Dimensional Vlasov Poisson System Research Articles

On the two and one-half dimensional Vlasov–Poisson system with an external magnetic field: Global well-posedness and stability of confined steady states

The time evolution of a two-component collisionless plasma is modelled by the Vlasov–Poisson system. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. We consider the case that an external magnetic field is present in order to confine the plasma in a given infinitely long cylinder. After discussing global well-posedness of the corresponding Cauchy problem, we construct stationary solutions whose support stays away from the wall of the confinement device. Then, in the main part of this work we investigate the stability of such steady states, both with respect to perturbations of the initial data, where we employ the energy-Casimir method, and also with respect to perturbations of the external magnetic field.

A truly forward semi-Lagrangian WENO scheme for the Vlasov-Poisson system

A class of modular high order forward semi-Lagrangian (FSL) schemes with a number of advantages (outlined below) for linear advection equations is extended to have high resolution, in the sense that sharp gradients can be captured without loss of accuracy, by means of weighted essentially non-oscillatory derivative (DWENO) calculations based on finite differences. Principally, our approach (the convected scheme, CS) differs from conventional high order FSL schemes in the means by which it reaches higher order accuracy. The high order CS progressed in this work reserves use of a low order histogram representation of the solution and a mass-conservative volume-weighted projection operator which ordinarily produces a second order accurate solution. A higher order version is obtained by compensating for the amount by which the CS solution and the exact solution disagree at every order in a Taylor series sense. Instead of including these error terms algebraically, which changes the projection operator itself, the novel idea of the CS is to incorporate this information geometrically as a commensurate adjustment to the volume-weighted proportion assigned to each cell. We leverage DWENO approximations to compute these Taylor series error terms more sensitively, and show the corresponding WENO convected schemes (WENO-CS) achieve the desired dual high order and high resolution behavior. Fifth, seventh, and ninth order WENO-CS base solvers are presented for the 1D constant speed advection equation prototype, and are employed in a Strang splitting approach for the numerical solution of linear and nonlinear two-dimensional hyperbolic equations. Results are showcased for 2D linear transport and rigid body rotation, as well as for classic benchmark problems for the (1+1)-dimensional Vlasov–Poisson system including Landau damping, and the two-stream instability. The schemes enjoy the usual benefits of semi-Lagrangian schemes (e.g. no CFL time step restriction) while further claiming distinct advantages afforded to forward schemes including automatic mass conservation, compact support, and therefore straightforward parallelization.

Moments propagation for weak solutions of the Vlasov–Poisson system in the three-dimensional torus

We consider weak solutions to the three dimensional Vlasov–Poisson system in the periodic case. We show that the velocity moments of order >3 can be propagated.

On global solutions to the Vlasov-Poisson system with radiation damping

In this paper, the dynamics of three dimensional Vlasov-Poisson system with radiation damping is investigated. We prove global existence of a classical as well as weak solution that propagates boundedness of velocity-space support or velocity-space moment of order two respectively. This kind of solutions possess finite mass but need not necessarily have finite kinetic energy. Moreover, uniqueness of the classical solution is also shown.

Classical solutions for the Vlasov–Poisson system with damping term

ABSTRACTWe investigate global existence of classical solution to the three dimensional Vlasov–Poisson system with radiation damping . We prove that a small perturbation of initial datum for a global classical solution verifying certain decay conditions also launches a global solution which has sharp decay. In addition, we prove that a quasineutral initial datum launches a global classical solution.

The two dimensional Vlasov-Poisson system with steady spatial asymptotics

We consider a two dimensional collisionless plasma interacting with a fixed background of positive charge, the density of which depends only upon velocity variable $v$ and decays as $|v| \to \infty $. Suppose that mobile negative ions balance the positive charge as spatial variable $|x|\to \infty $, then on the mesoscopic level the system is characterized by the two dimensional Vlasov-Poisson system with steady spatial asymptotics, whose total positive charge and total negative charge are both infinite. Smooth solutions with appropriate asymptotic behavior are shown to exist locally in time, and an 'almost optimal' criterion for the continuation of these solutions is established.

On the Asymptotic Limit of the Three Dimensional Vlasov–Poisson System for Large Magnetic Field: Formal Derivation

This paper establishes the long time asymptotic limit of the three dimensional Vlasov-Poisson equation with strong external magnetic field. The guiding center approximation is investigated in the three dimensional case with a non-constant magnetic field. In the long time asymptotic limit, the motion can be split in two parts : one stationary flow along the lines of the magnetic field and the guiding center motion in the orthogonal plane of the magnetic field where classical drift velocities are recovered. We discuss in particular the effect of nonconstant external magnetic fields.

Global Existence to the Vlasov–Poisson System and Propagation of Moments Without Assumption of Finite Kinetic Energy

We consider classical as well as weak solutions to the three dimensional Vlasov–Poisson system. Without assuming finiteness of kinetic energy, we prove global existence of classical solutions by assuming the initial datum is smooth enough and has a compact velocity-spatial support, which will be specified in Theorem 1.1. We also establish some propagation results for low moments of weak solutions.

Non linear stability of spherical gravitational systems described by the Vlasov-Poisson equation

In this work, we prove the nonlinear stability of galaxy models derived from the three dimensional gravitational Vlasov Poisson system, which is a canonical model in astrophysics to describe the dynamics of galactic clusters.

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 ℝ3. In the periodic case, we show that velocity moments of order > 14/3 are propagated.

