A new symmetric hyperbolic formulation and the local Cauchy problem for the Einstein–Yang–Mills system in the temporal gauge

Source abstract

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 wave gauge (for the metric) and temporal gauge (for the Yang–Mills field); (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. 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 [C. Liu, T. A. Oliynyk, and J. Wang, J. Eur. Math. Soc. 28(10), 4559–4637 (2026)].