Two blow-up regimes for L 2 supercritical nonlinear Schrödinger equations
Séminaire Équations aux dérivées partielles (Polytechnique) (2009-2010), Talk no. 2, 11 p.

We consider the focusing nonlinear Schrödinger equations i t u+Δu+u|u| p-1 =0. We prove the existence of two finite time blow up dynamics in the supercritical case and provide for each a qualitative description of the singularity formation near the blow up time.

@article{SEDP_2009-2010____A2_0,
     author = {Merle, Frank and Rapha\"el, Pierre and Szeftel, J\'er\'emie},
     title = {Two blow-up regimes for $L^2$ supercritical nonlinear Schr\"odinger equations},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2009-2010},
     note = {talk:2},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_2009-2010____A2_0}
}
Merle, Frank; Raphaël, Pierre; Szeftel, Jérémie. Two blow-up regimes for $L^2$ supercritical nonlinear Schrödinger equations. Séminaire Équations aux dérivées partielles (Polytechnique) (2009-2010), Talk no. 2, 11 p. http://www.numdam.org/item/SEDP_2009-2010____A2_0/

[1] Bourgain, J.; Wang, W., Construction of blow-up solutions for the nonlinear Schrödinger with critical nonlinearity. Ann, Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no 1-2, 197-215. | Numdam | MR 1655515 | Zbl 1043.35137

[2] Fibich, G.; Gavish, N.; Wang, X.P., Singular ring solutions of critical and supercritical nonlinear Schrödinger equations, Physica D: Nonlinear Phenomena, 231 (2007), no. 1, 55–86. | MR 2370365 | Zbl 1118.35043

[3] Gidas, B.; Ni, W.M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209—243. | MR 544879 | Zbl 0425.35020

[4] 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 533218 | Zbl 0396.35028

[5] Kopell, N.; Landman, M., Spatial structure of the focusing singularity of the nonlinear Schrödinger equation: a geometrical analysis, SIAM J. Appl. Math. 55 (1995), no. 5, 1297–1323. | MR 1349311 | Zbl 0836.34041

[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 969899 | Zbl 0676.35032

[7] Merle, F.; Raphaël, P., Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Ann. Math. 161 (2005), no. 1, 157–222. | MR 2150386 | Zbl pre02204253

[8] Merle, F.; Raphaël, P., Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13 (2003), 591-642. | MR 1995801 | Zbl 1061.35135

[9] Merle, F.; Raphaël, P., On universality of blow up profile for L 2 critical nonlinear Schrödinger equation, Invent. Math. 156, 565-672 (2004). | MR 2061329 | Zbl 1067.35110

[10] Merle, F.; Raphaël, P., Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), no. 1, 37–90. | MR 2169042 | Zbl 1075.35077

[11] Merle, F.; Raphaël, P., Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys. 253 (2005), no. 3, 675–704. | MR 2116733 | Zbl 1062.35137

[12] Merle, F.; Raphaël, P.; Szeftel, J., Stable self similar blow up dynamics for slightly L 2 supercritical NLS equations, submitted.

[13] Perelman, G., On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D, Ann. Henri. Poincaré, 2 (2001), 605-673. | MR 1826598 | Zbl 1007.35087

[14] Raphaël, P., Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation, Math. Ann. 331 (2005), 577–609. | MR 2122541 | Zbl 1082.35143

[15] Raphaël, P., Existence and stability of a solution blowing up on a sphere for a L 2 supercritical nonlinear Schrödinger equation, Duke Math. J. 134 (2006), no. 2, 199–258. | MR 2248831 | Zbl 1117.35077

[16] Raphaël, P., Szeftel, J., Standing ring blow up solutions to the quintic NLS in dimension N, to appear in Comm. Math. Phys. | MR 2525647

[17] Sulem, C.; Sulem, P.L., The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. | MR 1696311 | Zbl 0928.35157

[18] Weinstein, M.I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567—576. | MR 691044 | Zbl 0527.35023

[19] 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 406174