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

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- by Frank Merle and Pierre Raphael
- J. Amer. Math. Soc.
**19**(2006), 37-90 - DOI: https://doi.org/10.1090/S0894-0347-05-00499-6
- Published electronically: September 1, 2005
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## 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

- G. D. Akrivis, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney,
*Numerical approximation of blow-up of radially symmetric solutions of the nonlinear Schrödinger equation*, SIAM J. Sci. Comput.**25**(2003), no. 1, 186–212. MR**2047201**, DOI 10.1137/S1064827597332041 - H. Berestycki and P.-L. Lions,
*Nonlinear scalar field equations. I. Existence of a ground state*, Arch. Rational Mech. Anal.**82**(1983), no. 4, 313–345. MR**695535**, DOI 10.1007/BF00250555 - J. Bourgain,
*Global solutions of nonlinear Schrödinger equations*, American Mathematical Society Colloquium Publications, vol. 46, American Mathematical Society, Providence, RI, 1999. MR**1691575**, DOI 10.1090/coll/046 - Jean Bourgain and W. Wang,
*Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**25**(1997), no. 1-2, 197–215 (1998). Dedicated to Ennio De Giorgi. MR**1655515** - S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov,
*Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation*, Phys. D**57**(1992), no. 1-2, 96–160. MR**1169619**, DOI 10.1016/0167-2789(92)90090-A
FMR Fibich, G.; Merle, F.; Raphaël, P., Numerical proof of a spectral property related to the singularity formation for the $L^2$ critical nonlinear Schrödinger equation, in preparation.
- J. Ginibre and G. Velo,
*On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case*, J. Functional Analysis**32**(1979), no. 1, 1–32. MR**533218**, DOI 10.1016/0022-1236(79)90076-4 - Man Kam Kwong,
*Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\textbf {R}^n$*, Arch. Rational Mech. Anal.**105**(1989), no. 3, 243–266. MR**969899**, DOI 10.1007/BF00251502 - M. J. Landman, G. C. Papanicolaou, C. Sulem, and P.-L. Sulem,
*Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension*, Phys. Rev. A (3)**38**(1988), no. 8, 3837–3843. MR**966356**, DOI 10.1103/PhysRevA.38.3837 - Mihai Mariş,
*Existence of nonstationary bubbles in higher dimensions*, J. Math. Pures Appl. (9)**81**(2002), no. 12, 1207–1239 (English, with English and French summaries). MR**1952162**, DOI 10.1016/S0021-7824(02)01274-6 - Y. Martel and F. Merle,
*Instability of solitons for the critical generalized Korteweg-de Vries equation*, Geom. Funct. Anal.**11**(2001), no. 1, 74–123. MR**1829643**, DOI 10.1007/PL00001673 - Kevin McLeod,
*Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $\textbf {R}^n$. II*, Trans. Amer. Math. Soc.**339**(1993), no. 2, 495–505. MR**1201323**, DOI 10.1090/S0002-9947-1993-1201323-X - F. Merle,
*Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power*, Duke Math. J.**69**(1993), no. 2, 427–454. MR**1203233**, DOI 10.1215/S0012-7094-93-06919-0 - Frank Merle and Pierre Raphael,
*Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation*, Journées “Équations aux Dérivées Partielles” (Forges-les-Eaux, 2002) Univ. Nantes, Nantes, 2002, pp. Exp. No. XII, 5. MR**1968208** - F. Merle and P. Raphael,
*Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation*, Geom. Funct. Anal.**13**(2003), no. 3, 591–642. MR**1995801**, DOI 10.1007/s00039-003-0424-9 - Frank Merle and Pierre Raphael,
*On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation*, Invent. Math.**156**(2004), no. 3, 565–672. MR**2061329**, DOI 10.1007/s00222-003-0346-z - Frank Merle and Pierre Raphael,
*Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation*, Comm. Math. Phys.**253**(2005), no. 3, 675–704. MR**2116733**, DOI 10.1007/s00220-004-1198-0 - Dmitry Pelinovsky,
*Radiative effects to the adiabatic dynamics of envelope-wave solitons*, Phys. D**119**(1998), no. 3-4, 301–313. MR**1639310**, DOI 10.1016/S0167-2789(98)00037-2
P Perelman, G., On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D, Ann. Henri Poincaré, 2 (2001), 605-673.
- Pierre Raphael,
*Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation*, Math. Ann.**331**(2005), no. 3, 577–609. MR**2122541**, DOI 10.1007/s00208-004-0596-0 - Catherine Sulem and Pierre-Louis Sulem,
*The nonlinear Schrödinger equation*, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. MR**1696311** - Michael I. Weinstein,
*Modulational stability of ground states of nonlinear Schrödinger equations*, SIAM J. Math. Anal.**16**(1985), no. 3, 472–491. MR**783974**, DOI 10.1137/0516034 - Michael I. Weinstein,
*Nonlinear Schrödinger equations and sharp interpolation estimates*, Comm. Math. Phys.**87**(1982/83), no. 4, 567–576. MR**691044**, DOI 10.1007/BF01208265

## Bibliographic 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
- Received by editor(s): March 2, 2004
- Published electronically: September 1, 2005
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2169042