AI 中文总结
研究洛伦兹规范下陈 - 西蒙斯规范\(O(3)\)西格玛模型低正则性柯西问题，通过识别零结构和建立能量估计，在一维和二维分别为特定初始数据建立局部适定性，改进前人结果并接近尺度不变指数。
AI 中文摘要
本文研究\(\mathbb{R}^{1 + d}\)（\(d = 1,2\)）中洛伦兹规范下陈 - 西蒙斯规范\(O(3)\)西格玛模型的低正则性柯西问题。对于\(d = 1\)，在\(s_1>\frac{1}{2}\)时，为初始数据\((\boldsymbol{\phi}_0,\mathbf{A}_0)\in H^{s_1}(\mathbb{R})\times H^{s_1 - 1}(\mathbb{R})\)建立了局部适定性，比之前结果提高了四分之一个导数且接近尺度不变正则性。对于\(d = 2\)，在\(s_2>1\)时，为初始数据\((\boldsymbol{\phi}_0,\mathbf{A}_0)\in H^{s_2}(\mathbb{R}^2)\times H^{s_2-\frac{3}{4}}(\mathbb{R}^2)\)建立了局部适定性，改进了之前结果并使正则性阈值接近尺度不变指数。分析依赖两个主要因素，二维中识别导数非线性的完整零结构，一维中在特定函数空间建立直接能量估计。
英文摘要
In this paper, we study the low-regularity Cauchy problem for the Chern--Simons gauged $O(3)$ sigma model in $\mathbb{R}^{1+d}$ ($d=1,2$) under the Lorenz gauge. For $d=1$, we establish local well-posedness for initial data $(\boldsymbolϕ_0,\mathbf{A}_0)\in H^{s_1}(\mathbb{R})\times H^{s_1-1}(\mathbb{R})$ with $s_1>\frac12$. This improves the previous result of Jin and Huh \cite{HJ} by one quarter of a derivative and is almost optimal in view of the scaling-invariant regularities $\dot H^{1/2}(\mathbb{R})$ for the matter field and $\dot H^{-1/2}(\mathbb{R})$ for the gauge field. For $d=2$, we establish local well-posedness for initial data $(\boldsymbolϕ_0,\mathbf{A}_0)\in H^{s_2}(\mathbb{R}^2)\times H^{s_2-\frac34}(\mathbb{R}^2)$ with $s_2>1$. This improves the previous result of Jin and Zhang \cite{JZ} by one quarter of a derivative and brings the regularity threshold close to the scaling-invariant exponents $\dot H^{1}(\mathbb{R}^2)$ and $\dot H^{0}(\mathbb{R}^2)$ for the matter and gauge fields, respectively. The analysis relies on two main ingredients. In two space dimensions, we identify the complete null structure of the derivative nonlinearities, allowing the entire system to be treated within a unified null-form framework. In one space dimension, we establish a direct energy estimate in the function space introduced by Keel and Tao, avoiding the finite-propagation reduction to a small-data problem and enabling the low-regularity iteration for general initial data.
Comments29 pages
