [20] C. E. Kenig, G. Ponce & L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J.40 (1991), 33–69. Zbl0738.35022MR1101221
  21. [21] J. E. Lin & W. A. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal.30 (1978), 245–263. Zbl0395.35070MR515228
  22. [22] T. Ozawa & Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations11 (1998), 201–222. Zbl1008.35070MR1741843
  23. [23] D. Salort, Dispersion and Strichartz inequalities for the one-dimensional Schrödinger equation with variable coefficients, Int. Math. Res. Not.2005 (2005), 687–700. Zbl1160.35509MR2146323
  24. [24] P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J.55 (1987), 699–715. Zbl0631.42010
  25. [25] G. Staffilani & D. Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations27 (2002), 1337–1372. Zbl1010.35015MR1924470
  26. [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. Zbl0372.35001MR512086
  27. [27] T. Tao, A. Vargas & L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc.11 (1998), 967–1000. Zbl0924.42008MR1625056
  28. [28] H. Triebel, Theory of function spaces, Monographs in Mathematics 78, Birkhäuser, 1983. Zbl0546.46027MR781540
  29. [29] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc.102 (1988), 874–878. Zbl0654.42014MR934859
  30. [30] L. Vega & N. Visciglia, On the local smoothing for the Schrödinger equation (electronic), Proc. Amer. Math. Soc.135 (2007), 119–128. Zbl1173.35107MR2280200

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.