Chern--Simons--Higgs方程的非简单爆破：先验分析与构造
Non-simple blow-up for the Chern--Simons--Higgs equation: A priori analysis and constructions
- Ulsan National Institute of Science and Technology (UNIST)（蔚山科学技术院）
- University of Florida（佛罗里达大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文首次证明Chern--Simons--Higgs方程非简单爆破解的存在性，涵盖有限高度情形，并揭示簇的等质量、正多边形排列及质量间隙准则。
AI中文摘要：
自对偶Chern--Simons--Higgs方程中的爆破具有丰富的多尺度结构，但详细理论主要集中于简单情形，即每个集中点由一个完整轮廓解析。相比之下，非简单爆破在解析上更为精细：多个相互作用的正则核必须在它们向单个涡旋坍缩的同时被同时解析，同时保持各自尺度上的分离。由于这一困难，此类解的存在性一直是一个长期未解决的开放问题。据我们所知，我们证明了该方程非简单爆破解的首个存在性结果，特别涵盖了先前未解决的有限高度Chern--Simons情形。我们还证明了非简单簇具有相等的极限核质量、共同的轮廓类型以及正多边形排列。我们确定了它们可能的总质量，并推导出一个质量间隙准则，该准则排除了某些涡旋构型下的此类爆破。我们对一般有限高度簇的定量轮廓和矩估计指导了近似解的设计，并阐明了有限高度构造背后的核相互作用。在单位圆盘上的一个独立构造沿具有变化Dirichlet迹的序列产生了两个平均场核。
英文摘要：
Blow-up in the self-dual Chern--Simons--Higgs equation has a rich multiscale structure, but the detailed theory has largely focused on the simple regime, where one entire profile resolves each concentration point. By contrast, non-simple blow-up is analytically more delicate: several interacting regular cores must be resolved simultaneously as they collapse toward a single vortex while remaining separated on their own scales. Because of this difficulty, the existence of such solutions has remained a long-standing open problem. To the best of our knowledge, we prove the first existence result for non-simple blow-up solutions of this equation, covering in particular the previously unresolved finite-height Chern--Simons regime. We also show that non-simple clusters have equal limiting core masses, a common profile type, and regular-polygon arrangements. We determine their possible total masses and derive a mass-gap criterion that rules out such blow-up for certain vortex configurations. Our quantitative profile and moment estimates for general finite-height clusters guide the design of the approximate solution and clarify the core interactions underlying the finite-height construction. A separate construction on the unit disk produces two mean-field cores along a sequence with varying Dirichlet traces.
