arXiv subjects

Gerhard Rein

Publications and source records attributed to Gerhard Rein.

65 records · Page 4Linked to original sources

arXiv1998

Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics

Equilibrium states in galactic dynamics can be described as stationary solutions of the Vlasov-Poisson system, which is the non-relativistic case, or of the Vlasov-Einstein system, which is the relativistic case. To obtain spherically symmetric stationary solutions the distribution function of the particles (stars) on phase space is taken to be a function of the particle energy and angular momentum. We give a new condition on this function which guarantees that the resulting steady state has finite mass and compact support both for the non-relativistic and the relativistic case. The condition is local in the sense that only the asymptotic behaviour for energy values close to the maximal energy value in the particle distribution needs to be prescribed.

gr-qc↗

arXiv1998

Static shells for the Vlasov-Poisson and Vlasov-Einstein systems

We prove the existence of static, spherically symmetric solutions of the stellar dynamic Vlasov-Poisson and Vlasov-Einstein systems, which have the property that their spatial support is a finite, spherically symmetric shell with a vacuum region at the center.

math-ph↗

arXiv1998

Flat steady states in stellar dynamics - existence and stability

We consider a special case of the three dimensional Vlasov-Poisson system where the particles are restricted to a plane, a situation that is used in astrophysics to model extremely flattened galaxies. We prove the existence of steady states of this system. They are obtained as minimizers of an energy-Casimir functional from which fact a certain dynamical stability property is deduced. From a mathematics point of view these steady states provide examples of partially singular solutions of the three dimensional Vlasov-Poisson system.

math-ph↗

arXiv1998

Stationary and static stellar dynamic models with axial symmetry

We study the existence of stationary solutions of the Vlasov-Poisson system with finite radius and finite mass in the stellar dynamics case. So far, the existence of such solutions is known only under the assumption of spherical symmetry. Using the implicit function theorem we show that certain stationary, spherically symmetric solutions can be embedded in one parameter families of stationary, axially symmetric solutions with finite radius and finite mass. In general, these new steady states have non-vanishing average velocity field, but they can also be constructed such that their velocity field does vanish, in which case they are called static.

math-ph↗

arXiv1998

Stable steady states in stellar dynamics

We prove the existence and nonlinear stability of steady states of the Vlasov-Poisson system in the stellar dynamics case. The steady states are obtained as minimizers of an energy-Casimir functional from which fact their dynamical stability is deduced. The analysis applies to some of the well-known polytropic steady states, but it also considerably extends the class of known steady states.

math-ph↗

arXiv1998

Critical collapse of collisionless matter - a numerical investigation

In recent years the threshold of black hole formation in spherically symmetric gravitational collapse has been studied for a variety of matter models. In this paper the corresponding issue is investigated for a matter model significantly different from those considered so far in this context. We study the transition from dispersion to black hole formation in the collapse of collisionless matter when the initial data is scaled. This is done by means of a numerical code similar to those commonly used in plasma physics. The result is that for the initial data for which the solutions were computed, most of the matter falls into the black hole whenever a black hole is formed. This results in a discontinuity in the mass of the black hole at the onset of black hole formation.

gr-qc↗

arXiv1996

Nonlinear stability of homogeneous models in Newtonian cosmology

We consider the Vlasov-Poisson system in a cosmological setting and prove nonlinear stability of homogeneous solutions against small, spatially periodic perturbations in the sup-norm of the spatial mass density. This result is connected with the question of how large scale structures such as galaxies have evolved out of the homogeneous state of the early universe.

gr-qc↗

arXiv1994

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

The Vlasov-Einstein system describes a self-gravitating, collisionless gas within the framework of general relativity. We investigate the initial value problem in a cosmological setting with spherical, plane, or hyperbolic symmetry and prove 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.

gr-qc↗

arXiv1993

Global existence of classical solutions to the Vlasov-Poisson system in a three dimensional, cosmological setting

The initial value problem for the Vlasov-Poisson system is by now well understood in the case of an isolated system where, by definition, the distribution function of the particles as well as the gravitational potential vanish at spatial infinity. Here we start with homogeneous solutions, which have a spatially constant, non-zero mass density and which describe the mass distribution in a Newtonian model of the universe. These homogeneous states can be constructed explicitly, and we consider deviations from such homogeneous states, which then satisfy a modified version of the Vlasov-Poisson system. We prove global existence and uniqueness of classical solutions to the corresponding initial value problem for initial data which represent spatially periodic deviations from homogeneous states.

gr-qc↗

arXiv1993

A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system

In a previous paper two of the authors (G. R. and A. D. R.) showed that there exist global, classical solutions of the spherically symmetric Vlasov-Einstein system for small initial data. The present paper continues this investigation and allows also large initial data. It is shown that if a solution of the spherically symmetric Vlasov-Einstein system develops a singularity at all then the first singularity has to appear at the center of symmetry. The result adds weight to the conjecture that cosmic censorship holds if one replaces dust as matter model for which naked singularities do form by a collisionless gas described by the Vlasov equation. The main tool is an estimate which shows that a solution is global if all the matter remains away from the center of symmetry.

gr-qc↗

arXiv1993

Static solutions of the spherically symmetric Vlasov-Einstein system

The Vlasov-Einstein system describes the evolution of an ensemble of particles (such as stars in a galaxy, galaxies in a galaxy cluster etc.) interacting only by the gravitational field which they create collectively and which obeys Einstein's field equations. The matter distribution is described by the Vlasov or Liouville equation for a collisionless gas. Recent investigations seem to indicate that such a matter model is particularly suited in a general relativistic setting and may avoid the formation of naked singularities, as opposed to other matter models. In the present note we consider the Vlasov-Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular center and may have isotropic or anisotropic pressure, and solutions, which have a Schwarzschild-singularity at the center. The paper extends previous work, where smooth, globally defined solutions with regular center and isotropic pressure were considered.

gr-qc↗