# Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation

Journées équations aux dérivées partielles (2002)

- Volume: 161, Issue: 1, page 1-5
- ISSN: 0752-0360

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topMerle, Frank, and Raphael, Pierre. "Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation." Journées équations aux dérivées partielles 161.1 (2002): 1-5. <http://eudml.org/doc/93423>.

@article{Merle2002,

abstract = {We consider the critical nonlinear Schrödinger equation $iu_t=-\Delta u-|u|^\{\{4\over N\}\}u$ with initial condition $u(0,x)=u_0$ in dimension $N$. For $u_0\in H^1$, local existence in time of solutions on an interval $[0,T)$ is known, and there exists finite time blow up solutions, that is $u_0$ such that $\lim _\{t\rightarrow T<+\infty \}|u_x(t)|_\{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.},

author = {Merle, Frank, Raphael, Pierre},

journal = {Journées équations aux dérivées partielles},

keywords = {sufficient conditions for blow-up for solutions; critical nonlinear Schrödinger equation},

language = {eng},

number = {1},

pages = {1-5},

publisher = {Université de Nantes},

title = {Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation},

url = {http://eudml.org/doc/93423},

volume = {161},

year = {2002},

}

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

PB - Université de Nantes

VL - 161

IS - 1

SP - 1

EP - 5

AB - We consider the critical nonlinear Schrödinger equation $iu_t=-\Delta u-|u|^{{4\over N}}u$ with initial condition $u(0,x)=u_0$ in dimension $N$. For $u_0\in H^1$, local existence in time of solutions on an interval $[0,T)$ is known, and there exists finite time blow up solutions, that is $u_0$ such that $\lim _{t\rightarrow T<+\infty }|u_x(t)|_{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.

LA - eng

KW - sufficient conditions for blow-up for solutions; critical nonlinear Schrödinger equation

UR - http://eudml.org/doc/93423

ER -

## References

top- [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). Zbl1043.35137MR1655515
- [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. Zbl0396.35028MR533218
- [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. MR966356
- [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. Zbl0996.35064MR1896235
- [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. Zbl0808.35141MR1203233
- [6] Merle, F.; Raphael, P., Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, preprint. Zbl1185.35263
- [7] Merle, F.; Raphael, P., Sharp upper bound on the blow up rate for critical nonlinear Schrodinger equation, preprint. Zbl1061.35135MR1995801
- [8] Perelman, G., On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D, to appear in Annale Henri Poincare. Zbl1061.35523
- [9] Weinstein, M.I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983) Zbl0527.35023MR691044

## Citations in EuDML Documents

top- Pierre Raphaël, Igor Rodnianski, Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang-Mills Problems
- Frank Merle, Pierre Raphaël, Jérémie Szeftel, Two blow-up regimes for ${L}^{2}$ supercritical nonlinear Schrödinger equations
- Rémi Carles, Changing blow-up time in nonlinear Schrödinger equations

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