Asymptotic Stability of Minkowski Space-Time with non-compactly supported massless Vlasov matter

We prove the global asymptotic stability of the Minkowski space for the massless Einstein-Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in $v$ is optimal. The present proof is based on vector field and weighted vector field techniques for Vlasov fields, as developed in previous work of Fajman, Joudioux, and Smulevici, and heavily relies on several structural properties of the massless Vlasov equation, similar to the null and weak null conditions. To deal with the weak decay rate of the metric, we propagate well-chosen hierarchized weighted energy norms which reflect the strong decay properties satisfied by the particle density far from the light cone. A particular analytical difficulty arises at top order, when we do not have access to improved pointwise decay estimates for certain metric components. This difficulty is resolved using a novel hierarchy in the massless Einstein-Vlasov system, which exploits the propagation of different growth rates for the energy norms of different metric components

POSSIBLE SOURCES OF ACOUSTIC EMISSION DURING STATIC BENDING TEST OF WOOD SPECIMENS

The paper is dedicated to identifi cation of sources of acoustic emission generated during static bending test of wood specimen. Information on wood structure, wood failure behavior and computer-generated fi nite element method (FEM) simulation of static bending test were used to localize and estimate power of individual acoustic emission sources. Signifi cant acoustic emission sources are expected in the specimen in two areas: under the upper central support (during the entire bending test run) and in the tension portion of the specimen at the centre of lower baseline (at the fracture/destruction time). Strong acoustic emission sources were registered in tension portion of the specimen in direct connection with fi nal fracture of specimen

Nonlinear stability of the Milne model with matter

We show that any 3+1-dimensional Milne model is future nonlinearly, asymptotically stable in the set of solutions to the Einstein-Vlasov system. For the analysis of the Einstein equations we use the constant-mean-curvature-spatial-harmonic gauge. For the distribution function the proof makes use of geometric L^2-estimates based on the Sasaki-metric. The resulting estimates on the energy momentum tensor are then upgraded by employing the natural continuity equation for the energy density

Averaging with a time-dependent perturbation parameter

Motivated by recent problems in mathematical cosmology, in which temporal averaging methods are applied in order to analyse the future asymptotics of models which exhibit oscillatory behaviour, we provide a theorem concerning the large-time behaviour for solutions of a general class of systems. We thus propose our result to be applicable to a wide range of problems in spatially homogenous cosmology with oscillatory behaviour. Mathematically the theorem builds up on the standard theory of averaging in non-linear dynamical systems.11Nsciescopu

The Stability of the Minkowski space for the Einstein-Vlasov system

We prove the global stability of the Minkowski space viewed as the trivial solution of the Einstein-Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques developed in [FJS15; FJS17]. In particular, the initial support in the velocity variable does not need to be compact. To control the effect of the large velocities, we identify and exploit several structural properties of the Vlasov equation to prove that the worst non-linear terms in the Vlasov equation either enjoy a form of the null condition or can be controlled using the wave coordinate gauge. The basic propagation estimates for the Vlasov field are then obtained using only weak interior decay for the metric components. Since some of the error terms are not time-integrable, several hierarchies in the commuted equations are exploited to close the top order estimates. For the Einstein equations, we use wave coordinates and the main new difficulty arises from the commutation of the energy-momentum tensor, which needs to be rewritten using the modified vector fields

Models for Self-Gravitating Photon Shells and Geons

We prove existence of spherically symmetric, static, self-gravitating photon shells as solutions to the massless Einstein-Vlasov system. The solutions are highly relativistic in the sense that the ratio 2m(r) / r is close to 8 / 9, where m(r) is the Hawking mass and r is the area radius. In 1955 Wheeler constructed, by numerical means, so-called idealized spherically symmetric geons, i.e., solutions of the Einstein-Maxwell equations for which the energy momentum tensor is spherically symmetric on a time average. The structure of these solutions is such that the electromagnetic field is confined to a thin shell for which the ratio 2m / r is close to 8 / 9, i.e., the solutions are highly relativistic photon shells. The solutions presented in this work provide an alternative model for photon shells or idealized spherically symmetric geons