2021/11/08 by Chao Liu, Jinhua Wang, Liu, Chao +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph) #gr-qc #math-ph #math.AP #math.DG #math.MP
paper · pdf · doi:10.48550/arxiv.2111.04540
27 pages, comments are all welcome
openalex publication_date 2021/11/08 · arxiv created 2021/11/13 · arxiv updated 2021/11/16 · openalex created_date 2021/11/22 · openalex updated_date 2026/07/28
Motivated by the future stability problem of solutions of the Einstein--Yang--Mills (EYM) system with arbitrary dimension, we aim to (1) construct a tensorial symmetric hyperbolic formulation for the (n+1)-dimensional EYM system in the temporal gauge; (2) establish the local well-posedness for the Cauchy problem of EYM equations in the temporal gauge using this tensorial symmetric hyperbolic system. By introducing certain auxiliary variables, we extend essentially the (n+1)-dimensional Yang--Mills system to a tensorial symmetric hyperbolic system. On the contrary, this symmetric hyperbolic system with data satisfying some constraints (extending the Yang--Mills constraints) reduces to the Yang--Mills system. Consequently, an equivalence between the EYM and the tensorial symmetric hyperbolic system with a class of specific data set is concluded. Furthermore, a general symmetric hyperbolic system over tensor bundles is studied, with which, we conclude the local well-posedness of the EYM system. It turns out the idea of symmetric hyperbolic formulation of the Yang--Mills field is very useful in prompting a tensorial Fuchsian formalism and proving the future stability for the EYM system with arbitrary dimension (i.e., this new symmetric hyperbolic formulation of EYM manifests well behaved lower order terms for long time evolution), see our companion article [21] with Todd A. Oliynyk.