arXiv2022
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 $$ \partial_tu=\mathfrak{a}_1u\timesΔu-\mathfrak{a}_2u\times(u\timesΔu),\quad x\in\mathbb{R}^2, $$ where $u\in\mathbb{S}^2$, $\mathfrak{a}_1+i\mathfrak{a}_2\in\mathbb{C}$ with $\mathfrak{a}_2\geq0$ 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 $ν>1$, there exists a 1-equivariant finite-time blowup solution of the form $$ u(x,t)=ϕ(λ(t)x)+ζ(x,t),\quad λ(t)=t^{-1/2-ν}, $$ where $ϕ$ is a lowest energy steady state and $ζ(t)$ 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).