Gravitational Collapse and the Vlasov–Poisson System

A self-gravitating homogeneous ball of a fluid with pressure zero where the fluid particles are initially at rest collapses to a point in finite time. We prove that this gravitational collapse can be approximated arbitrarily closely by suitable solutions of the Vlasov–Poisson system which are known to exist globally in time.     

Stable Ground States for the Relativistic Gravitational Vlasov–Poisson System

We consider the three dimensional gravitational Vlasov–Poisson (GVP) system in both classical and relativistic cases. The classical problem is subcritical in the natural energy space and the stability of a large class of ground states has been derived by various authors. The relativistic problem is critical and displays finite time blow up solutions. Using standard concentration compactness techniques, we however show that the breaking of the scaling symmetry allows the existence of stable relativistic ground states. A new feature in our analysis which applies both to the classical and relativistic problem is that the orbital stability of the ground states does not rely as usual on an argument of uniqueness of suitable minimizers—which is mostly unknown—but on strong rigidity properties of the transport flow, and this extends the class of minimizers for which orbital stability is now proved.     

Moment Propagation for Weak Solutions to the Vlasov–Poisson System

We consider weak solutions to the Cauchy problem for the three dimensional Vlasov–Poisson system of equations. We obtain a propagation result for any velocity moment of order > 2 as well as a uniqueness statement in ?³. In the periodic case, we show that velocity moments of order > 14/3 are propagated.     

Stationary Spherically Symmetric Solutions of the Vlasov–Poisson System in the Three-Dimensional Case

Doklady Mathematics - We consider the three-dimensional stationary Vlasov–Poisson system of equations with respect to the distribution function of the gravitating matter $$f =...     

Stability of rarefaction waves for the non-cutoff Vlasov–Poisson–Boltzmann system with physical boundary

In this paper, we are concerned with the Vlasov–Poisson–Boltzmann (VPB) system in three-dimensional spatial space without angular cutoff in a rectangular duct with or without physical boundary conditions. Near a local Maxwellian with macroscopic quantities given by rarefaction wave solution of one-dimensional compressible Euler equations, we establish the time-asymptotic stability of planar rarefaction wave solutions for the Cauchy problem to VPB system with periodic or specular-reflection boundary condition. In particular, we successfully introduce physical boundaries, namely, specular-reflection boundary, to the models describing wave patterns of kinetic equations. Moreover, we treat the non-cutoff collision kernel instead of the cutoff one. As a simplified model, we also consider the stability and large time behavior of the rarefaction wave solution for the Boltzmann equation.     

On Classical Solutions to the First Mixed Problem for the Vlasov–Poisson System in an Infinite Cylinder

Doklady Mathematics - The first mixed problem for the Vlasov–Poisson system in an infinite cylinder is considered. This problem describes the kinetics of charged particles in a...     

Existence,Uniqueness and Asymptotic Behavior for the Vlasov–Poisson System with Radiation Damping

We investigate the Cauchy problem for the Vlasov–Poisson system with radiation damping.By virtue of energy estimate and a refined velocity average lemma, we establish the global existence of nonnegative weak solution and asymptotic behavior under the condition that initial data have finite mass and energy. Furthermore, by building a Gronwall inequality about the distance between the Lagrangian flows associated to the weak solutions, we can prove the uniqueness of weak solution when the initial data have a higher order velocity moment.     

A generalized Fourier–Hermite method for the Vlasov–Poisson system

BIT Numerical Mathematics - A generalized Fourier–Hermite semi-discretization for the Vlasov–Poisson equation is introduced. The formulation of the method includes as special cases the...     

Regularity Results for the Spherically Symmetric Einstein–Vlasov System

The spherically symmetric Einstein–Vlasov system is considered in Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem is the issue of global existence for initial data without size restrictions. The main purpose of the present work is to propose a method of approach for general initial data, which improves the regularity of the terms that need to be estimated compared to previous methods. We prove that global existence holds outside the center in both these coordinate systems. In the Schwarzschild case we improve the bound on the momentum support obtained in Rein et al. (Commun Math Phys 168:467–478, 1995) for compact initial data. The improvement implies that we can admit non-compact data with both ingoing and outgoing matter. This extends one of the results in Andréasson and Rein (Math Proc Camb Phil Soc 149:173–188, 2010). In particular our method avoids the difficult task of treating the pointwise matter terms. Furthermore, we show that singularities never form in Schwarzschild time for ingoing matter as long as 3m ≤ r. This removes an additional assumption made in Andréasson (Indiana Univ Math J 56:523–552, 2007). Our result in maximal-isotropic coordinates is analogous to the result in Rendall (Banach Center Publ 41:35–68, 1997), but our method is different and it improves the regularity of the terms that need to be estimated for proving global existence in general.     

Vlasov–Poisson equations for a two-component plasma in a half-space

The Vlasov–Poisson equations for a two-component high-temperature plasma with an external magnetic field in a half-space are considered. The electric field potential satisfies the Dirichlet condition on the boundary, and the initial density distributions of charged particles satisfy the Cauchy conditions. Sufficient conditions for the induction of the external magnetic field and the initial charged-particle density distributions are obtained that guarantee the existence of a classical solution for which the supports of the charged-particle density distributions are located at some distance from the boundary.     

Decay of the two-species Vlasov–Poisson–Boltzmann system

We establish the time decay rates of the solution to the Cauchy problem for the two-species Vlasov–Poisson–Boltzmann system near Maxwellians via a refined pure energy method. The total density of two species of particles decays at the optimal algebraic rate as the Boltzmann equation, but the disparity between two species and the electric field decay at an exponential rate. This phenomenon reveals the essential difference when compared to the one-species Vlasov–Poisson–Boltzmann system or the Navier–Stokes–Poisson equations in which the electric field decays at the optimal algebraic rate, and compared to the Vlasov–Boltzmann system in which the disparity between two species decays at the optimal algebraic rate. Our achievement heavily relies on a reformulation of the problem which well displays the cancelation property of the two-species system, and our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.     

Global Existence of Classical Solutions of the Nordström–Vlasov System in Two Space Dimensions

ABSTRACT

The dynamics of a self-gravitating ensemble of collisionless particles is modeled by the Nordström–Vlasov system in the framework of the Nordström scalar theory of gravitation. For this system in two space dimensions, integral representations of the first-order derivatives of the field are derived. Using these representations we show global existence of smooth solutions for large data.