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Gerhard Rein

Publications and source records attributed to Gerhard Rein.

At least 55 records · Page 3Linked to original sources
arXiv2005

Nonlinear Stability of Newtonian Galaxies and Stars from a Mathematical Perspective

The stability of equilibrium configurations of galaxies or stars are time honored problems in astrophysics. We present mathematical results on these problems which have in recent years been obtained by Yan Guo and the author in the context of the Vlasov-Poisson and the Euler-Poisson model. Based on a careful analysis of the minimization properties of conserved quantities--the total energy and so-called Casimir functionals--non-linear stability results are obtained for a wide class of equilibria.

astro-ph↗
arXiv2004

Global weak solutions to the Nordström-Vlasov system

The Nordström-Vlasov system is a Lorentz invariant model for a self-gravitating collisionless gas. We establish suitable a-priori-bounds on the solutions of this system, which together with energy estimates and the smoothing effect of ``momentum averaging'' yield the existence of global weak solutions to the corresponding initial value problem. In the process we improve the continuation criterion for classical solutions which was derived recently. The weak solutions are shown to preserve mass.

math-ph↗
arXiv2004

Global weak solutions to the relativistic Vlasov-Maxwell system revisited

In 1989, R. DiPerna and P.-L. Lions established the existence of global weak solutions to the Vlasov-Maxwell system. In the present notes we give a somewhat simplified proof of this result for the relativistic version of this system, the main purpose being to make this important result of kinetic theory more easily accessible to newcomers in the field. We show that the weak solutions preserve the total charge.

math-ph↗
arXiv2004

On Classical Solutions of the Relativistic Vlasov-Klein-Gordon System

We consider a collisionless ensemble of classical particles coupled to a Klein-Gordon field. For the resulting nonlinear system of partial differential equations, the relativistic Vlasov-Klein-Gordon system, we prove local-in-time existence of classical solutions and a continuation criterion which says that a solution can blow up only if the particle momenta become large. We also show that classical solutions are global in time in the one-dimensional case.

math.AP↗
arXiv2003

Global classical solutions to the spherically symmetric Nordström-Vlasov system

Classical solutions of the spherically symmetric Nordström-Vlasov system are shown to exist globally in time. The main motivation for investigating the mathematical properties of the Nordström-Vlasov system is its relation to the Einstein-Vlasov system. The former is not a physically correct model, but it is expected to capture some of the typical features of the latter, which constitutes a physically satisfactory, relativistic model but is mathematically much more complex. We show that classical solutions of the spherically symmetric Nordström-Vlasov system exist globally in time for compactly supported initial data under the additional condition that there is a lower bound on the modulus of the angular momentum of the initial particle system. We emphasize that this is not a smallness condition and that our result holds for arbitrary large initial data satisfying this hypothesis.

gr-qc↗
arXiv2003

On classical solutions of the Nordström-Vlasov system

The Nordström-Vlasov system describes the dynamics of a self-gravitating ensemble of collisionless particles in the framework of the Nordström scalar theory of gravitation. We prove existence and uniqueness of classical solutions of the Cauchy problem in three dimensions and establish a condition which guarantees that the solution is global in time. Moreover, we show that if one changes the sign of the source term in the field equation, which heuristically corresponds to the case of a repulsive gravitational force, then solutions blow up in finite time for a large class of initial data. Finally, we prove global existence of classical solutions for the one dimensional version of the system with the correct sign in the field equation.

math-ph↗
arXiv2002

Non-linear stability of gaseous stars

We construct steady states of the Euler-Poisson system with a barotropic equation of state as minimizers of a suitably defined energy functional. Their minimizing property implies the non-linear stability of such states against general, i.e., not necessarily spherically symmetric perturbations. The mathematical approach is based on previous stability results for the Vlasov-Poisson system by Y. Guo and the author, exploiting the energy-Casimir technique. The analysis is conditional in the sense that it assumes the existence of solutions to the initial value problem for the Euler-Poisson system which preserve mass and energy. The relation between the Euler-Poisson and the Vlasov-Poisson system in this context is also explored.

math-ph↗
arXiv2002

Convergence of a particle-in-cell scheme for the spherically symmetric Vlasov-Einstein system

We consider spherically symmetric, asymptotically flat space-times with a collisionless gas as matter model. Many properties of the resulting Vlasov-Einstein system are not yet accessible by purely analytical means. We present a discretized version of this system which is suitable for numerical implementation and is based on the particle-in-cell technique. Convergence of the resulting approximate solutions to the exact solution is proven and error bounds are provided.

