Concentric bubbles concentrating in finite time for the energy critical wave maps equation

Abstract

We show that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ to $\mathbb{S}^2$ and restricted to the co-rotational setting with co-rotation index $k = 2$ admits finite time blow up solutions of finite energy on $(0, t_0]\times \mathbb{R}^2$, $t_0>0$, and concentrating two concentric bubble profiles at the frequency scales $\lambda_1(t) = e^{\alpha(t)},\,\alpha(t)\sim \big|\log t\big|^{\beta+1}$, as well as $\lambda_2(t) = t^{-1}\cdot \big|\log t\big|^{\beta}$. The parameter $\beta>\frac32$ can be chosen arbitrarily. This shows that soliton resolution scenarios with finite time blow up and $N = 2$ collapsing profiles, i. e. bubble trees, do occur for this equation.