Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation


Authors: Frank Merle and Pierre Raphael
Journal: J. Amer. Math. Soc. 19 (2006), 37-90
MSC (2000): Primary 35Q55; Secondary 35Q51, 35B05
DOI: https://doi.org/10.1090/S0894-0347-05-00499-6
Published electronically: September 1, 2005
MathSciNet review: 2169042
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the $L^2$ critical nonlinear Schrödinger equation $iu_t=-\Delta u-|u|^{\frac {4}{N}}u$ with initial condition in the energy space $u(0,x)=u_0\in H^1$ and study the dynamics of finite time blow-up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in $L^2_{loc}$ a sharp and stable upper bound on the blow-up rate: $|\nabla u(t)|_{L^2}\leq C\left (\frac {\log |\log (T-t)|}{T-t}\right )^{\frac {1}{2}}$. In an earlier paper, the authors then addressed the question of a lower bound on the blow-up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is, $\lim _{t\to T}\sqrt {T-t}|\nabla u(t)|_{L^2}=+\infty .$ In this paper, we prove the sharp lower bound \[ |\nabla u(t)|_{L^2}\geq C \left (\frac {\log |\log (T-t)|}{T-t}\right )^{\frac {1}{2}}\] by exhibiting the dispersive structure in the scaling invariant space $L^2$ for this log-log regime. In addition, we will extend to the pure energy space $H^1$ a dynamical characterization of the solitons among the zero energy solutions.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 35Q55, 35Q51, 35B05

Retrieve articles in all journals with MSC (2000): 35Q55, 35Q51, 35B05


Additional Information

Frank Merle
Affiliation: Université de Cergy–Pontoise, Cergy-Pointoise, France, Institute for Advanced Studies and Centre National de la Recherche Scientifique
MR Author ID: 123710

Pierre Raphael
Affiliation: Université de Cergy–Pontoise, Cergy-Pointoise, France, Institute for Advanced Studies and Centre National de la Recherche Scientifique

Keywords: Critical Schrödinger equation, finite time blowup, blow-up rate, log-log law.
Received by editor(s): March 2, 2004
Published electronically: September 1, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.