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Mass concentration phenomena for the $ L^2$-critical nonlinear Schrödinger equation

Authors: Pascal Bégout and Ana Vargas
Journal: Trans. Amer. Math. Soc. 359 (2007), 5257-5282
MSC (2000): Primary 35B05, 35B33, 35B40, 35Q55, 42B10
Published electronically: May 16, 2007
MathSciNet review: 2327030
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Abstract: In this paper, we show that any solution of the nonlinear Schrödinger equation $ iu_t+\Delta u\pm\vert u\vert^\frac{4}{N}u=0,$ which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on Bourgain's (1998), which has established this result in the bidimensional spatial case, and on a generalization of Strichartz's inequality, where the bidimensional spatial case was proved by Moyua, Vargas and Vega (1999). We also generalize to higher dimensions the results in Keraani (2006) and Merle and Vega (1998).

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  • 1. H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), 131-175.MR 1705001 (2000i:35123)
  • 2. J. Bourgain, On the restriction and the multiplier problem in $ \mathbb{R}^3$, Geometric aspects of functional analysis (1989-90), Lecture Notes in Math. #1469, Springer, Berlin, 1991, 179-191. MR 1122623 (92m:42017)
  • 3. J. Bourgain, Refinements of Strichartz' inequality and applications to $ 2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices 5 (1998) 253-283.MR 1616917 (99f:35184)
  • 4. R. Carles and S. Keraani, On the role of quadratic oscillations in nonlinear Schrödinger equation II. The $ L^2$-critical case, Trans. Amer. Math. Soc. 359 (2007), 33-62.
  • 5. T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York, 2003.MR 2002047 (2004j:35266)
  • 6. T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, Partial Differential Equations and Attractors, T.L. Gill and W.W. Zachary (eds.), Lecture Notes in Mathematics #1394, Springer, Berlin, 1989, 18-29.MR 1021011 (91a:35149)
  • 7. J. Colliander, S. Raynor, C. Sulem and J. D. Wright, Ground state mass concentration in the $ L\sp 2$-critical nonlinear Schrödinger equation below $ H\sp 1$, Math. Res. Lett. 12 (2005), no. 2-3, 357-375.MR 2150890 (2006e:35300)
  • 8. A. Córdoba, The Kakeya maximal function and the spherical summation multipliers, Amer. J. Math. 99 (1977) 1-22. MR 0447949 (56:6259)
  • 9. P. Gérard, Description du défault de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998) 213-233 (electronic).MR 1632171 (99h:46051)
  • 10. J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985) 309-327.MR 0801582 (87b:35150)
  • 11. R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977) 1794-1797.MR 0460850 (57:842)
  • 12. T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, International Math. Research Notices 46 (2005) 2815-2828. MR 2180464
  • 13. S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal. 235 (2006), 171-192. MR 2216444
  • 14. F. Merle and Y. Tsutsumi, $ {L}\sp 2$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations 84 (1990) 205-214. MR 1047566 (91e:35194)
  • 15. F. Merle and L. Vega Compactness at blow-up time for $ L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, International Math. Research Notices 8 (1998) 399-425. MR 1628235 (99d:35156)
  • 16. A. Moyua, A. Vargas and L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, International Math. Research Notices 16 (1996) 793-815. MR 1413873 (97k:42042)
  • 17. A. Moyua, A. Vargas and L. Vega, Restriction theorems and maximal operators related to oscillatory integrals in $ \mathbb{R}\sp 3$, Duke Math. J. 96 (1999) 547-574.MR 1671214 (2000b:42017)
  • 18. H. Nawa, ``Mass concentration'' phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity, Funkcial. Ekvac. 35 (1992), no. 1, 1-18.MR 1172417 (93h:35193)
  • 19. K. Rogers, A. Vargas, A Refinement of the Strichartz Inequality on the Saddle and Applications, J. Funct. Anal. 241 (2006), no. 1, 212-231. MR 2264250
  • 20. E. M. Stein, Oscillatory integrals in Fourier analysis, Beijing Lectures in Harmonic Analysis, Annals of Math. Study #112 Princeton University Press, 1986. MR 0864375 (88g:42022)
  • 21. R. S. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equation, Duke Math. J. 44 (1977) 705-713.MR 0512086 (58:23577)
  • 22. T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003) 1359-1384. MR 2033842 (2004m:47111)
  • 23. T. Tao, A. Vargas and L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998) 967-1000.MR 1625056 (99f:42026)
  • 24. P. A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477-478. MR 0358216 (50:10681)
  • 25. Y. Tsutsumi, $ {L}\sp 2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987) 115-125.MR 0915266 (89c:35143)
  • 26. N. Tzirakis, Mass concentration phenomenon for the quintic nonlinear Schrödinger equation in 1D, SIAM J. Math. Anal. 37 (2006), 1923-1946 (electronic). MR 2213400
  • 27. S. N. Vlasov, V. A. Petrischev, V. I. Talanov, Averaged description of wave beams in linear and nonlinear media (the method of moments), Izv. Vys. Uchebn. Zaved. Radiofiz. 14 1353 [Radiophys. Quantum Electron. 14 1062-1070 (1974)].
  • 28. M. Visan, X. Zhang, On the blowup for the $ L^2$-critical focusing nonlinear Schrödinger equation in higher dimensions below the energy class. To appear in SIAM J. Math. Anal.

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Additional Information

Pascal Bégout
Affiliation: Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte Courrier 187, 4, place Jussieu, 75252 Paris Cedex 05, France
Address at time of publication: Département de Mathématiques, Laboratoire d’Analyse et Probabilités, Université d’Évry Val d’Essone, Boulevard François Mitterrand, 91025 Évry Cedex, France

Ana Vargas
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid 28049 Madrid, Spain

Keywords: Schr\"{o}dinger equations, restriction theorems, Strichartz's estimate, blow-up
Received by editor(s): July 20, 2005
Published electronically: May 16, 2007
Additional Notes: This research was partially supported by the European network HPRN–CT–2001–00273–HARP (Harmonic analysis and related problems). The second author was also supported by Grant MTM2004–00678 of the MEC (Spain).
Article copyright: © Copyright 2007 American Mathematical Society

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