Blowup dynamics for smooth equivariant solutions to energy critical Landau-Lifschitz flow

In this paper, we study the energy critical 1-equivariant Landau-Lifschitz flow mapping ${\mathbb{R}}^2$ to ${\mathbb{S}}^2$ with arbitrary given coefficients ${\rho }_1\in \mathbb{R}$, ${\rho }_2>0$. We prove that there exists a codimension one smooth well-localized set of initial data arbitrarily close to the ground state which generates type-II finite-time blowup solutions, and give a precise description of the corresponding singularity formation. In our proof, both the Schr"odinger part and the heat part play important roles in the construction of approximate solutions and the mixed energy/Morawetz functional. However, the blowup rate is independent of the coefficients.