Abstract
For the Schrödinger map equation \({u_t = u \times \triangle u \, {\rm in} \, \mathbb{R}^{2+1}}\) , with values in S 2, we prove for any \({\nu > 1}\) the existence of equivariant finite time blow up solutions of the form \({u(x, t) = \phi(\lambda(t) x) + \zeta(x, t)}\) , where \({\phi}\) is a lowest energy steady state, \({\lambda(t) = t^{-1/2-\nu}}\) and \({\zeta(t)}\) is arbitrary small in \({\dot H^1 \cap \dot H^2}\) .
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Perelman, G. Blow Up Dynamics for Equivariant Critical Schrödinger Maps. Commun. Math. Phys. 330, 69–105 (2014). https://doi.org/10.1007/s00220-014-1916-1
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DOI: https://doi.org/10.1007/s00220-014-1916-1