Weakly Regular Einstein–Euler Spacetimes with Gowdy Symmetry: The Global Areal Foliation

We consider weakly regular Gowdy-symmetric spacetimes on T3 satisfying the Einstein–Euler equations of general relativity, and we solve the initial value problem when the initial data set has bounded variation, only so that the corresponding spacetime may contain impulsive gravitational waves as well as shock waves. By analyzing both future expanding and future contracting spacetimes, we establish the existence of a global foliation by spacelike hypersurfaces so that the time function coincides with the area of the surfaces of symmetry and asymptotically approaches infinity in the expanding case and zero in the contracting case. More precisely, the latter property in the contracting case holds provided the mass density does not exceed a certain threshold, which is a natural assumption since certain exceptional data with sufficiently large mass density are known to give rise to a Cauchy horizon, on which the area function attains a positive value. An earlier result by LeFloch and Rendall assumed a different class of weak regularity and did not determine the range of the area function in the contracting case. Our method of proof is based on a version of the random choice scheme adapted to the Einstein equations for the symmetry and regularity class under consideration. We also analyze the Einstein constraint equations under weak regularity.