论文标题
Landau-Lifshitz-Gilbert方程式的有限时间奇异性形成二
Finite-time singularity formations for the Landau-Lifshitz-Gilbert equation in dimension two
论文作者
论文摘要
我们从$ {\ mathbb r}^2 $构建有限的时间爆破解决方案，向landau-lifshitz-gilbert方程式（llg）构建有限的爆炸解决方案，构建$ {\ Mathbb r}^2 $ to $ s^2 $ \ begin {equation*} \ begin {cases} case} u_t = a（case} u_t = a（ΔU+| \ nabla U |^2U |^2U）-b u |^2U | {\ MathBb r}^2 \ times（0，t），u（\ cdot，0）= u_0 \ in s^2＆\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\ {\ Mathbb r}^2，\ end end {cases} \ end {cases} \ end End {eend {quemation*} r} $。 Given any prescribed $N$ points in $\mathbb{R}^2$ and small $T>0$, we prove that there exists regular initial data such that the solution blows up precisely at these points at finite time $t=T$, taking around each point the profile of sharply scaled degree 1 harmonic map with the type II blow-up speed \begin{equation*} \| \nabla u\|_{L^\infty } \sim \frac{|\ln(T-t)|^2}{ T-t } \ \mbox{ as } \ t\to T. \end{equation*} The proof is based on the {\em parabolic inner-outer gluing method}, developed in \cite{17HMF} for谐波图流（HMF）。但是，分散的存在的直接结果是{\ em缺乏最大原理}适合数量，这使得分析即使在线性化水平上也更加精致。为了克服这一困难，我们利用了两种关键的技术成分：首先，对于内部问题，我们采用{\ em扭曲的傅里叶变换}的工具，由Krieger，Miao，Schlag和Tataru \ Cite \ Cite {Krieger09duke，KMS20WM}开发。其次，外部问题的线性理论是通过在非差异形式的基本解决方案的基本解决方案的基本估计中实现的，该解决方案在太空中具有DINI平均振荡的系数（$ \ Mathsf {dmo_x} $），这是由Dong，Kim和Lee \ cite \ cite {Dong22-dong22-Non-divergence证明的。
We construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert equation (LLG) from ${\mathbb R}^2$ into $S^2$ \begin{equation*} \begin{cases} u_t= a(Δu+|\nabla u|^2u) -b u\wedge Δu &\ \mbox{ in }\ {\mathbb R}^2\times(0,T), u(\cdot,0) = u_0\in S^2 &\ \mbox{ in }\ {\mathbb R}^2, \end{cases} \end{equation*} where $a^2+b^2=1,~a > 0,~ b\in {\mathbb R}$. Given any prescribed $N$ points in $\mathbb{R}^2$ and small $T>0$, we prove that there exists regular initial data such that the solution blows up precisely at these points at finite time $t=T$, taking around each point the profile of sharply scaled degree 1 harmonic map with the type II blow-up speed \begin{equation*} \| \nabla u\|_{L^\infty } \sim \frac{|\ln(T-t)|^2}{ T-t } \ \mbox{ as } \ t\to T. \end{equation*} The proof is based on the {\em parabolic inner-outer gluing method}, developed in \cite{17HMF} for Harmonic Map Flow (HMF). However, a direct consequence of the presence of dispersion is the {\em lack of maximum principle} for suitable quantities, which makes the analysis more delicate even at the linearized level. To overcome this difficulty, we make use of two key technical ingredients: first, for the inner problem we employ the tool of {\em distorted Fourier transform}, as developed by Krieger, Miao, Schlag and Tataru \cite{Krieger09Duke,KMS20WM}. Second, the linear theory for the outer problem is achieved by means of the sub-Gaussian estimate for the fundamental solution of parabolic system in non-divergence form with coefficients of Dini mean oscillation in space ($\mathsf{DMO_x}$), which was proved by Dong, Kim and Lee \cite{dong22-non-divergence} recently.
