发表机构
Zhejiang University; Peking University（浙江大学; 北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究二维抛物型向量Allen-Cahn系统的渐近行为，将Bethuel椭圆紧性定理推广到抛物情形，证明能量收敛到1-可修正界面，并建立双迁移率系统，扩展Ilmanen框架至向量值系统。
AI 中文摘要
我们研究二维空间中具有有限个非退化势阱的固定光滑势的抛物型向量 Allen-Cahn 方程。我们的结果为 Bethuel 椭圆紧性定理提供了抛物型推广：在几乎所有正时刻，扩散能量、势能以及梯度张量收敛到集中在可数 1-可修正界面上的测度，并满足所有 Bethuel 型关系。在标量情形下，切向缺陷消失，并恢复出通常的平均曲率流结构。对于一般向量势，极限应力和耗散产生一个双迁移率系统：法向能量通量平衡加权曲率，而切向内部迁移率输运残余切向能量。因此，该结果将 Ilmanen 的 Allen-Cahn 到 Brakke 框架从标量系统推广到向量值系统，并在局部化能量不等式中附加了一个非负耗散缺陷。
英文摘要
We study the parabolic vectorial Allen--Cahn equation in two space dimensions for a fixed smooth potential with finitely many non-degenerate wells. Our result provides a parabolic extension of Bethuel's elliptic compactness theorem: at almost every positive time, the diffuse energy, potential energy, and gradient tensor converge to measures concentrated on a countably \(1\)-rectifiable interface and satisfy all the Bethuel-type relations. In the scalar case, the tangential defect vanishes and the usual mean-curvature-flow structure is recovered. For general vectorial potentials, the limiting stress and dissipation yield a two-mobility system: the normal energy flux balances weighted curvature, while a tangential internal mobility transports residual tangential energy. Thus the result extends the Allen--Cahn-to-Brakke framework of Ilmanen from scalar to vector-valued systems, with an additional non-negative dissipation defect in the localized energy inequality.
Comments64 pages
