# Compensated compactness and corrector stress tensor for the Einstein equations in $T_{2}$ symmetry

### Bruno Le Floch

Ecole Normale Supérieure, Paris, France### Philippe G. LeFloch

Sorbonne Université, Paris, France

## Abstract

We consider the Einstein equations in $T_{2}$ symmetry, either for vacuum spacetimes or coupled to the Euler equations for a compressible fluid, and we introduce the notion of $T_{2}$ areal flows on $T_{3}$ 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 $T_{2}$ areal flows is sequentially compact in a natural topology. In order to handle general initial data we propose a ‘‘relaxed’’ notion of $T_{2}$ 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.

## Cite this article

Bruno Le Floch, Philippe G. LeFloch, Compensated compactness and corrector stress tensor for the Einstein equations in $T_{2}$ symmetry. Port. Math. 77 (2020), no. 3/4, pp. 409–421

DOI 10.4171/PM/2057