Solutions globales des équations d’Einstein-Maxwell

Loizelet, Julien

doi:10.5802/afst.1212

Cited by 20 publications

(16 citation statements)

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“…New mathematical difficulties are present since the governing equations for the matter are now transport equations, though in the case considered here, where 1 The analogue of the Christodoulou-Klainerman theorem has been shown by Zipser [2] to hold for electromagnetic matter described by the Maxwell equations. See also Loizelet [24]. Speck [34] shows a similar result for the Einstein equations coupled to a large family of nonlinear electromagnetic equations.…”

Section: Introductionmentioning

confidence: 64%

The Global Nonlinear Stability of Minkowski Space for the Massless Einstein–Vlasov System

Taylor

2017

Ann. PDE

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Minkowski space is shown to be globally stable as a solution to the EinsteinVlasov system in the case when all particles have zero mass. The proof proceeds by showing that the matter must be supported in the "wave zone", and then proving a small data semi-global existence result for the characteristic initial value problem for the massless Einstein-Vlasov system in this region. This relies on weighted estimates for the solution which, for the Vlasov part, are obtained by introducing the Sasaki metric on the mass shell and estimating Jacobi fields with respect to this metric by geometric quantities on the spacetime. The stability of Minkowski space result for the vacuum Einstein equations is then appealed to for the remaining regions.

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Section: Introductionmentioning

confidence: 64%

The Global Nonlinear Stability of Minkowski Space for the Massless Einstein–Vlasov System

Taylor

2017

Ann. PDE

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“…A second proof, which relied on wave coordinates and allowed for the presence of a (small) scalar field, was given by Lindblad-Rodnianski in [54,55]. Various extensions of these results can be found in [13,57,79]. The analysis in all of these proofs is based on the dispersive decay properties of solutions to wave-like equations on R 1+3 , together with careful studies of the special structure of the nonlinearities present in the gauges that were employed.…”

Section: Global Solutions Without Symmetry Assumptionsmentioning

confidence: 99%

Stable Big Bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems

Rodnianski

Speck

2018

Sel. Math. New Ser.

121

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We prove a stable singularity formation result, without symmetry assumptions, for solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Our results apply to small perturbations of the spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) solution with topology (0, ∞)×T 3 . The FLRW solution models a spatially uniform scalar-field/stiff fluid evolving in a spacetime that expands towards the future and that has a "Big Bang" singularity at {0} × T 3 , where its curvature blows up. We place "initial" data on a Cauchy hypersurface Σ ′ 1 that are close, as measured by a Sobolev norm, to the FLRW data induced on {1} × T 3 . We then study the asymptotic behavior of the perturbed solution in the collapsing direction and prove that its basic qualitative and quantitative features closely resemble those of the FLRW solution.In particular, we construct constant mean curvature-transported spatial coordinates for the perturbed solution covering (t, x) ∈ (0, 1] × T 3 and show that it also has a Big Bang at {0} × T 3 , where its curvature blows up. The blow-up confirms Penrose's Strong Cosmic Censorship hypothesis for the "past-half" of near-FLRW solutions. Furthermore, we show that the Einstein equations are dominated by kinetic (time derivative) terms that induce approximately monotonic behavior near the Big Bang, and consequently, various time-rescaled components of the solution converge to functions of x as t ↓ 0.The most difficult aspect of the proof is showing that the solution exists for (t, x) ∈ (0, 1] × T 3 , and to this end, we derive a hierarchy of energy estimates that are allowed to mildly blow-up as t ↓ 0. To close these estimates, it is essential that we are able to rule out more singular energy blow-up, which is in turn tied to the most important ingredient in our analysis: an L 2 −type energy approximate monotonicity inequality that holds for near-FLRW solutions. In the companion article [73], we used the approximate monotonicity to prove a stability result for solutions to linearized versions of the equations. The present article shows that the linear stability result can be upgraded to control the nonlinear terms.1 Globally hyperbolic spacetimes are those containing a Cauchy hypersurface. 2 The assumptions needed in Hawking's theorem are i) the matter model verifies the strong energy condition, which is T µν − 1 2 (g −1 ) αβ T αβ T µν X µ X ν ≥ 0 whenever X is timelike; ii) k a a < −C everywhere on the initial Cauchy hypersurface, where k is the second fundamental form of Σ t (see (3.6)) and C > 0 is a constant. More precisely, Hawking's theorem shows that no past-directed timelike curve can have length greater than 3 C . In our main results, we derive a slightly sharper version of this estimate that is tailored to the solutions addressed in this article (see inequality (15.6)). 3 As we describe below, the relevant time coordinate t for the perturbed solution has level sets with mean curvature −(1 3)t −1 . 4 Roughly, this conjecture asserts that the maximal globally hyperbolic development...

