Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation
Journées équations aux dérivées partielles (2002), article no. 12, 5 p.

We consider the critical nonlinear Schrödinger equation $i{u}_{t}=-\Delta u-{|u|}^{\frac{4}{N}}u$ with initial condition $u\left(0,x\right)={u}_{0}$ in dimension $N$. For ${u}_{0}\in {H}^{1}$, local existence in time of solutions on an interval $\left[0,T\right)$ is known, and there exists finite time blow up solutions, that is ${u}_{0}$ such that ${lim}_{t\to T<+\infty }{|{u}_{x}\left(t\right)|}_{{L}^{2}}=+\infty$. This is the smallest power in the nonlinearity for which blow up occurs, and is critical in this sense. The question we address is to understand the blow up dynamic. Even though there exists an explicit example of blow up solution and a class of initial data known to lead to blow up, no general understanding of the blow up dynamic is known. At first, we propose in this paper a general setting to study and understand small in a certain sense blow up solutions. Blow up in finite time follows for the whole class of initial data in ${H}^{1}$ with strictly negative energy, and one is able to prove a control from above of the blow up rate below the one of the known explicit explosive solution, which has strictly positive energy.

@article{JEDP_2002____A12_0,
author = {Merle, Frank and Raphael, Pierre},
title = {Blow up dynamic and upper bound on the blow up rate for critical nonlinear {Schr\"odinger} equation},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {12},
publisher = {Universit\'e de Nantes},
year = {2002},
doi = {10.5802/jedp.610},
mrnumber = {1968208},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/jedp.610/}
}
TY  - JOUR
AU  - Merle, Frank
AU  - Raphael, Pierre
TI  - Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation
JO  - Journées équations aux dérivées partielles
PY  - 2002
DA  - 2002///
PB  - Université de Nantes
UR  - http://archive.numdam.org/articles/10.5802/jedp.610/
UR  - https://www.ams.org/mathscinet-getitem?mr=1968208
UR  - https://doi.org/10.5802/jedp.610
DO  - 10.5802/jedp.610
LA  - en
ID  - JEDP_2002____A12_0
ER  - 
%0 Journal Article
%A Merle, Frank
%A Raphael, Pierre
%T Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation
%J Journées équations aux dérivées partielles
%D 2002
%I Université de Nantes
%U https://doi.org/10.5802/jedp.610
%R 10.5802/jedp.610
%G en
%F JEDP_2002____A12_0
Merle, Frank; Raphael, Pierre. Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation. Journées équations aux dérivées partielles (2002), article  no. 12, 5 p. doi : 10.5802/jedp.610. http://archive.numdam.org/articles/10.5802/jedp.610/

[1] Bourgain, J.; Wang, W., 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). | Numdam | MR | Zbl

[2] Ginibre, J.; Velo, G., On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32 (1979), no. 1, 1-32. | MR | Zbl

[3] Landman, M. J.; Papanicolaou, G. C.; Sulem, C.; Sulem, P.-L., Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension. Phys. Rev. A (3) 38 (1988), no. 8, 3837-3843. | MR

[4] Martel, Y.; Merle, F., Blow up in finite time and dynamics of blow up solutions for the L 2 -critical generalized KdV equation, to appear in J. Amer. Math. Soc. | MR | Zbl

[5] Merle, F., 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 | Zbl

[6] Merle, F.; Raphael, P., Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, preprint.

[7] Merle, F.; Raphael, P., Sharp upper bound on the blow up rate for critical nonlinear Schrodinger equation, preprint. | MR

[8] Perelman, G., On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D, to appear in Annale Henri Poincare.

[9] Weinstein, M.I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983) | MR | Zbl

Cited by Sources: