发表机构
University of California, Berkeley（加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文在DeTurck规范下证明了Manton的Chern-Simons-Schrödinger方程的全局适定性，并利用几何Littlewood-Paley理论获得自对偶涡旋的轨道稳定性。
AI 中文摘要
Chern-Simons-Schrödinger方程（在时间规范下）作为$\mathbb R^2$上阿贝尔希格斯能量的Hamilton流出现。Manton（arXiv:hep-th/9701027）将该方程引入为能量临界点（即涡旋）动力学的模型。他猜想，对于一定范围的耦合常数，Chern-Simons-Schrödinger流下的涡旋运动可以被Jaffe-Taubes（1980）和Samols（1992）构造的自对偶涡旋模空间上的一阶常微分方程有效捕获。作为严格证明Manton猜想的第一步，我们在DeTurck规范下在自然能量空间中提出Cauchy问题并证明全局适定性。我们还作为适定性理论和我们先前工作（arXiv:2603.14900）中稳定性结果的推论，获得了在自对偶耦合附近Chern-Simons-Schrödinger流下自对偶涡旋的轨道稳定性。我们分析的核心在于发展基于caloric规范中协变热方程的几何Littlewood-Paley理论，我们用它来对Chern-Simons-Schrödinger方程进行仿积风格的分解。
英文摘要
The Chern-Simons-Schrödinger equation (in the temporal gauge) arises as the Hamiltonian flow of the abelian Higgs energy on $\mathbb R^2$. Manton (arXiv:hep-th/9701027) introduced the equation as a model for the dynamics of the critical points of the energy, known as vortices. He conjectured that, for a certain range of coupling constants, the vortex motion under the Chern-Simons-Schrödinger flow can be effectively captured by a first-order ODE on the moduli space of self-dual vortices constructed by Jaffe-Taubes (1980) and Samols (1992). As a first step towards a rigorous proof of Manton's conjecture, we formulate the Cauchy problem in DeTurck gauge within the natural energy space and prove global well-posedness. We also obtain, as a corollary of the well-posedness theory and the stability results in our previous work (arXiv:2603.24900), orbital stability of the self-dual vortices under the Chern-Simons-Schrödinger flow near self-dual coupling. The heart of our analysis lies in developing a geometric Littlewood-Paley theory based on the covariant heat equation in caloric gauge, which we use to perform a paradifferential-style decomposition of the Chern-Simons-Schrödinger equation.
Comments111 pages, 2 figures
