AI 中文总结
该研究将三维共旋转调和映射热流的稳定自相似剖面结果推广到4、5、6维，借助严格计算机辅助证明了稳定收缩子存在，还得到共旋转类自相似剖面的有限余维稳定性。
AI 中文摘要
我们研究从欧几里得空间$\boldsymbol{\text{R}}^d$到$d$维单位球面$\boldsymbol{\text{S}}^d$的调和映射热流在超临界维数$d \boldsymbol{\text{∈}} \boldsymbol{\text{\textit{\textbf{\text{3,4,5,6}}}}$时的奇点形成问题。众所周知，在这些维数中存在无穷多个自相似解，它们是有限时间内正则性丧失的实例。本文将文献\textbf{\text{\textit{\textbf{BieDon18}}}}、\textbf{\text{\textit{\textbf{BieDonSch17}}}}中关于$d=3$的结果推广到高空间维数$d \boldsymbol{\text{∈}} \boldsymbol{\text{\textit{\textbf{\text{4,5,6}}}}$，证明了存在一个单调递增的自相似剖面$f_0$，其在小共旋转扰动下渐近稳定。为构造该解并求解谱问题，我们使用了严格的计算机辅助方法。作为稳定性分析的副产品，我们还得到了共旋转类中任意自相似剖面的有限余维稳定性。
英文摘要
We study singularity formation for the heat flow of harmonic maps from $\R^d$ into $\mathbb{S}^d$ in supercritical dimensions $d \in \{3,4,5,6\}$. It is well known that in each of these dimensions there exist infinitely many self-similar solutions that provide examples of loss of regularity in finite time. In this paper, we extend the results of \cite{BieDon18}, \cite{BieDonSch17} for $d=3$ to higher space dimensions $d \in \{4,5,6\}$ and prove the existence of a monotonically increasing self-similar profile $f_0$, which is asymptotically stable under small corotational perturbations. To construct the solution and resolve the spectral problem, we use rigorous computer assistance. As a byproduct of our stability analysis, we also obtain finite-codimension stability of arbitrary self-similar profiles within the corotational class.
