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Cosmological solutions of the Vlasov-Einstein system with spherical, plane, and hyperbolic symmetry

Cosmological solutions of the Vlasov-Einstein system with spherical, plane, and hyperbolic symmetry,10.1017/S0305004100074569,Mathematical Proceedings

Cosmological solutions of the Vlasov-Einstein system with spherical, plane, and hyperbolic symmetry   (Citations: 33)
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The Vlasov-Einstein system describes a self-gravitating, collisionless gas within the framework of general relativity. The author investigates the initial value problem in a cosmological setting with spherical, plane, or hyperbolic symmetry and proves that for small initial data solutions exist up to a spacetime singularity which is a curvature and a crushing singularity. An important tool in the analysis is a local existence result with a continuation criterion saying that solutions can be extended as long as the momenta in the support of the phase-space distribution of the matter remain bounded.
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    • ...contexts: Gowdy spacetimes [17], plane symmetric spacetimes with a massless scalar field [18], polarized and half-polarized U(1) symmetric vacuum spacetimes [19], spacetimes with collisionless matter and spherical, plane or hyperbolic symmetry [20], and a particular subset of general Gowdy spacetimes [21]...

    Thibault Damouret al. Describing general cosmological singularities in Iwasawa variables

    • ...The method, which has been used for instance in [8] for the Einstein-Vlasov system in plane symmetry, consists on constructing an iteration and proving its convergence...

    S. B. TCHAPNDAet al. THE PLANE SYMMETRIC EINSTEIN-DUST SYSTEM WITH POSITIVE COSMOLOGICAL CO...

    • ...The study of the surface-symmetric cosmological solutions of the Einstein-Vlasov system considered here was initiated in [32, 29]...
    • ...Past inextendibility for small data under various assumptions on k and � has been shown in [29, 37]...

    Mihalis Dafermoset al. Strong cosmic censorship for surface-symmetric cosmological spacetimes...

    • ...To this end we maintain the notation in [7, 12, 13], and follow their work wherever possible...
    • ...The first step consists on generalizing the local existence result in [7, theorem 3.1] to the case of charged particles under study:...
    • ...The regularity properties required for a solution are as in [7]...
    • ...Instead of considering the subsystem (1.12)-(1.15), (1.19), an idea used in [7], that we follow here, is to consider an auxiliary system consisting of...
    • ...As in [7], the solution of the auxiliary system above is used to construct a sequence of iterative solutions...

    Sophonie Blaise Tchapnda. On Surface-Symmetric Spacetimes with Collisionless and Charged Matter

    • ...In the case where the matter is given by the Vlasov equation alone certain results on the initial singularity have been obtained by Rein [7]...
    • ...Using results of [5] it was shown that the initial singularity is a curvature singularity (the Kretschmann scalar blows up uniformly there) and a crushing singularity (the mean curvature of the hypersurfaces of constant t blows up uniformly as t → 0). For matter described by the Vlasov equation alone the results already mentioned can be combined with theorems in [7] to show that t = 0 is a crushing singularity where the Kretschmann scalar ...
    • ...coordinates. The functions � and µ are periodic in r with period 1. It has been shown in [7] that due to the symmetry, f can be written as a function of...
    • ...It seems that if the estimates could be improved slightly they would allow a bootstrap argument on the bound for Q similar to that used in [7]...
    • ...Proof We can use the following expression for the Kretschman scalar from [7]...
    • ...Proof We use the same argument as in [7] and obtain the following : K(t,r) = −(ú � + 2t)e −µ ; ú � = e2µ � 4�t� − k+e...
    • ...are the generalized Kasner exponents and a(r) a continuous function of r. Proof We have as in [7]...
    • ...For the Einstein-Vlasov system this has been done in [7] for small data...
    • ...In Proposition 3.2 an analogue of some parts of the proof of the small data theorem in [7] is obtained but this is not sufficient in order to determine the asymptotic behaviour...

    David Tegankonget al. On the nature of initial singularities for solutions of the Einstein-V...

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