阿贝尔杨-米尔斯-希格斯模型中一度涡旋的渐近稳定性
Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model
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中文总结 AI 辅助
本文作为三篇系列论文的核心部分，证明了阿贝尔杨-米尔斯-希格斯模型中一度涡旋在加权索伯列夫空间小等变扰动下的渐近稳定性，通过多种分析方法克服了耦合动力学的长时间分析难点。
中文摘要 AI 辅助
我们证明了在自对偶耦合下，(1+2)维阿贝尔杨-米尔斯-希格斯模型中一度涡旋在加权索伯列夫空间中小等变扰动下的渐近稳定性。阿贝尔杨-米尔斯-希格斯模型是(1+2)维闵可夫斯基空间上的经典相对论场论，描述复值场与电磁势的耦合，且存在称为涡旋的拓扑孤子。本文是三篇论文系列的最终核心部分。在本文采用的正交规范条件下，涡旋的扰动由动力学变量的非线性克莱因-戈登方程组与电磁势时间分量的椭圆方程耦合所支配。线性化算子具有连续谱[1,∞)和一个重数为2的正间隙本征值（内部模式），其谱性质、相关的畸变傅里叶理论及线性衰减估计已在两篇配套论文中得到发展。主要难点在于辐射-内部模式耦合动力学的长时间分析：二维空间中克莱因-戈登辐射的色散衰减相对较弱，而内部模式仅在非线性辐射阻尼决定的长时间尺度上衰减；同时，辐射的克莱因-戈登方程包含非空间局域化的变系数二次相互作用，无法用微扰法处理，需借助范式分析。我们通过结合好-坏分解、平-锐分解以及相对于平克莱因-戈登流的时空共振分析来证明辐射的衰减，平分析与带势克莱因-戈登流之间的过渡通过由畸变傅里叶理论导出的ILED和转移估计实现。
英文摘要
We prove asymptotic stability of the degree-one vortex in the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at the self-dual coupling, for small equivariant perturbations in weighted Sobolev spaces. The abelian Yang-Mills-Higgs model is a classical relativistic field theory on $(1+2)$-dimensional Minkowski space, describing a complex-valued field coupled to an electromagnetic potential and admitting topological solitons known as vortices. This paper is the final and main part of a three-paper series. Under the orthogonal gauge condition used here, perturbations of the vortex are governed by a system of nonlinear Klein-Gordon equations for the dynamical variables, coupled to an elliptic equation for the temporal component of the electromagnetic potential. The linearized operator has continuous spectrum $[1,\infty)$ and a single positive gap eigenvalue (internal mode) of multiplicity two, whose spectral properties, associated distorted Fourier theory, and linear decay estimates are developed in the two companion papers. The main difficulty is the long-time analysis of the coupled radiation--internal-mode dynamics. In two space dimensions the dispersive decay of the Klein-Gordon radiation is relatively weak, while the internal mode decays only on the long time scale dictated by nonlinear radiation damping. At the same time, the Klein-Gordon equations for the radiation contain non-spatially localized variable coefficient quadratic interactions, which cannot be treated perturbatively and require a normal form analysis. We prove decay of the radiation by combining a good-bad decomposition, a flat-sharp decomposition, and a space-time resonance analysis carried out relative to the flat Klein-Gordon flow. The passage between the flat analysis and the Klein-Gordon flow with potential is achieved through ILED and transference estimates derived from the distorted Fourier theory.
