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 $\hat{I}{»}_1(t)\gg \hat{I}{»}_2(t)\gg \dots \gg \hat{I}{»}_n(t)$, where $\hat{I}{»}_n(t)=t^{-1}|\log t{|}^2$, and $\hat{I}{»}_j(t)\gtrsim \exp ({\int }_t^{t_0}\hat{I}{»}_{j+1}(s)ds)$, for any $j<n$. Here ${\hat{I}}^2>\frac{3}{2}$ 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.

72 pages