Orbital stability of spherical galactic models

We consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture (Binney, Tremaine in Galactic Dynamics, Princeton University Press, Princeton, 1987) is the stability of spherical models which are nonincreasing radially symmetric steady states solutions. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov (Sov. Astron. 4:859–867, 1961). In the previous work (Lemou et al. in A new variational approach to the stability of gravitational systems, submitted, 2011), we derived the stability of anisotropic models under spherically symmetric perturbations using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics literature (Lynden-Bell in Mon. Not. R. Astron. Soc. 144:189–217, 1969; Gardner in Phys. Fluids 6:839–840, 1963; Wiechen et al. in Mon. Not. R. Astron. Soc. 223:623–646, 1988; Aly in Mon. Not. R. Astron. Soc. 241:15, 1989). In this work, we show how this approach combined with a new generalized Antonov type coercivity property implies the orbital stability of spherical models under general perturbations.

A New Variational Approach to the Stability of Gravitational Systems

We consider the three dimensional gravitational Vlasov Poisson system which describes the mechanical state of a stellar system subject to its own gravity. A well-known conjecture in astrophysics is that the steady state solutions which are nonincreasing functions of their microscopic energy are nonlinearly stable by the flow. This was proved at the linear level by several authors based on the pioneering work by Antonov in 1961. Since then, standard variational techniques based on concentration compactness methods as introduced by P.-L. Lions in 1983 have led to the nonlinear stability of subclasses of stationary solutions of ground state type. In this paper, inspired by pioneering works from the physics litterature (MNRAS 241:15, 1989), (Mon. Not. R. Astr. Soc. 144:189–217, 1969), (Mon. Not. R. Ast. Soc. 223:623–646, 1988) we use the monotonicity of the Hamiltonian under generalized symmetric rearrangement transformations to prove that non increasing steady solutions are the local minimizer of the Hamiltonian under equimeasurable constraints, and extract compactness from suitable minimizing sequences. This implies the nonlinear stability of nonincreasing anisotropic steady states under radially symmetric perturbations.

Transient growth in stable linearized Vlasov–Maxwell plasmas

Large amplitude transient growth of kinetic scale perturbations in stable collisionless magnetized plasmas has recently been demonstrated using a linearized Landau fluid model. Initial perturbations with lengthscales of the order of the ion gyroradius were shown to have transient timescales that in some cases were long compared to the ion gyroperiod, Ωit⪢1. Moreover, it was suggested that such perturbations are not rare but instead form a large class within the set of all possible initial conditions. For collisionless plasmas, the Vlasov–Maxwell equations provide a more complete description of kinetic physics and the existence of transient growth of solutions for the linearized Vlasov–Maxwell system is an interesting question. The existence of transient growth of solutions is demonstrated here for a special case of the Vlasov–Maxwell equations, namely, the one dimensional Vlasov–Poisson system. The analysis is different from the standard approach of nonmodal analysis since the initial value problem is described by a Volterra integral equation of the second kind, reflecting the fact that the time evolution of the system depends on the memory of the state from time zero through time t. For the case of a thermal equilibrium plasma, it is shown how initial conditions may be constructed to obtain solutions that grow linearly in time; the duration of this growth is the time required for a thermal electron to traverse the wavelength of the initial perturbation, a timescale that can last for many plasma periods 2π/ωpe, thus demonstrating the existence of transient growth of solutions for the linearized Vlasov–Poisson system. The results suggest that the phenomenon of transient growth may be a common feature of the linearized Vlasov–Maxwell system as well as for Landau fluid models.

A Direct and Accurate Adaptive Semi-Lagrangian Scheme for the Vlasov-Poisson Equation

A Direct and Accurate Adaptive Semi-Lagrangian Scheme for the Vlasov-Poisson EquationThis article aims at giving a simplified presentation of a new adaptive semi-Lagrangian scheme for solving the (1 + 1)-dimensional Vlasov-Poisson system, which was developed in 2005 with Michel Mehrenberger and first described in (Campos Pinto and Mehrenberger, 2007). The main steps of the analysis are also given, which yield the first error estimate for an adaptive scheme in the context of the Vlasov equation. This article focuses on a key feature of our method, which is a new algorithm to transport multiscale meshes along a smooth flow, in a way that can be saidoptimalin the sense that it satisfies both accuracy and complexity estimates which are likely to lead to optimal convergence rates for the whole numerical scheme. From the regularity analysis of the numerical solution and how it gets transported by the numerical flow, it is shown that the accuracy of our scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step. As a consequence, the numerical solutions are proved to converge inL∞towards the exact ones as ε and δttend to zero, and in addition to the numerical tests presented in (Campos Pinto and Mehrenberger, 2007), some complexity bounds are established which are likely to prove the optimality of the meshes.

THE CONVERGENCE OF THE PARTICLE METHOD FOR THE VLASOV–POISSON SYSTEM WITH EQUALLY SPACED INITIAL DATA POINTS

A proof of convergence of particle methods is given for the Vlasov–Poisson system in three dimensions. This proof applies to some classes of initial functions not included in previous analyses of this problem. A semidiscrete problem is considered discretized in space but not in time with equally spaced initial data points. The convergence is proved for initial functions that are two and three times differentiable. Error estimates are obtained for these cases that depend on the two parameters, the interparticle distance and the mollification parameter in the approximate electric field. To gain further insight into the form such error bounds may take some computations are then done on the one dimensional Vlasov–Poisson system. Experimentally determined error bounds are obtained for particle-in-cell methods. Based on the computed error estimates in one dimension some conclusions are drawn regarding the accuracy of the type of error bounds obtained for the system in three dimensions.

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.