Remarks on global existence and compactness for L 2 solutions in the critical nonlinear schrödinger equation in 2D
Gonzalez, Luis Vega
Journées équations aux dérivées partielles, (1998), p. 1-9 / Harvested from Numdam

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 (𝐑 2 ). 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.

Publié le : 1998-01-01
@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},
     year = {1998},
     pages = {1-9},
     zbl = {01808722},
     language = {en},
     url = {http://dml.mathdoc.fr/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), pp. 1-9. http://gdmltest.u-ga.fr/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