Sur la dynamique explosive des solutions de l’équation de Schrödinger non linéaire
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2004-2005), Talk no. 3, 11 p.
Raphaël, Pierre 1

1 Université de Paris Sud et CNRS
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     author = {Rapha\"el, Pierre},
     title = {Sur la dynamique explosive des solutions de l{\textquoteright}\'equation de {Schr\"odinger} non lin\'eaire},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
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     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
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Raphaël, Pierre. Sur la dynamique explosive des solutions de l’équation de Schrödinger non linéaire. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2004-2005), Talk no. 3, 11 p. http://archive.numdam.org/item/SEDP_2004-2005____A3_0/

[1] Berestycki, H. ; Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. | MR | Zbl

[2] Bourgain, J. ; Wang, W., Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 197–215 (1998). | Numdam | MR | Zbl

[3] Ginibre, J. ; Velo, G., On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32 (1979), no. 1, 1–32. | MR | Zbl

[4] Glassey, R.T., On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18, 1794-1797 (1977). | MR | Zbl

[5] Glangetas, L. ; Merle, F., Existence of self-similar blow-up solutions for Zakharov equation in dimension two. I. Comm. Math. Phys. 160 (1994), no. 1, 173–215. | MR | Zbl

[6] Kwong, M. K., Uniqueness of positive solutions of Δu-u+u p =0 in R n . Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. | MR | Zbl

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

[8] Martel, Y. ; Merle, F., 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, 235–280. | MR | Zbl

[9] Merle, F., Construction of solutions with exactly k blow up points for the Schrödinger equation with critical nonlinearity, J. Diff. Eq. 84 (1990), no. 2, 223-240. | MR | Zbl

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

[11] Merle, F., Lower bounds for the blow up rate of solutions of the Zakharov equations in dimension two, Comm. Pure. Appl. Math. 49 (1996), n0. 8, 765-794. | MR | Zbl

[12] Merle, F. ; Raphaël, P., Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, to appear in Annals of Math. | Numdam | MR | Zbl

[13] Merle, F. ; Raphaël, P., Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation, Geom. Funct .Ana 13 (2003), 591–642. | MR | Zbl

[14] Merle, F. ; Raphaël, P., On Universality of Blow up Profile for L 2 critical nonlinear Schrödinger equation, Invent. Math. 156, 565-672 (2004). | MR | Zbl

[15] Merle, F. ; Raphaël, P., Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation, preprint.

[16] Merle, F. ; Raphaël, P., Profiles and quantization of the blow up mass for critical non linear Schrödinger equation, to appear in Comm. Math. Phys. | MR | Zbl

[17] Perelman, G., On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D, Ann. Henri. Poincaré, 2 (2001), 605-673. | MR | Zbl

[18] Raphaël, P., Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation, to appear in Math. Annalen. | Zbl

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

[20] Weinstein, M.I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567—576. | MR | Zbl

[21] Zakharov, V.E. ; Shabat, A.B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media, Sov. Phys. JETP 34 (1972), 62—69. | MR