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Weakly regular Einstein-Euler spacetimes with Gowdy symmetry: the global areal foliation. (English) Zbl 1266.83027

Summary: We consider weakly regular Gowdy-symmetric spacetimes on \(T^3\) 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 P. G. LeFloch and A. D. Rendall [Arch. Ration. Mech. Anal. 201, No. 3, 841–870 (2011; Zbl 1256.35165)] 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.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C15 Exact solutions to problems in general relativity and gravitational theory
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
76E20 Stability and instability of geophysical and astrophysical flows
53Z05 Applications of differential geometry to physics
35Q75 PDEs in connection with relativity and gravitational theory

Citations:

Zbl 1256.35165
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References:

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