发表机构
School of Arts and Sciences, North China Institute of Aerospace Engineering（华北航天工程学院文理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种耦合Green势的调和映射热流，通过能量源重构共形因子实现重整化，证明全局解并分解极限能量为气泡树，防止有限时间爆破。
AI 中文摘要
我们为从闭黎曼曲面出发的映射引入了一个Green势热流。受Parker--Wolfson重整化~\ncite{ParkerWolfson1993,Parker1996}的启发，在每一时刻根据当前映射所确定的基于能量的源来重构共形因子。由此产生的抛物-椭圆系统将重整化纳入演化过程，同时保持Dirichlet能量泛函及其调和临界点不变。我们建立了$H^3$初值的局部适定性，以及基于能量集中的延拓准则。对于足够大的耦合，我们排除了有限时间气泡形成，并获得了全局解。在无穷时间，能量恒等式仅提供张力场的加权$L^2$控制。将集中图的规则区域局部化，在指定时间的子序列上得到调和轮廓。我们在流的包图中表示这些轮廓，将它们组织成有限气泡树，并将极限能量分解为调和分量能量和颈部能量。一个临界归一化的径向例子说明了改进后的流如何能防止有限时间爆破并产生具有无颈部性质的调和气泡。
英文摘要
We introduce a Green--potential heat flow for maps from a closed Riemannian surface. Inspired by Parker--Wolfson renormalization~\cite{ParkerWolfson1993,Parker1996}, the conformal factor is reconstructed at each time from an energy-based source determined by the current map. The resulting parabolic--elliptic system incorporates renormalization into the evolution while leaving the Dirichlet energy functional and its harmonic critical points unchanged. We establish local well-posedness for $H^3$ initial data and a continuation criterion in terms of energy concentration. For sufficiently large coupling, we exclude finite-time bubbling and obtain global solutions. At infinite time, the energy identity provides only weighted $L^2$ control of the tension field. Localizing to regular regions of concentration charts yields harmonic profiles along subsequences of prescribed times. We represent these profiles in the flow's packet charts, organize them into a finite bubble tree, and decompose the limiting energy into harmonic component energies and neck energies. A critically normalized radial example illustrates how the modified flow can prevent finite-time blow-up and produce harmonic bubbles with no-neck property.
