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Zbl 1213.35139
Merle, Frank; Raphaël, Pierre; Rodnianski, Igor
Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map. (Dynamique explosive de solutions régulières équivariantes de l'application de Schrödinger map.) (English. Abridged French version)
[J] C. R., Math., Acad. Sci. Paris 349, No. 5-6, 279-283 (2011). ISSN 1631-073X

Summary: We consider the energy critical Schrödinger map $\partial_tu=u\wedge\Delta u$ 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 $$u(t,x)- e^{\Theta^*R}Q_1\bigg(\frac{x}{\lambda(t)}\bigg)\to u^* \quad\text{in }\dot H^1\text{ as }t\to T,$$ where $\Theta^*\in\Bbb R$, $u^*\in\dot H^1$, $R$ is a rotation and the concentration rate is given for some $\kappa(u)>0$ by $$\lambda(t)= \kappa(u) \frac{T-t}{|\log(T-t)|^2} \big(1+o(1)\big) \quad\text{as }t\to T.$$

MSC 2000:: *35B44
35K59
35K15 Second order parabolic equations, initial value problems

Keywords: singularity formation; concentration of universal bubble

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