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

Abstract

We consider the Einstein equations in T2 symmetry, either for vacuum spacetimes or coupled to the Euler equations for a compressible fluid, and we introduce the notion of T2 areal flows on T3 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 T2 areal flows is sequentially compact in a natural topology. In order to handle general initial data we propose a relaxed notion of T2 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 T2 orbits generically approaches infinity in the future-expanding regime. In the future-contracting regime, the volume of the T3 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 T2 vacuum spacetimes.