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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) Zbl 1213.35139

Summary: We consider the energy critical Schrödinger map t u=uΔu 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 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 Θ * R Q 1 x λ(t)u * inH ˙ 1 astT,

where Θ * , u * H ˙ 1 , R is a rotation and the concentration rate is given for some κ(u)>0 by

λ(t)=κ(u)T-t |log(T-t)| 2 1 + o ( 1 )astT·

MSC:
35B44Blow-up (PDE)
35K59Quasilinear parabolic equations
35K15Second order parabolic equations, initial value problems
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