Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Local and global well-posedness for the critical Schrödinger-Debye system


Authors: Adán J. Corcho, Filipe Oliveira and Jorge Drumond Silva
Journal: Proc. Amer. Math. Soc. 141 (2013), 3485-3499
MSC (2010): Primary 35Q55, 35Q60; Secondary 35B65
DOI: https://doi.org/10.1090/S0002-9939-2013-11612-6
Published electronically: June 18, 2013
MathSciNet review: 3080171
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish local well-posedness results for the Initial Value Problem associated to the Schrödinger-Debye system in dimensions $ N=2, 3$ for data in $ H^s\times H^{\ell }$, with $ s$ and $ \ell $ satisfying $ \max \{0, s-1\} \le \ell \le \min \{2s, s+1\}$. In particular, these include the energy space $ H^1\times L^2$. Our results improve the previous ones obtained by B. Bidégaray, and by A. J. Corcho and F. Linares. Moreover, in the critical case ($ N=2$) and for initial data in $ H^1\times L^2$, we prove that solutions exist for all times, thus providing a negative answer to the open problem mentioned by G. Fibich and G. C. Papanicolau concerning the formation of singularities for these solutions.


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

  • 1. C. Besse and B. Bidégaray, Numerical study of self-focusing solutions to the Schrödinger-Debye system, ESAIM: M2AN, 35 (2001), 35-55. MR 1811980 (2002b:78023)
  • 2. B. Bidégaray, On the Cauchy problem for systems occurring in nonlinear optics, Adv. Diff. Equat., 3 (1998), 473-496. MR 1751953 (2001c:78026)
  • 3. B. Bidégaray, The Cauchy problem for Schrödinger-Debye equations, Math. Models Methods Appl. Sci., 10 (2000), 307-315. MR 1753113 (2001a:78041)
  • 4. T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Lecture Notes in Math., 1394 (1989), 18-29. MR 1021011 (91a:35149)
  • 5. T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $ H^s$, Nonlinear Anal. TMA, 14 (1990), 807-836. MR 1055532 (91j:35252)
  • 6. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Differential Equations, 26 (2001), 1-7. MR 1824796 (2001m:35269)
  • 7. J. Colliander and T. Roy, Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on $ \mathbb{R}^2$, Comm. Pure and Applied Anal., 10 (2011), 397-414. MR 2754279 (2011m:35348)
  • 8. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $ \mathbb{R}^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014. MR 2053757 (2005b:35257)
  • 9. A. J. Corcho and F. Linares, Well-Posedness for the Schrödinger-Debye Equation, Contemporary Mathematics, 362 (2004), 113-131. MR 2091494 (2005k:35373)
  • 10. A. J. Corcho and C. Matheus, Sharp bilinear estimates and well-posedness for the
    1-D Schrödinger-Debye system,
    Differential and Integral Equations, 22 (2009), 357-391. MR 2492826 (2009m:35478)
  • 11. G. Fibich and G. C. Papanicolau, Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension, SIAM J. Appl. Math., 60 (2000), 183-240. MR 1740841 (2000j:78013)
  • 12. J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I and II, J. Funct. Anal., 32 (1979), 1-32, 33-72. MR 0533218 (82c:35057), MR 533219 (82c:35058)
  • 13. J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. MR 1491547 (2000c:35220)
  • 14. C. E. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 10 (1993), 255-288. MR 1230709 (94h:35238)
  • 15. C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. MR 1211741 (94h:35229)
  • 16. A. C. Newell and J. V. Moloney, Nonlinear Optics, Addison-Wesley, 1992. MR 1163192 (93i:78010)
  • 17. Y. Tsutsumi, $ L^2$-solutions for nonlinear Schrödinger equations and nonlinear group, Funkcialaj Ekvacioj, 30 (1987), 115-125. MR 915266 (89c:35143)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35Q55, 35Q60, 35B65

Retrieve articles in all journals with MSC (2010): 35Q55, 35Q60, 35B65


Additional Information

Adán J. Corcho
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro-UFRJ, Ilha do Fundão, 21945-970, Rio de Janeiro-RJ, Brazil
Email: adan@im.ufrj.br

Filipe Oliveira
Affiliation: Centro de Matemática e Aplicações, FCT-UNL, Monte da Caparica, Portugal
Email: fso@fct.unl.pt

Jorge Drumond Silva
Affiliation: Center for Mathematical Analysis, Geometry and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
Email: jsilva@math.ist.utl.pt

DOI: https://doi.org/10.1090/S0002-9939-2013-11612-6
Keywords: Perturbed nonlinear Schr\"odinger equation, Cauchy problem, global well-posedness
Received by editor(s): February 14, 2011
Received by editor(s) in revised form: December 15, 2011
Published electronically: June 18, 2013
Additional Notes: The first author was supported by CAPES and CNPq (Edital Universal-482129/2009-3), Brazil
The second author was partially supported by FCT/Portugal through Financiamento Base 2008-ISFL-1-297
The third author was partially supported by the Center for Mathematical Analysis, Geometry and Dynamical Systems through the Fundação para a Ciência e Tecnologia (FCT/Portugal) program POCTI/FEDER
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society