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

Abstract

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: Le Floch B. and P. G. LeFloch, Arch. Rational Mech. Anal. 233 (2019), 45–86.