Finite-energy spacetimes with torus symmetry. Cauchy stability of Einstein-Euler and Einstein-Navier-Stokes areal flows

Abstract

We establish finite-energy Cauchy stability for vacuum and matter spacetimes with $T^2$ symmetry on $T^3$, under general constitutive equations satisfying hyperbolicity and mild asymptotic conditions at vacuum and large mass-energy density. First, using the area of the symmetry orbits as a time function, we introduce the JKL formulation of Einstein areal flows}, a first-order evolution-constraint system coupling a wave-map structure for the geometry to hyperbolic matter balance laws and weighted transport equations for the twists and the momentum tangent to the symmetry orbits. Second, we also define two classes of hyperbolic Einstein-Navier-Stokes models, constructed from (as we call them) a Navier-Stokes potential and a relaxation rate map. The proposed particle-production model has divergence-free matter stress and non-negative particle-number production, while the proposed dissipated-energy model conserves particle number and dissipates fluid mass-energy, while an auxiliary stress restores the total stress-energy conservation required by the contracted Bianchi identity. For regular finite-energy (Einstein, Euler, Navier-Stokes) flows, we establish maximum and invariant-domain principles, geometric and energy monotonicity formulas, spacelike and timelike energy estimates, and total-variation estimates. Third, together, these estimates yield finite-energy Cauchy stability in the future expanding regime and, in the contracting regime, until the spacelike volume collapses. They are independent of the viscosity (or relaxation) mechanisms and cover vacuum, unbounded mass-energy density, and arbitrary finite rapidity. They also extend to weakly regular Einstein-Euler areal flows satisfying a reference mathematical entropy inequality.