Blowup dynamics for equivariant critical Landau--Lifshitz flow

The existence of finite time blowup solutions for the two-dimensional Landau–Lifshitz equation is a long-standing problem, which exists in the literature at least since 2001 (E, Mathematics Unlimited–2001 and Beyond, Springer, Berlin, P.410, 2001). A more refined description in the equivariant class is given in (van den Berg and Williams, European J. Appl. Math., 24(6), 912–948, 2013). In this paper, we consider the blowup dynamics of the Landau–Lifshitz equation $${\mathrm{\partial }}_{t}u={\mathfrak{a}}_{1}u\times \mathrm{\Delta }u-{\mathfrak{a}}_{2}u\times (u\times \mathrm{\Delta }u),x\in {\mathbb{R}}^2,$$ where $u\in {\mathbb{S}}^2$, ${\mathfrak{a}}_1+i{\mathfrak{a}}_2\in \mathbb{C}$ with ${\mathfrak{a}}_2\ge 0$ and ${\mathfrak{a}}_1+{\mathfrak{a}}_2=1$. We prove the existence of 1-equivariant Krieger–Schlag–Tataru type blowup solutions near the lowest energy steady state. More precisely, we prove that for any $\nu >1$, there exists a 1-equivariant finite-time blowup solution of the form $$u(x,t)=\phi (\lambda (t)x)+\zeta (x,t),\lambda (t)=t^{-1\mathrm{/}2-\nu },$$ where $\phi$ is a lowest energy steady state and $\zeta (t)$ is arbitrary small in ${\dot{H}}^1\cap {\dot{H}}^2$. The proof is accomplished by renormalizing the blowup profile and a perturbative analysis in the spirit of (Krieger, Schlag and Tataru, Invent. Math., 171(3), 543–615, 2008), (Perelman, Comm. Math. Phys., 330(1), 69–105, 2014) and (Ortoleva and Perelman, Algebra i Analiz, 25(2), 271–294, 2013).