Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schrödinger and Hartree equations


Authors: Jean Ginibre and Giorgio Velo
Journal: Quart. Appl. Math. 68 (2010), 113-134
MSC (2000): Primary 35P25; Secondary 35B40, 35Q40, 35Q55
DOI: https://doi.org/10.1090/S0033-569X-09-01141-9
Published electronically: October 26, 2009
MathSciNet review: 2598884
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Recently several authors have developed multilinear and in particular quadratic extensions of the classical Morawetz inequality. Those extensions provide (among other results) an easy proof of asymptotic completeness in the energy space for nonlinear Schrödinger equations in arbitrary space dimension and for Hartree equations in space dimension greater than two in the noncritical cases. We give a pedagogical review of the latter results.


References [Enhancements On Off] (What's this?)

  • 1. J. Bourgain, Global solutions of non linear Schrödinger equations, AMS Colloq. Pub. 46, 1999.
  • 2. T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Math. 10 AMS, 2003. MR 2002047 (2004j:35266)
  • 3. J. Colliander, M. Grillakis and N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $ \mathbb{R}^{2}$, Int. Math. Res. Not. IMRN 2007, no. 23, Art. ID mm090, 30pp. MR 2377216
  • 4. J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Preprint, math.AP/0807.0871.
  • 5. J. Colliander, H. Holmer, M. Visan and X. Zhang, Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $ \mathbb{R}$, Commun. Pure Appl. Anal. 7 (2008), 467-489. MR 2379437 (2009c:35433)
  • 6. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering in the energy space for the critical nonlinear Schrödinger equation in $ \mathbb{R}^3$, Annals Math. 166 (2007), 1-100. MR 2359021
  • 7. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions to a nonlinear Schrödinger equation on $ \mathbb{R}^3$, Comm. Pure Appl. Math. 57 (2004) no. 8, 987-1014. MR 2053757 (2005b:35257)
  • 8. D. De Silva, N. Pavlovic, G. Staffilani and N. Tzirakis, Global well-posedness and polynomial bounds for the defocusing $ L^2$-critical nonlinear Schrödinger equation in higher dimensions, Commun. Pure Appl. Anal. 6 (2007), 1023-1041. MR 2341818
  • 9. D. De Silva, N. Pavlovic, G. Staffilani and N. Tzirakis, Global well-posedness and polynomial bounds for the defocusing $ L^2$-critical nonlinear Schrödinger equation in $ \mathbb{R}$, Comm. Partial Differential Equations 33 (2008), no. 7-9, 1395-1429. MR 2450163
  • 10. Y. Fang and M. Grillakis, On the global existence of rough solutions to the cubic defocusing Schrödinger equation in $ \mathbb{R}^{2+1}$, J. Hyp. Diff. Eq. 4 (2007), 233-257. MR 2329384 (2008c:35303)
  • 11. D. Fang and Z. Han, The nonlinear Schrödinger equations with combined nonlinearities of power-type and Hartree-type, Preprint, math.AP/0808.1622.
  • 12. J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pure Appl. 64 (1985), 363-401. MR 839728 (87i:35171)
  • 13. J. Ginibre and G. Velo, Scattering theory in the energy space for a class of Hartree equations, in Nonlinear Wave Equations, Y. Guo Ed., Contemporary Mathematics 263, AMS 2000. MR 1777634 (2001g:35205)
  • 14. J. Holmer and N. Tzirakis, Asymptotically linear solutions in $ H^1$ of the 2D defocusing nonlinear Schrödinger and Hartree equations, Preprint, math.AP/0805.2925.
  • 15. J. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal. 30 (1978), 245-263. MR 515228 (80k:35056)
  • 16. C. Miao, G. Xu and L. Zhao, Global wellposedness and scattering for the defocusing $ H^{1/2}$ subcritical Hartree equation in $ \mathbb{R}^d$, Preprint, math.AP/0805.3378.
  • 17. C. Morawetz, Time decay for the nonlinear Klein-Gordon equation, Proc. Roy. Soc. A206 (1968), 291-296. MR 0234136 (38:2455)
  • 18. C. Morawetz and W. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math. 25 (1972), 1-31. MR 0303097 (46:2239)
  • 19. K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal. 169 (1999), 201-225. MR 1726753 (2000m:35141)
  • 20. K. Nakanishi, Energy scattering for Hartree equations, Math. Res. Lett. 6 (1999), no. 1, 107-118. MR 1682697 (2000d:35173)
  • 21. F. Planchon and L. Vega, Bilinear virial identities and applications, Preprint, math.AP/0712.4076. To appear in Ann. Scient. ENS.
  • 22. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1967. MR 0290095 (44:7280)
  • 23. T. Tao, Nonlinear dispersive equations, local and global analysis, Regional. Conf. Ser. Math. 106, AMS 2006. MR 2233925 (2008i:35211)
  • 24. T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power type non linearities, Comm. Partial Diff. Eq. 32 (2007), 1281-1343. MR 2354495
  • 25. M. Visan and X. Zhang, Global wellposedness and scattering for a class of nonlinear Schrödinger equations below the energy space, Diff. Int. Eq. 22 (2009), 99-124.

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35P25, 35B40, 35Q40, 35Q55

Retrieve articles in all journals with MSC (2000): 35P25, 35B40, 35Q40, 35Q55


Additional Information

Jean Ginibre
Affiliation: Laboratoire de Physique Théorique, Université de Paris XI, Bâtiment 210, F-91405 ORSAY Cedex, France
Email: Jean.Ginibre@th.u-psud.fr

Giorgio Velo
Affiliation: Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, Italy
Email: Velo@bo.infn.it

DOI: https://doi.org/10.1090/S0033-569X-09-01141-9
Received by editor(s): July 3, 2008
Received by editor(s) in revised form: October 7, 2008
Published electronically: October 26, 2009
Additional Notes: First author’s sponsoring institution: Unité Mixte de Recherche (CNRS) UMR 8627
Dedicated: Dedicated to Professor Walter Strauss on his 70th birthday
Article copyright: © Copyright 2009 Brown University

American Mathematical Society