# Remarks on global existence and compactness for ${L}^{2}$ solutions in the critical nonlinear schrödinger equation in 2D

Journées équations aux dérivées partielles (1998)

- page 1-9
- ISSN: 0752-0360

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topGonzalez, 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): 1-9. <http://eudml.org/doc/93355>.

@article{Gonzalez1998,

abstract = {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 $L^2(\mathbf \{R\}^2)$. 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.},

author = {Gonzalez, Luis Vega},

journal = {Journées équations aux dérivées partielles},

keywords = {compactness of blow up solutions; critical nonlinear Schrödinger equation; concentration},

language = {eng},

pages = {1-9},

publisher = {Université de Nantes},

title = {Remarks on global existence and compactness for $L^2$ solutions in the critical nonlinear schrödinger equation in 2D},

url = {http://eudml.org/doc/93355},

year = {1998},

}

TY - JOUR

AU - Gonzalez, Luis Vega

TI - Remarks on global existence and compactness for $L^2$ solutions in the critical nonlinear schrödinger equation in 2D

JO - Journées équations aux dérivées partielles

PY - 1998

PB - Université de Nantes

SP - 1

EP - 9

AB - 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 $L^2(\mathbf {R}^2)$. 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.

LA - eng

KW - compactness of blow up solutions; critical nonlinear Schrödinger equation; concentration

UR - http://eudml.org/doc/93355

ER -

## References

top- [B1] J. Bourgain, Some new estimates on oscillatory integrals, Essays on Fourier Analysis in Honour of E. Stein, Princeton UP 42 (1995), 83-112 Zbl0840.42007MR96c:42028
- [B2] J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity, Preprint Zbl0917.35126
- [B-L] H. Berestycki, P.L. LionsNonlinear scalar field equations, Arch. Rat. Mech. Anal., 82 (1983), 313-375 Zbl0533.35029MR84h:35054a
- [C] T. CazenaveAn introduction to nonlinear Schrödinger equations, Textos de Metodos Matematicos 26 (Rio de Janeiro)
- [C-W] T. Cazenave, F. WeisslerSome 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 Zbl0694.35170MR91a:35149
- [G] R.T GlasseyOn the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys. 18 (1977), 1794-1797 Zbl0372.35009MR57 #842
- [G-V] J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z 170, (1980), 109-136 Zbl0407.35063MR82c:35018
- [K] M.K. KwongUniqueness of positive solutions of Δu - u + up = 0 in RN, Arch. Rat. Mech. Ann. 105, (1989), 243-266 Zbl0676.35032MR90d:35015
- [M1] F. MerleDetermination of blow-up solutions with minimal mass for non-linear Schrödinger equations with critical power, Duke Math. J., 69, (2) (1993), 427-454 Zbl0808.35141MR94b:35262
- [M2] F. MerleLower 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 Zbl0856.35014MR97d:35210
- [MV] F. Merle, L. VegaCompactness at blow-up time for L2 solutions of the critical nonlinear Schrödinger equation. To appear in IMRN, 1998 Zbl0913.35126
- [MVV] A. Moyua, A. Vargas, L. VegaRestriction theorems and maximal operators related to oscillatory integrals in ℝ³ to appear in Duke Math. J. Zbl0946.42011
- [St] R. StrichartzRestriction of Fourier transforms to quadratic surfaces and decay of solutions to wave equations, Duke Math J., 44, (1977), 705-714 Zbl0372.35001MR58 #23577
- [W] M.I. WeinsteinOn the structure and formation of singularities of solutions to nonlinear dispersive equations Comm. P.D.E. 11, (1986), 545-565 Zbl0596.35022MR87i:35026
- [ZSS] V.E. Zakharov, V. V. Sobolev, and V.S. SynachCharacter of the singularity and stochastic phenomena in self-focusing, Zh. Eksper. Teoret. Fiz. 14 (1971), 390-393

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