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

Luis Vega Gonzalez

Journées équations aux dérivées partielles (1998)

  • page 1-9
  • ISSN: 0752-0360

Abstract

top
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.

How to cite

top

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): 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
  1. [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
  2. [B2] J. Bourgain, Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity, Preprint Zbl0917.35126
  3. [B-L] H. Berestycki, P.L. LionsNonlinear scalar field equations, Arch. Rat. Mech. Anal., 82 (1983), 313-375 Zbl0533.35029MR84h:35054a
  4. [C] T. CazenaveAn introduction to nonlinear Schrödinger equations, Textos de Metodos Matematicos 26 (Rio de Janeiro) 
  5. [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
  6. [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
  7. [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
  8. [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
  9. [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
  10. [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
  11. [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
  12. [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
  13. [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
  14. [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
  15. [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 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.