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“…There has also been work on the problem of the stability of Minkowski space for the Einstein equations coupled to various other matter models [4], [46], [17], [20], [26], [27], [28], [41].…”

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confidence: 99%

Global Stability of Minkowski Space for the Einstein–Vlasov System in the Harmonic Gauge

Lindblad

Taylor

2019

Arch Rational Mech Anal

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Minkowski space is shown to be globally stable as a solution to the massive Einstein-Vlasov system. The proof is based on a harmonic gauge in which the equations reduce to a system of quasilinear wave equations for the metric, satisfying the weak null condition, coupled to a transport equation for the Vlasov particle distribution function. Central to the proof is a collection of vector fields used to control the particle distribution function, a function of both spacetime and momentum variables. The vector fields are derived using a general procedure, are adapted to the geometry of the solution and reduce to the generators of the symmetries of Minkowski space when restricted to acting on spacetime functions. Moreover, when specialising to the case of vacuum, the proof provides a simplification of previous stability works. ContentsDate: November 7, 2017. 1 6. The Einstein equations 51 6.1. Weak L ∞ decay estimates 52 6.2. The sharp decay estimates for the first order derivatives 55 6.3. The commutators and Lie derivatives 59 6.4. The sharp L ∞ decay estimates for higher order low derivatives 62 6.5. The energy estimate 63 6.6. Higher order L 2 Energy estimates 64 7. The continuity argument and the proof of Theorem 1.2 68 References 701 Note that the Einstein equations (1.1) are equivalent to Ric(g) αβ = T αβ − 1 2 g αβ trg T . 4where Γ α βγ are the Christoffel symbols of the metric g with respect to a given coordinate chart (t, x 1 , x 2 , x 3 ). Define X(s, t, x, p ) 0 = s and P (s, t, x, p ) 0 = 1. The notation X(s), P (s) will sometimes be used for X(s, t, x, p ), P (s, t, x, p ) when it is clear from the context which point (t, x, p ) is meant, and the notation X (s) = (s, X(s)) will sometimes be used.It follows that the Vlasov equation (1.3) can be rewritten as,for all s. The notation (y, q) will be used to denote points in the mass shell over the initial hypersurface, P| t=0 . In Theorem 1.2 it is assumed that f 0 has compact support; |y| ≤ K and |q| ≤ K ′ for (y, q) ∈ supp(f 0 ), for some constants K, K ′ . Under a relatively mild smallness assumptions on h = g − m, see Proposition 2.1, it follows that there exists c < 1, depending only on K ′ , such that solutions of the Vlasov equation satisfyfor |I| ≤ N . Here the constants D k depend only on C ′ N , K, K ′ and c, and ε 0 depends only on c.Remark 1.4. The proof of Theorem 1.3 still applies when a ≥ 1, though the theorem is only used in the proof of Theorem 1.2 for some fixed 1 2 < a <1. The case of a = 1 is omitted in order to avoid logarithmic factors. The proof of an appropriate result when a > 1 is much simpler, although, when Theorem 1.3 is used in the proof of Theorem 1.2, one could not hope for the assumptions (1.17) to hold with a > 1, see Section 1.5.2.Remark 1.5. In Section 5 a better L 2 estimate in terms of t behaviour, compared with the L 2 estimate of Theorem 1.3, is shown to hold for Z I T µν , which involves one extra derivative of Γ. See Proposition 5.8. It is important however to use the L 2 estimate which does not lose a derivative in t...

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Solutions globales des équations d’Einstein-Maxwell

Cited by 20 publications

References 13 publications

The Global Nonlinear Stability of Minkowski Space for the Massless Einstein–Vlasov System

The Global Nonlinear Stability of Minkowski Space for the Massless Einstein–Vlasov System

Stable Big Bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems

Global Stability of Minkowski Space for the Einstein–Vlasov System in the Harmonic Gauge

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