Remarks on global existence and compactness for ${L}^{2}$ solutions in the critical nonlinear schrödinger equation in 2D
Journées équations aux dérivées partielles (1998), article no. 13, 9 p.

In the talk we shall present some recent results obtained with F. Merle about compactness of blow up solutions of the critical nonlinear Schrödinger equation for initial data in ${L}^{2}\left({𝐑}^{2}\right)$. They are based on and are complementary to some previous work of J. Bourgain about the concentration of the solution when it approaches to the blow up time.

@article{JEDP_1998____A13_0,
author = {Gonzalez, Luis Vega},
title = {Remarks on global existence and compactness for $L^2$ solutions in the critical nonlinear schr\"odinger equation in 2D},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
publisher = {Universit\'e de Nantes},
year = {1998},
zbl = {01808722},
language = {en},
url = {http://www.numdam.org/item/JEDP_1998____A13_0}
}

Gonzalez, Luis Vega. Remarks on global existence and compactness for $L^2$ solutions in the critical nonlinear schrödinger equation in 2D. Journées équations aux dérivées partielles (1998), article  no. 13, 9 p. http://www.numdam.org/item/JEDP_1998____A13_0/

[B1] J. Bourgain, Some new estimates on oscillatory integrals, Essays on Fourier Analysis in Honour of E. Stein, Princeton UP 42 (1995), 83-112 | MR 96c:42028 | Zbl 0840.42007

[B2] J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity, Preprint | Zbl 0917.35126

[B-L] H. Berestycki, P.L. Lions Nonlinear scalar field equations, Arch. Rat. Mech. Anal., 82 (1983), 313-375 | MR 84h:35054a | Zbl 0533.35029

[C] T. Cazenave An introduction to nonlinear Schrödinger equations, Textos de Metodos Matematicos 26 (Rio de Janeiro)

[C-W] T. Cazenave, F. Weissler Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors, Lect. Notes in Math., 1394, Spr. Ver., 1989, 18-29 | MR 91a:35149 | Zbl 0694.35170

[G] R.T Glassey On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys. 18 (1977), 1794-1797 | MR 57 #842 | Zbl 0372.35009

[G-V] J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z 170, (1980), 109-136 | MR 82c:35018 | Zbl 0407.35063

[K] M.K. Kwong Uniqueness of positive solutions of Δu - u + up = 0 in RN, Arch. Rat. Mech. Ann. 105, (1989), 243-266 | MR 90d:35015 | Zbl 0676.35032

[M1] F. Merle Determination of blow-up solutions with minimal mass for non-linear Schrödinger equations with critical power, Duke Math. J., 69, (2) (1993), 427-454 | MR 94b:35262 | Zbl 0808.35141

[M2] F. Merle Lower bounds for the blow-up rate of solutions of the Zakharov equation in dimension two Comm. Pure and Appl. Math, Vol. XLIX, (1996), 8, 765-794 | MR 97d:35210 | Zbl 0856.35014

[MV] F. Merle, L. Vega Compactness at blow-up time for L2 solutions of the critical nonlinear Schrödinger equation. To appear in IMRN, 1998 | Zbl 0913.35126

[MVV] A. Moyua, A. Vargas, L. Vega Restriction theorems and maximal operators related to oscillatory integrals in ℝ³ to appear in Duke Math. J. | Zbl 0946.42011

[St] R. Strichartz Restriction of Fourier transforms to quadratic surfaces and decay of solutions to wave equations, Duke Math J., 44, (1977), 705-714 | MR 58 #23577 | Zbl 0372.35001

[W] M.I. Weinstein On the structure and formation of singularities of solutions to nonlinear dispersive equations Comm. P.D.E. 11, (1986), 545-565 | MR 87i:35026 | Zbl 0596.35022

[ZSS] V.E. Zakharov, V. V. Sobolev, and V.S. Synach Character of the singularity and stochastic phenomena in self-focusing, Zh. Eksper. Teoret. Fiz. 14 (1971), 390-393