Infinite time blow-up solutions to the energy critical wave maps equation

Abstract

We consider the wave maps problem with domain ${\mathbb{R}}^{2+1}$ and target ${\mathbb{S}}^2$ in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from ${\mathbb{R}}^2$ to ${\mathbb{S}}^2$, with polar angle equal to $Q_1(r)=2\arctan (r)$. By applying the scaling symmetry of the equation, $Q_{\lambda }(r)=Q_1(r\lambda )$ is also a harmonic map, and the family of all such $Q_{\lambda }$ are the unique minimizers of the harmonic map energy among finite energy, 1-equivariant, topological degree one maps. In this work, we construct infinite time blowup solutions along the $Q_{\lambda }$ family. More precisely, for $b>0$, and for all ${\lambda }_{0,0,b}\in C^{\infty }([100,\infty ))$ satisfying, for some $C_l,C_{m,k}>0$, $$\frac{C_l}{{\log }^b(t)}\le {\lambda }_{0,0,b}(t)\le \frac{C_m}{{\log }^b(t)},|{\lambda }_{0,0,b}^{(k)}(t)|\le \frac{C_{m,k}}{t^k{\log }^{b+1}(t)},k\ge 1t\ge 100$$ there exists a wave map with the following properties. If $u_b$ denotes the polar angle of the wave map into ${\mathbb{S}}^2$, we have $$u_b(t,r)=Q_{\frac{1}{{\lambda }_b(t)}}(r)+v_2(t,r)+v_e(t,r),t\ge T_0$$ where $$-{\partial }_{tt}{v}_2+{\partial }_{rr}{v}_2+\frac{1}{r}{\partial }_r{v}_2-\frac{v_2}{r^2}=0$$ $$||{\partial }_t(Q_{\frac{1}{{\lambda }_b(t)}}+v_e)|{|}_{L^2(rdr)}^2+||\frac{v_e}{r}|{|}_{L^2(rdr)}^2+||{\partial }_r{v}_e|{|}_{L^2(rdr)}^2\le \frac{C}{t^2{\log }^{2b}(t)},t\ge T_0$$ and $${\lambda }_b(t)={\lambda }_{0,0,b}(t)+O\left(\frac{1}{{\log }^b(t)\sqrt{\log (\log (t))}}\right)$$

This is the version accepted for publication. No major changes relative to v2