Concentric bubbles concentrating in finite time for the energy critical wave maps equation
Jacek Jendrej, Joachim Krieger
TL;DR
This work proves the existence of finite-time blow-up for the energy-critical co-rotational Wave Maps equation into $\\mathbb{S}^2$ with $k=2$, featuring two concentric collapsing bubbles of scales $\\lambda_2(t)=t^{-1}\\lvert\log t\rvert^{\\beta}$ and $\\lambda_1(t)=e^{\\alpha(t)}$ with $\\alpha(t)\\sim\lvert\log t\rvert^{\\beta+1}$ for any $\\beta>\\tfrac{3}{2}$. The authors develop a two-scale outer-inner construction, building an outer profile $\\tilde{Q}_2$ and a highly accurate two-bubble approximate solution $u_N=Q(\\lambda_1 r)-\\tilde{Q}_2+v_N$, with $v_N$ formed from iterative corrections that are controlled via a spectral/dispersion framework and a nonlinear fixed-point argument. A modulation equation couples the inner and outer scales, and a refined perturbative analysis yields a residual error $e_N$ decaying like $\\tau^{-N}$, enabling the passage from an approximate to an actual solution. The results demonstrate the occurrence of bubble-tree type finite-time blow-up beyond the threshold energy and provide a robust framework potentially extensible to more concentric bubbles. The analysis draws on distorted Fourier methods, two-scale expansions, and careful modulation-control to manage the interplay between inner high-frequency and outer low-frequency bubbles.
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 $λ_1(t) = e^{α(t)},\,α(t)\sim \big|\log t\big|^{β+1}$, as well as $λ_2(t) = t^{-1}\cdot \big|\log t\big|^β$. The parameter $β>\frac32$ 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.