Quantized slow blowup dynamics for the energy-critical Schrödinger map flow

We study the Cauchy problem for the energy-critical Schrödinger map into the 2-sphere: $${\mathrm{\partial }}_{t}u=u\times \mathrm{\Delta }u,(t,x)\in {\mathbb{R}}^{1+2},u(t,x)\in {\mathbb{S}}^2$$ for equivariant initial data of homotopy degree $k=1$. Given any integer $L\in {\mathbb{N}}_{+}$, we exhibit a set of initial data arbitrarily close to the ground state harmonic map in the energy norm such that the corresponding solution blows up in finite time with a quantized rate $$\lambda (t)\sim \frac{{(T-t)}^L}{{\mid \log (T-t)\mid }^{\frac{2L}{2L-1}}}\text{quadas}t\uparrow T\mathrm{.}$$ Our proof is inspired by the techniques developed for the energy-critical harmonic map heat flow in a similar context by Rapha{ë}l and Schweyer~[Anal. PDE 7 (2014), 1713–1805]. However, its adaptation to a Schrödinger-type equation requires the introduction of a new energy-Virial-Morawetz Lyapunov functional tailored to quantized slow blowup dynamics.