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

Bruno Le Floch (1) , Philippe G. LeFloch (2)

1 IPM - institut de Physique Théorique Philippe Meyer
2 LJLL - Laboratoire Jacques-Louis Lions

We consider the Einstein equations in T2\mathbb T^2T2 symmetry, either for vacuum spacetimes or coupled to the Euler equations for a compressible fluid, and we introduce the notion of T2\mathbb T^2T2 areal flows on T3\mathbb T^3T3 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 T2\mathbb T^2T2 areal flows is sequentially compact in a natural topology. In order to handle general initial data we propose a ‘‘relaxed’’ notion of T2\mathbb T^2T2 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.

Domaines: Relativité Générale et Cosmologie Quantique [gr-qc]

Bruno Le Floch, Philippe G. LeFloch. Compensated compactness and corrector stress tensor for the Einstein equations in $\mathbb T^2$ symmetry. Portug.Math., 2020, 77 (3), pp.409-421. ⟨10.4171/pm/2057⟩. ⟨hal-02447870⟩