math.NA↗
arXiv2002

Global Weak Solutions of the Relativistic Vlasov-Klein-Gordon System

We consider an ensemble of classical particles coupled to a Klein-Gordon field. For the resulting nonlinear system of partial differential equations, which we call the relativistic Vlasov-Klein-Gordon system, we prove the existence of global weak solutions for initial data satisfying a size restriction. The latter becomes necessary since the energy of the system is indefinite, and only for restricted data a-priori bounds on the solutions can be derived from conservation of energy.

math.AP↗
arXiv2002

Isotropic steady states in galactic dynamics revised

The present paper completes our earlier results on nonlinear stability of stationary solutions of the Vlasov-Poisson system in the stellar dynamics case. By minimizing the energy under a mass-Casimir constraint we construct a large class of isotropic, spherically symmetric steady states and prove their nonlinear stability against general, i. e., not necessarily symmetric perturbations. The class is optimal in a certain sense, in particular, it includes all polytropes of finite mass with decreasing dependence on the particle energy.

math-ph↗
arXiv2002

Stable Models of Elliptical Galaxies

We construct stable axially symmetric models of elliptical galaxies. The particle density on phase space for these models depends monotonically on the particle energy and on the third component of the angular momentum. They are obtained as minimizers of suitably defined energy-Casimir functionals, and this implies their nonlinear stability. Since our analysis proceeds from a rigorous but purely mathematical point of view it should be interesting to determine if any of our models match observational data in astrophysics. The main purpose of these notes is to initiate some exchange of information between the astrophysics and the mathematics communities.

astro-ph↗
arXiv2001

On the Einstein-Vlasov system with hyperbolic symmetry

It is shown that a spacetime with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry can be globally covered by compact hypersurfaces on which the mean curvature is constant and by compact hypersurfaces on which the area radius is constant. Results for the related cases of spherical and plane symmetry are reviewed and extended. The prospects of using the global time coordinates obtained in this way to investigate the global geometry of the spacetimes concerned are discussed.

gr-qc↗
arXiv2001

Reduction and a concentration-compactness principle for energy-Casimir functionals

Energy-Casimir functionals are a useful tool for the construction of steady states and the analysis of their nonlinear stability properties for a variety of conservative systems in mathematical physics. Recently, Y. Guo and the author employed them to construct stable steady states for the Vlasov-Poisson system in stellar dynamics, where the energy-Casimir functionals act on number density functions on phase space. In the present paper we construct natural, reduced functionals which act on mass densities on space and study compactness properties and the existence of minimizers in this context. This puts the techniques developed by Y.Guo and the author into a more general framework. We recover the concentration-compactness principle due to P.L.Lions in a more specific setting and connect our stability analysis with the one of G.Wolansky.

math-ph↗
arXiv2001

Existence and nonlinear stability of steady states of the Schrödinger-Poisson system

We consider the Schrödinger-Poisson system in the attractive (plasma physics) Coulomb case. Given a steady state from a certain class we prove its nonlinear stability, using an appropriately defined energy-Casimir functional as Lyapunov function. To obtain such steady states we start with a given Casimir functional and construct a new functional which is in some sense dual to the corresponding energy-Casimir functional. This dual functional has a unique maximizer which is a steady state of the Schrödinger-Poisson system and lies in the stability class. The steady states are parametrized by the equation of state, giving the occupation probabilities of the quantum states as a strictly decreasing function of their energy levels.

math-ph↗
arXiv1999

Time-dependent rescalings and Lyapunov functionals for the Vlasov-Poisson and Euler-Poisson systems, and for related models of kinetic equations, fluid dynamics and quantum physics

We investigate rescaling transformations for the Vlasov-Poisson and Euler-Poisson systems and derive in the plasma physics case Lyapunov functionals which can be used to analyze dispersion effects. The method is also used for studying the long time behaviour of the solutions and can be applied to other models in kinetic theory (2-dimensional symmetric Vlasov-Poisson system with an external magnetic field), in fluid dynamics (Euler system for gases) and in quantum physics (Schrödinger-Poisson system, nonlinear Schrödinger equation).

math.AP↗
arXiv1999

Stability of spherically symmetric steady states in galactic dynamics against general perturbations

Certain steady states of the Vlasov-Poisson system can be characterized as minimizers of an energy-Casimir functional, and this fact implies a nonlinear stability property of such steady states. In previous investigations by Y. Guo and the author stability was obtained only with respect to spherically symmetric perturbations. In the present investigation we show how to remove this unphysical restriction.

math-ph↗
arXiv1999

Existence and stability of Camm type steady states in galactic dynamics

We prove the existence and nonlinear stability of Camm type steady states of the Vlasov-Poisson system in the gravitational case. The paper demonstrates the effectiveness of an approach to the existence and stability problem for steady states, which was used in previous work by the authors: The steady states are obtained as minimizers of an energy-Casimir functional, and from this fact their dynamical stability is deduced.

math-ph↗