Dynamics for the corotational energy-critical wave map equation with quantized blow-up rates

We consider the wave maps from ${\mathbb{R}}^{1+2}$ into ${\mathbb{S}}^2\subset {\mathbb{R}}^3\mathrm{.}$ Under an additional assumption of $k$-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation: \begin{equation*} \partial_t^2 u-\partial_r^2 u-\frac{\partial_r u}{r}+k^2 \frac{\sin(2u)}{2r^2}=0. \end{equation*} Given any integer $k\ge 1$ and any integer $m\ge 2k,$ we exhibit a set of initial data $(u_0,u_1)$ with energy arbitrarily close to that of the ground state solution $Q$, such that the corresponding solution $u$ blows up in finite time by concentrating its energy. To be precise, the solution $u$ satisfies \begin{equation*} \lim\limits_{t\rightarrow T} \left|\left(u(t,r)-Q\left(\frac{r}{λ(t)}\right)-u_1^®, \partial_t u-u_2^®\right)\right|{H\times L^2}=0 \end{equation*} with a quantized speed \begin{equation*} λ(t)=c(u_0,u_1)(1+o{t\to T}(1))\frac{(T-t)^{\frac{m}{k}}}{|\log(T-t)|^{\frac{m}{k(m-k)}}}, \end{equation*} where $\mathrm{\parallel }u{\mathrm{\parallel }}_H:={\int }_{{\mathbb{R}}^2}(\mathrm{\mid }{\mathrm{\partial }}_{r}u{\mathrm{\mid }}^2+\frac{\mathrm{\mid }u{\mathrm{\mid }}^2}{r^2})\mathrm{.}$