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

HTML articles powered by AMS MathViewer

- by Adán J. Corcho, Filipe Oliveira and Jorge Drumond Silva PDF
- Proc. Amer. Math. Soc.
**141**(2013), 3485-3499 Request permission

## 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

- Christophe Besse and Brigitte Bidégaray,
*Numerical study of self-focusing solutions to the Schrödinger-Debye system*, M2AN Math. Model. Numer. Anal.**35**(2001), no. 1, 35–55 (English, with English and French summaries). MR**1811980**, DOI 10.1051/m2an:2001106 - Brigitte Bidégaray,
*On the Cauchy problem for some systems occurring in nonlinear optics*, Adv. Differential Equations**3**(1998), no. 3, 473–496. MR**1751953** - Brigitte Bidégaray,
*The Cauchy problem for Schrödinger-Debye equations*, Math. Models Methods Appl. Sci.**10**(2000), no. 3, 307–315. MR**1753113**, DOI 10.1142/S0218202500000185 - Thierry Cazenave and Fred B. Weissler,
*Some remarks on the nonlinear Schrödinger equation in the critical case*, Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987) Lecture Notes in Math., vol. 1394, Springer, Berlin, 1989, pp. 18–29. MR**1021011**, DOI 10.1007/BFb0086749 - Thierry Cazenave and Fred B. Weissler,
*The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$*, Nonlinear Anal.**14**(1990), no. 10, 807–836. MR**1055532**, DOI 10.1016/0362-546X(90)90023-A - 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 (2001), No. 26, 7. MR**1824796** - James Colliander and Tristan Roy,
*Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on $\Bbb R^2$*, Commun. Pure Appl. Anal.**10**(2011), no. 2, 397–414. MR**2754279**, DOI 10.3934/cpaa.2011.10.397 - J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao,
*Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\Bbb R^3$*, Comm. Pure Appl. Math.**57**(2004), no. 8, 987–1014. MR**2053757**, DOI 10.1002/cpa.20029 - A. J. Corcho and F. Linares,
*Well-posedness for the Schrödinger-Debye equation*, Partial differential equations and inverse problems, Contemp. Math., vol. 362, Amer. Math. Soc., Providence, RI, 2004, pp. 113–131. MR**2091494**, DOI 10.1090/conm/362/06608 - Adán J. Corcho and Carlos Matheus,
*Sharp bilinear estimates and well posedness for the 1-D Schrödinger-Debye system*, Differential Integral Equations**22**(2009), no. 3-4, 357–391. MR**2492826** - Gadi Fibich and George Papanicolaou,
*Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension*, SIAM J. Appl. Math.**60**(2000), no. 1, 183–240. MR**1740841**, DOI 10.1137/S0036139997322407 - J. Ginibre and G. Velo,
*On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case*, J. Functional Analysis**32**(1979), no. 1, 1–32. MR**533218**, DOI 10.1016/0022-1236(79)90076-4 - J. Ginibre, Y. Tsutsumi, and G. Velo,
*On the Cauchy problem for the Zakharov system*, J. Funct. Anal.**151**(1997), no. 2, 384–436. MR**1491547**, DOI 10.1006/jfan.1997.3148 - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*Small solutions to nonlinear Schrödinger equations*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**10**(1993), no. 3, 255–288 (English, with English and French summaries). MR**1230709**, DOI 10.1016/S0294-1449(16)30213-X - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle*, Comm. Pure Appl. Math.**46**(1993), no. 4, 527–620. MR**1211741**, DOI 10.1002/cpa.3160460405 - Alan C. Newell and Jerome V. Moloney,
*Nonlinear optics*, Advanced Topics in the Interdisciplinary Mathematical Sciences, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1992. MR**1163192** - Yoshio Tsutsumi,
*$L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups*, Funkcial. Ekvac.**30**(1987), no. 1, 115–125. MR**915266**

## 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
- 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
- © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3080171