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

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 and 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: Le Floch B. and LeFloch P.G., Arch. Rational Mech. Anal. 233 (2019), 45-86. In addition, we determine the global geometry of the corresponding future Cauchy developments and we prove that the area of the $\mathbb{T}^2$ orbits generically approaches infinity in the future-expanding regime. In the future-contracting regime, the volume of the $\mathbb{T}^3$ spacelike slices approaches zero and, for generic initial data, the area of the orbits of symmetry approaches zero in Gowdy symmetric matter spacetimes and in $\mathbb{T}^2$ vacuum spacetimes