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

Abstract

We study the Cauchy problem for the energy-critical Schrödinger map into the 2-sphere: \[ \partial_{t}u = u\timesΔu, \quad (t,x)\in\mathbb{R}^{1+2}, \quad u(t,x)\in\mathbb{S}^2 \] 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 \[ λ(t)\sim \frac{\left(T-t\right)^L}{\left|\log(T-t)\right|^{\frac{2L}{2L-1}}} \quad\mbox{as}\ \ t\uparrow T. \] 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.