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

TL;DR

This paper constructs finite time blow-up solutions for the energy critical wave maps equation, demonstrating the formation of two concentric bubble profiles with specific scaling behaviors, thus confirming the occurrence of bubble trees in this setting.

Contribution

It provides the first explicit construction of finite time blow-up solutions with two concentric bubbles for the energy critical wave maps equation in the co-rotational setting.

Findings

Finite time blow-up solutions with two concentric bubbles are constructed.

The bubble scales follow specific logarithmic and polynomial behaviors.

Bubble trees with multiple collapsing profiles occur for this 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.