Title: Global solutions to cubic Dirac and Dirac-Klein-Gordon systems on spacetimes close to the Minkowski space Show Chinese title

Title: 三维时空上接近闵可夫斯基空间的三次狄拉克和狄拉克-克莱因-戈登系统的整体解

Abstract: We establish global existence and derive sharp pointwise decay estimates of solutions to cubic Dirac and Dirac-Klein-Gordon systems on a curved background, close to the Minkowski spacetime. By squaring the Dirac operator, we reduce the analysis to a nonlinear wave-type equation involving spinorial connections, and apply energy estimates based on vector field methods and the hyperboloidal foliation framework, introduced by LeFloch-Ma. A key difficulty arises from the commutator structure of the Dirac operator, which exhibits significantly different behaviour from that of scalar field equations and requires refined control throughout the analysis, particularly due to the spacetime-dependent gamma matrices, which reduce to constant matrices in the flat Minkowski spacetime. Show Chinese abstract

Abstract: 我们建立了在接近闵可夫斯基时空的弯曲背景上三次狄拉克和狄拉克-克莱因-戈登系统的解的整体存在性，并推导出解的精确点态衰减估计。 通过对方程中的狄拉克算子进行平方，我们将分析简化为涉及旋量连接的非线性波动方程，并应用基于向量场方法和由LeFloch-Ma引入的双曲层化框架的能量估计。 一个关键难点来自于狄拉克算子的对易子结构，其表现出与标量场方程显著不同的行为，需要在整个分析过程中进行精细控制，特别是由于时空依赖的伽马矩阵，它们在平坦的闵可夫斯基时空中退化为常数矩阵。