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 $C_{l}, C_{m,k}>0$, $$\frac{C_{l}}{\log^{b}(t)} \leq \lambda_{0,0,b}(t) \leq \frac{C_{m}}{\log^{b}(t)}, \quad |\lambda_{0,0,b}^{(k)}(t)| \leq \frac{C_{m,k}}{t^{k} \log^{b+1}(t) }, k\geq 1 \quad t \geq 100$$ 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 $$u_{b}(t,r) = Q_{\frac{1}{\lambda_{b}(t)}}(r) + v_{2}(t,r) + v_{e}(t,r), \quad t \geq T_{0}$$ where $$-\partial_{tt}v_{2}+\partial_{rr}v_{2}+\frac{1}{r}\partial_{r}v_{2}-\frac{v_{2}}{r^{2}}=0$$ $$||\partial_{t}(Q_{\frac{1}{\lambda_{b}(t)}}+v_{e})||_{L^{2}(r dr)}^{2}+||\frac{v_{e}}{r}||_{L^{2}(r dr)}^{2} + ||\partial_{r}v_{e}||_{L^{2}(r dr)}^{2} \leq \frac{C}{t^{2} \log^{2b}(t)}, \quad t \geq T_{0}$$ and $$\lambda_{b}(t) = \lambda_{0,0,b}(t) + O\left(\frac{1}{\log^{b}(t) \sqrt{\log(\log(t))}}\right)$$