PORTUGALIAEMATHEMATICA
Vol. 62, No. 3, pp. 349-370 (2005)
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PORTUGALIAE MATHEMATICA Vol. 62, No. 3, pp. 349-370 (2005) |
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Shock waves and gravitational waves in matter spacetimes with Gowdy symmetryP.G. LeFloch and J.M. StewartLaboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, UMR 7598,Université Pierre et Marie Curie, B.C. 187, 75252 Paris Cedex 05 -- FRANCE E-mail: lefloch@ann.jussieu.fr Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Cambridge CB3 0WA -- UK E-mail: j.m.stewart@damtp.cam.ac.uk Abstract: In the context of classical general relativity, we consider the matter Einstein equations for perfect fluids on Gowdy spacetimes with plane-symmetry. Such spacetimes admit two commuting Killing vector fields and contain gravitational waves; the fluid variable may exhibit shock waves. We establish the existence of a bounded variation solution to the Cauchy problem, which is defined globally until either a true singularity occurs in the geometry (e.g. the vanishing of the area of the $2$-dimensional space-like orbits of the symmetry group) or a blow-up of the energy density takes place. Keywords: Einstein--Euler equations; Gowdy spacetimes; singularities; hyperbolic equations; conservation laws; shock waves; compensated compactness. Classification (MSC2000): 83C05, 83C35, 76L05, 35L65. Full text of the article:
Electronic version published on: 7 Mar 2008.
© 2005 Sociedade Portuguesa de Matemática
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