Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map

Merle, Frank; Raphaël, Pierre; Rodnianski, Igor

Comptes Rendus. Mathématique, Tome 349 (2011) no. 5-6, pp. 279-283.

Nous considérons lʼapplication de Schrödinger sur la 2-sphère énergie critique ${\partial }_{t}u=u\wedge \mathrm{\Delta }u$ pour des données initiales à symétrie équivariante et de degré $k=1$. Nous exhibons un ensemble de données initiales régulières arbitrairement proches dans la topologie invariante dʼéchelle ${\dot{H}}^1$ de lʼapplication harmonique dʼénergie minimale $Q_1$ qui engendrent des solutions explosives en temps fini. Nous donnons une description fine de la formation de singularité qui correspond à la concentration dʼune bulle universelle dʼénergie

$$u(t,x)-e^{{\Theta }^{⁎}R}{Q}_1\left(\frac{x}{\lambda (t)}\right)\to u^{⁎}\text{in}{\dot{H}}^1$$

où ${\Theta }^{⁎}\in \mathbb{R}$, $u^{⁎}\in {\dot{H}}^1$, R est une rotation et la vitesse de concentration est donnée pour une certain $\kappa (u)>0$ par :

$$\lambda (t)=\kappa (u)\frac{T-t}{{|\log (T-t)|}^2}(1+o(1))\text{quand}t\to T.$$

We consider the energy critical Schrödinger map ${\partial }_{t}u=u\wedge \mathrm{\Delta }u$ to the 2-sphere for equivariant initial data of homotopy number $k=1$. We show the existence of a set of smooth initial data arbitrarily close to the ground state harmonic map $Q_1$ in the scale invariant norm ${\dot{H}}^1$ which generate finite time blow up solutions. We give in addition a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy

$$u(t,x)-e^{{\Theta }^{⁎}R}{Q}_1\left(\frac{x}{\lambda (t)}\right)\to u^{⁎}\text{in}{\dot{H}}^1\text{as}t\to T$$

where ${\Theta }^{⁎}\in \mathbb{R}$, $u^{⁎}\in {\dot{H}}^1$, R is a rotation and the concentration rate is given for some $\kappa (u)>0$ by

$$\lambda (t)=\kappa (u)\frac{T-t}{{|\log (T-t)|}^2}(1+o(1))\text{as}t\to T.$$

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