Elliptic-hyperbolic systems and the Einstein equations
Lars Andersson, Vincent Moncrief
TL;DR
The paper develops a gauge-fixed formulation of the vacuum Einstein equations combining constant mean curvature with spatial harmonic coordinates (CMCSH), producing a coupled elliptic–hyperbolic system for (g,k) with elliptic defining equations for the lapse N and shift X. It establishes local strong well-posedness and a continuation principle for the Cauchy problem of this elliptic–hyperbolic system by proving robust energy estimates for the hyperbolic portion and elliptic estimates for the gauge sector, then extends these results to the nonlinear Einstein system. The analysis shows that, from initial data satisfying the Einstein constraints and CMCSH gauge, the evolution yields a vacuum spacetime and that the gauge/constraint quantities propagate consistently, thanks to a hyperbolic system for constraints with an energy bound. An isomorphism property for the elliptic part under suitable curvature assumptions guarantees well-posedness of the lapse/shift definitions, underpinning the robustness of the CMCSH formulation and its continuation behavior.
Abstract
The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasilinear elliptic--hyperbolic system of evolution equations. We prove that the Cauchy problem is locally strongly well--posed and that a continuation principle holds. For initial data satisfying the Einstein constraint and gauge conditions, the solutions to the elliptic-hyperbolic system defined by the gauge fixed Einstein evolution equations are shown to give vacuum spacetimes.