Compensated compactness and corrector stress tensor for the Einstein equations in \(\mathbb{T}^2\) symmetry (Q2043310)

Summary: We consider the Einstein equations in \(\mathbb{T}^2\) symmetry, either for vacuum spacetimes or coupled to the Euler equations for a compressible fluid, and we introduce the notion of \(\mathbb{T}^2\) areal flows on \(\mathbb{T}^3\) with finite total energy. By uncovering a hidden structure of the Einstein equations, we establish a compensated compactness framework which allows us to solve the global evolution problem for vacuum spacetimes as well as for self-gravitating compressible fluids. We study the stability and instability of such flows and prove that, when the initial data are well-prepared, any family of \(\mathbb{T}^2\) areal flows is sequentially compact in a natural topology. In order to handle general initial data we propose a ``relaxed'' notion of \(\mathbb{T}^2\) areal flows endowed with a corrector stress tensor (as we call it) which is a bounded measure generated by geometric oscillations and concentrations propagating at the speed of light. This generalizes a result for vacuum spacetimes in [the authors, Arch. Ration. Mech. Anal. 233, No. 1, 45--86 (2019; Zbl 1416.83034)].