Compensated compactness and corrector stress tensor for the Einstein equations in T-2 symmetry

摘要

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 selfgravitating 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.