Post-Newtonian Approximation of the Vlasov–Nordström System

ABSTRACT

We study the Nordström–Vlasov system, which describes the dynamics of a self-gravitating ensemble of collisionless particles in the framework of the Nordström scalar theory of gravitation. If the speed of light c is considered as a parameter, it is known that in the Newtonian limit c → ∞ the Vlasov–Poisson system is obtained. In this paper we determine a higher approximation and establish a pointwise error estimate of order 𝒪(c ^?4). Such an approximation is usually called a 1.5 post-Newtonian approximation.     

On the distribution of typical shortest-path lengths in connected random geometric graphs

Stationary point processes in ?² with two different types of points, say H and L, are considered where the points are located on the edge set G of a random geometric graph, which is assumed to be stationary and connected. Examples include the classical Poisson–Voronoi tessellation with bounded and convex cells, aggregate Voronoi tessellations induced by two (or more) independent Poisson processes whose cells can be nonconvex, and so-called β-skeletons being subgraphs of Poisson–Delaunay triangulations. The length of the shortest path along G from a point of type H to its closest neighbor of type L is investigated. Two different meanings of “closeness” are considered: either with respect to the Euclidean distance (e-closeness) or in a graph-theoretic sense, i.e., along the edges of G (g-closeness). For both scenarios, comparability and monotonicity properties of the corresponding typical shortest-path lengths C ^e? and C ^g? are analyzed. Furthermore, extending the results which have recently been derived for C ^e?, we show that the distribution of C ^g? converges to simple parametric limit distributions if the edge set G becomes unboundedly sparse or dense, i.e., a scaling factor κ converges to zero and infinity, respectively.     

Existence of stationary,collisionless plasmas in bounded domains

We consider a collisionless plasma, which consists of electrons and positively charged ions and is confined to a bounded domain in ?³. The distribution functions of the particles are assumed to satisfy specular reflections on the boundary of the domain and the boundary is assumed to be perfectly conducting. We establish the existence of stationary plasmas in the non-relativistic, electrostatic case described by the Vlasov–Poisson system as well as in the relativistic, electrodynamic case described by the relativistic Vlasov–Maxwell system.     

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.     

An asymptotically stable Particle-in-Cell (PIC) scheme for collisionless plasma simulations near quasineutrality

We propose a new Particle-in-Cell scheme for the Vlasov–Poisson equation. This scheme remains stable when the Debye length and plasma period tend to zero without any restriction on the size of the time and length step. It relies on a semi-implicit integration of the particle trajectories. The numerical integration cost is that of the standard explicit method thanks to the use of a reformulation of the Poisson equation. To cite this article: P. Degond et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Global weak solution of the vlasov–poisson system for small electrons mass

We are studying the existence and weak stability of a Vlasov–Poisson syste with two typs of particles , in which the electrons are supposed to be at thermal equilibrium. This modifies the source term in the Poisson equaitonm\, and estimates in the Marcinkiewicz space M³ for the potential are used to get the strong compactness of approximations using a new regularized kernal which preservs an approriate energy inequality.     

A Distributional Approach to Multiple Stochastic Integrals and Transformations of the Poisson Measure, II

We study a class of “nonpoissonian” transformations of the configuration space (over a space of the form G=S×?, where S is a complete separable metric space) and the corresponding transformations of the Poisson measure. For the Poisson measures of the Lévy-Khinchin type we find conditions which are sufficient to ensure that the transformed measure (which in general is nonpoissonian) is absolutely continuous with respect to the initial Poisson measure and derive an expression for the corresponding Radon-Nikodym derivative. To this end we use a distributional approach to Poisson multiple stochastic integrals. This is the second of a series of papers, as compared to the first part the space G is different and the intensity measure is more general, allowing a stronger singularity at the origin.     

Global existence for the Vlasov–Darwin system in ℝ3 for small initial data

We prove the global existence of weak solutions to the Vlasov–Darwin system in R3 for small initial data. The Vlasov–Darwin system is an approximation of the Vlasov–Maxwell model which is valid when the characteristic speed of the particles is smaller than the light velocity, but not too small. In contrast to the Vlasov–Maxwell system, the total energy conservation does not provide an L2‐bound on the transverse part of the electric field. This difficulty may be overcome by exploiting the underlying elliptic structure of the Darwin equations under a smallness assumption on the initial data. We finally investigate the convergence of the Vlasov–Darwin system towards the Vlasov–Poisson system. Copyright © 2003 John Wiley & Sons, Ltd.     

Boundary Layers Associated with Incompressible Navier-Stokes Equations: The Noncharacteristic Boundary Case

The goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly noncharacteristic, i.e., there is injection and/or suction everywhere at the boundary. Following earlier work on the boundary layer for linearized Navier-Stokes equations in the case where the boundaries are characteristic (no-slip at the boundary and non-permeable), we consider here the case where the boundary is permeable and thus noncharacteristic. The form of the boundary layer and convergence results are derived in two cases: linearized equation and full nonlinear equations. We prove that there exists a boundary layer at the outlet (downwind) of the form e^−Uz/ε where U is the speed of injection/suction at the boundary, z is the distance to the outlet of the channel, and ε is the kinematic viscosity. We improve an earlier result of S. N. Alekseenko (1994, Siberian Math. J.35, No. 2, 209-230) where the convergence in L² of the solutions of the Navier-Stokes equations to that of the Euler equations at vanishing viscosity was established. In the two dimensional case we are able to derive the physically relevant uniform in space (L^∞ norm) estimates of the boundary layer. The uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic Sobolev imbedding. To the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full (nonlinear) Navier-Stokes equations for incompressible fluids.     

