Explosion pour l'équation de Schrödinger au régime du « log log »
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 953, pp. 33-54.

On présente dans cet exposé des résultats récents de Merle et Raphael sur l’analyse des solutions explosives de l’équation de Schrödinger L 2 critique. On s’intéresse en particulier à leur preuve du fait que les solutions d’énergie négative (dont on savait qu’elles explosaient par l’argument du viriel) et dont la norme L 2 est proche de celle de l’état fondamental, explosent au régime du “log log”et que ce comportement est stable.

In this talk we present some recent results by Merle and Raphael on analysis of blow-up solution for the L 2 critical non linear Schrödinger equation. In particular, we focus on their proof of the fact that initial data with negative energy (which had been known to blow up by the viriel identity) and with L 2 norm close to the ground states’s L 2 norm, do blow up in the “log log” regime and that this behaviour is stable.

Classification : 35B30, 35B35, 35B65
Mot clés : equations de Schrödinger non linéaires, explosion
Keywords: non linear Schrödinger equations, blow-up
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Burq, Nicolas. Explosion pour l'équation de Schrödinger au régime du « log log », dans Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 953, pp. 33-54. http://www.numdam.org/item/SB_2005-2006__48__33_0/

[BP93] V. Buslaev & G. Perel'Man - “Scattering for the nonlinear Schrödinger equation : States close to a soliton”, St. Petersbg. Math. J. 4 (1993), no. 6, p. 1111-1142 (Russian, English). | MR | Zbl

[BP95] -, “On the stability of solitary waves for nonlinear Schrödinger equations”, Amer. Math. Soc. Transl., Ser. 2 164 (1995), no. 22, p. 75-98 (English). | Zbl

[BW97] J. Bourgain & W. Wang - “Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity.”, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., IV. Ser. 25 (1997), no. 1-2, p. 197-215 (English). | EuDML | Numdam | MR | Zbl

[FMR04] G. Fibich, F. Merle & P. Raphael - “Numerical proof of a spectral property related to the singularity formation for the L 2 critical non linear Schrödinger equation”, preprint, 2004. | Zbl

[GM94a] L. Glangetas & F. Merle - “Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two II”, Comm. Math. Phys. 160 (1994), no. 2, p. 349-389. | MR | Zbl

[GM94b] -, “Existence of self-similar blow-up solutions for Zakharov equation in dimension two I”, Comm. Math. Phys. 160 (1994), no. 1, p. 173-215. | MR | Zbl

[GV79] J. Ginibre & G. Velo - “On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case”, J. Funct. Anal. 32 (1979), no. 1, p. 1-32. | MR | Zbl

[JP93] R. Johnson & X. B. Pan - “On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation”, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 4, p. 763-782. | MR | Zbl

[Lio84] P.-L. Lions - “The concentration-compactness principle in the calculus of variations. The locally compact case. I”, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, p. 109-145. | EuDML | Numdam | MR | Zbl

[LPSS88] M. J. Landman, G. C. Papanicolaou, C. Sulem & P.-L. Sulem - “Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension”, Phys. Rev. A (3) 38 (1988), no. 8, p. 3837-3843. | MR

[Mer93] F. Merle - “Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power”, Duke Math. J. 69 (1993), no. 2, p. 427-454. | MR | Zbl

[Mer96a] -, “Blow-up results of virial type for Zakharov equations”, Comm. Math. Phys. 175 (1996), no. 2, p. 433-455. | MR | Zbl

[Mer96b] -, “Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two”, Comm. Pure Appl. Math. 49 (1996), no. 8, p. 765-794. | MR | Zbl

[MM00] Y. Martel & F. Merle - “A Liouville theorem for the critical generalized Korteweg-de Vries equation”, J. Math. Pures Appl. (9) 79 (2000), no. 4, p. 339-425. | MR | Zbl

[MM02a] -, “Blow up in finite time and dynamics of blow up solutions for the L 2 -critical generalized KdV equation”, J. Amer. Math. Soc. 15 (2002), no. 3, p. 617-664 (electronic). | MR | Zbl

[MM02b] -, “Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation”, Ann. of Math. (2) 155 (2002), no. 1, p. 235-280. | MR | Zbl

[MM04] -, “Review on blow up and asymptotic dynamics for critical and subcritical gKdV equations”, Noncompact problems at the intersection of geometry, analysis, and topology, Contemp. Math., vol. 350, Amer. Math. Soc., Providence, 2004, p. 157-177. | MR | Zbl

[MR03] F. Merle & P. Raphael - “Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation”, Geom. Funct. Anal. 13 (2003), no. 3, p. 591-642. | MR | Zbl

[MR04] -, “On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation”, Invent. Math. 156 (2004), no. 3, p. 565-672. | MR | Zbl

[MR05a] -, “The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation”, Ann. of Math. (2) 161 (2005), no. 1, p. 157-222. | MR | Zbl

[MR05b] -, “On one blow up point solutions to the critical nonlinear Schrödinger equation”, J. Hyperbolic Differ. Equ. 2 (2005), no. 4, p. 919-962. | MR | Zbl

[MR05c] -, “Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation”, Comm. Math. Phys. 253 (2005), no. 3, p. 675-704. | MR | Zbl

[Naw99] H. Nawa - “Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power”, Comm. Pure Appl. Math. 52 (1999), no. 2, p. 193-270. | MR | Zbl

[OT91] T. Ogawa & Y. Tsutsumi - “Blow-up of H 1 solution for the nonlinear Schrödinger equation”, J. Differential Equations 92 (1991), no. 2, p. 317-330. | MR | Zbl

[Per01] G. Perelman - “On the formation of singularities in solutions of the critical nonlinear Schrödinger equation”, Ann. Inst. H. Poincaré 2 (2001), no. 4, p. 605-673. | MR | Zbl

[Rap04] P. Raphael - “On the blow up phenomenon for the L 2 critical non linear Schrödinger equation”, preprint, 2004. | Zbl

[Rap05] -, “Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation”, Math. Ann. 331 (2005), no. 3, p. 577-609. | MR | Zbl

[SS99] C. Sulem & P.-L. Sulem - The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999, Self-focusing and wave collapse. | MR | Zbl

[Tzv05] N. Tzvetkov - “On the long time behaviour of KdV type equations”, Séminaire Bourbaki, vol. 2003/2004, Astérisque, vol. 299, 2005, exp. no 933. | EuDML | Numdam | Zbl

[Wei85] M. I. Weinstein - “Modulational stability of ground states of nonlinear Schrödinger equations”, SIAM J. Math. Anal. 16 (1985), no. 3, p. 472-491. | MR | Zbl

[Wei83] -, “Nonlinear Schrödinger equations and sharp interpolation estimates”, Comm. Math. Phys. 87 (1982/83), no. 4, p. 567-576. | MR | Zbl

[ZS71] V. E. Zakharov & A. B. Shabat - “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media”, Ž. Èksper. Teoret. Fiz. 61 (1971), no. 1, p. 118-134. | MR