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 ${\lambda }_1(t)=e^{\alpha (t)},\alpha (t)\sim |\log t{|}^{\beta +1}$, as well as ${\lambda }_2(t)=t^{-1}\cdot |\log t{|}^{\beta }$. The parameter $\beta >\frac{3}{2}$ 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.

69 pages