On the Explicit Determination of Certain Solutions of Periodic Differential Equations of Higher Order

We consider the class of differential equations $ y^{(k)}+\Sigma_{k- 2}^{\nu=1}A_{\nu}y^{(\nu)}+A_0(z)y=0\ {\rm where}\ A_{1},\dots,A_{k- 2}$ are constants, k ≥ 3 and where A₀(z) is a non-constant periodic entire function, which is a rational function of e ^z. In this paper we develop a method that enables us to decide if this equation can have solutions with few zeros, and we also present the construction of these solutions.     

Rate of Convergence Towards Semi-Relativistic Hartree Dynamics

We consider the semi-relativistic system of N gravitating Bosons with gravitation constant G. The time evolution of the system is described by the relativistic dispersion law, and we assume the mean-field scaling of the interaction where N → ∞ and G → 0 while GN = λ fixed. In the super-critical regime of large λ, we introduce the regularized interaction where the cutoff vanishes as N → ∞. We show that the difference between the many-body semi-relativistic Schrödinger dynamics and the corresponding semi-relativistic Hartree dynamics is at most of order N ^?1 for all λ, i.e., the result covers the sub-critical regime and the super-critical regime. The N dependence of the bound is optimal.     

Symmetry results for nonlinear elliptic operators with unbounded drift

We prove the one-dimensional symmetry of solutions to elliptic equations of the form ?div(e ^G(x) a(|?u|)?u) = f(u) e ^G(x), under suitable energy conditions. Our results holds without any restriction on the dimension of the ambient space.     

The Vlasov–Poisson–Fokker–Planck equation in an interval with kinetic absorbing boundary conditions

We study the initial–boundary value problem for the Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions. We first prove the existence of weak solutions of the linearized equation in an interval with absorbing boundary conditions. Moreover, the weak solution converges to zero exponentially in time. Then we extend the above results to the fully nonlinear Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions; the existence and the longtime behavior of weak solutions. Finally, we prove that the weak solution is actually a classical solution by showing the hypoellipticity of the solution away from the grazing set and the Hölder continuity of the solution up to the grazing set.     

On the characterization of trees with signed edge domination numbers 1, 2, 3, or 4

Let G=(V,E) be a simple graph. For an edge e of G, the closed edge-neighbourhood of e is the set N[e]={e^′∈E|e^′ is adjacent to e}∪{e}. A function f:E→{1,−1} is called a signed edge domination function (SEDF) of G if ∑_e^′∈N[e]f(e^′)≥1 for every edge e of G. The signed edge domination number of G is defined as . In this paper, we characterize all trees T with signed edge domination numbers 1, 2, 3, or 4.     

Time decy,propagarion of low moments and dispersive

We prove time decay estimates for several kinetic equations like the free lkansport, Boltzmann and Vlasov—Poisson Equations. We also consider solutions with infinite energy of the Vlasov—Poisson Equation and we show that low moments in the velocity variable are propagated. As a consequence, we prove that the potential energy becomes finite immediately and that the kinetic energy is locally finite. Our approach is based on new dispersive identities for transport equations.     

Algebraic Independence of Solutions of First-Order Rational Difference Equations

We prove a theorem on algebraic independence of solutions of first order rational difference equations. By the theorem, we are able to prove algebraic independence of x, the exponential function e ^x and the Weierstrass function ${\wp(x)}$ over ${\mathbb{C}}$ only by seeing degrees of polynomials associated with their double angle formulas. As a corollary, we obtain a result on unsolvability of a first-order rational difference equation by solutions of other first-order rational difference equations, which implies its irreducibility. Additionally, we introduce some applications to algebraic independence of functions f(x), f(x ²), . . . , f(x ⁿ ).     

Finite groups with an irreducible character of large degree

Let G be a finite group and d the degree of a complex irreducible character of G, then write |G| = d(d + e) where e is a nonnegative integer. We prove that |G| ≤ e⁴?e³ whenever e > 1. This bound is best possible and improves on several earlier related results.     

L∞‐estimates for the Vlasov–Poisson–Fokker–Planck equation

We consider the Vlasov–Poisson–Fokker–Planck equation in three dimensions as the backward Kolmogorov equation associated to a non‐linear diffusion process. In this way we derive new L^∞‐estimates on the spatial density which are uniform in the diffusion parameters. Copyright © 2000 John Wiley & Sons, Ltd.     

The nonvacuum Einstein flow on surfaces of nonnegative curvature

We prove future nonlinear stability of homogeneous solutions to the Einstein–Vlasov system with massive particles on manifolds with topology M = ?×Σ, where Σ is either 𝕊² or 𝕋². For the sphere this implies the existence of an open subset of the initial data manifold with elements of strictly positive scalar curvature, whose developments are future geodesically complete. In combination with an earlier result for hyperbolic surfaces we conclude future completeness for the Einstein–Vlasov system in 2+1 dimensions independent of the compact spatial topology for an open set of initial data.