Long finite time bubble trees for two co-rotational wave maps

Abstract

We show that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ into $\mathbb{S}^2$, restricted to the $k=2$ co-rotational setting, admits arbitrarily large numbers of concentrating concentric $n$ bubble profiles. For any $n\in\mathbb{N}$, we construct an $n$-bubble solution concentrating at scales $\lambda_1(t)\gg \lambda_2(t)\gg \ldots\gg \lambda_n(t)$, where $\lambda_n(t)=t^{-1}\vert \log t\vert^\beta$, and $\lambda_j(t)\gtrsim \exp( \int_t^{t_0} \lambda_{j+1}(s)ds)$, for any $j<n$. Here $\beta>\tfrac32$ is a parameter that can be chosen arbitrarily. This shows that, as far as finite time blow-up case is concerned, the entirety of cases postulated in the soliton resolution theorem indeed occur, provided the concentric collapsing bubbles have alternating signs.