发表机构
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对两相调和映射热流，在给定移动界面条件下证明了短时间强解的存在唯一性，并通过法向反射和Schauder不动点方法处理界面条件，应用于矩阵值尖锐界面系统。
AI 中文摘要
设 $N^+$ 和 $N^-$ 是欧几里得空间中不相交的紧致光滑子流形，设 $\Lambda\subset N^+\times N^-$ 是紧致光滑嵌入子流形，用于规定允许的单侧迹对。给定一族光滑分离超曲面 $\Gamma_t\subset\mathbb T^d$，我们研究两个移动相中流入 $N^\pm$ 的调和映射热流，满足 \\[ (u^+,u^-)\in\Lambda, \qquad \bigl(-\partial_{\nu_t}u^+,\partial_{\nu_t}u^-\bigr) \perp T_{(u^+,u^-)}\Lambda \quad\text{在 }\Gamma_t\text{ 上}. \\] 对于 $q>d+2$，满足 $t=0$ 时这些条件的 $W_q^{2-2/q}$ 初始数据生成唯一的 $W_q^{2,1}$ 短时间强解，该解在任意正时刻在移动界面附近光滑；满足各阶相容条件的光滑初始数据产生直到初始角点都光滑的解。我们通过法向反射将配对映射沿 $\Lambda$ 加倍，从而将界面条件转化为界面附近的 Dirichlet 问题，并在 Schauder 不动点论证中通过重叠时空 Schwarz 映射将该问题与两个体相相匹配。当界面独立地按平均曲率演化时，我们还获得了以其曲率和两个相梯度表示的爆破替代。作为应用，我们证明了 Fei、Lin、Wang 和 Zhang（\emph{Invent. Math.} 233 (2023), 1--80）收敛理论中出现的周期矩阵值尖锐界面系统的局部光滑可解性。
英文摘要
Let $N^+$ and $N^-$ be disjoint compact smooth submanifolds of a Euclidean space, and let $Λ\subset N^+\times N^-$ be a compact smooth embedded submanifold prescribing admissible pairs of one-sided traces. Given a smooth family of separating hypersurfaces $Γ_t\subset\mathbb T^d$, we study harmonic map heat flows into $N^\pm$ in the two moving phases, subject to \[ (u^+,u^-)\inΛ, \qquad \bigl(-\partial_{ν_t}u^+,\partial_{ν_t}u^-\bigr) \perp T_{(u^+,u^-)}Λ \quad\text{on }Γ_t. \] For $q>d+2$, initial data in $W_q^{2-2/q}$ satisfying these conditions at $t=0$ generate a unique short-time strong solution in $W_q^{2,1}$, which is smooth up to the moving interface for every positive time; smooth initial data satisfying the compatibility conditions of every order yield solutions that are smooth up to the initial corner. We double the paired map across $Λ$ by normal reflection, which turns the interface conditions into a Dirichlet problem near the interface, and match this problem with the two bulk phases by an overlapping space-time Schwarz map inside a Schauder fixed-point argument. When the interface evolves independently by mean curvature, we also obtain a blow-up alternative in terms of its curvature and the two phase gradients. As an application, we prove local smooth solvability for the periodic matrix-valued sharp-interface system appearing in the convergence theory of Fei, Lin, Wang, and Zhang (\emph{Invent. Math.} 233 (2023), 1--80).
Comments42 pages, 1 figure
