The paper treats the validity problem of the nonrelativistic Vlasov-Poisson equation in $d\geq 2$ dimensions. It is shown that the Vlasov-Poisson dynamics can be derived as a combined mean field and point-particle limit of an N-particle Coulomb system of extended charges. This requires a sufficiently fast convergence of the initial empirical distributions. If the electron radius decreases slower than $N^{-\frac{1}{d(d+2)}}$, the corresponding initial configurations are typical. This result entails propagation of molecular chaos for the respective dynamics
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.
We study the asymptotic regime of strong electric fields that leads from the Vlasov-Poisson system to the Incompressible Euler equations. We also deal with the Vlasov-Poisson-Fokker-Planck system which induces dissipative effects. The originality consists in considering a situation with a finite total charge confined by a strong external field. In turn, the limiting equation is set in a bounded domain, the shape of which is determined by the external confining potential. The analysis extends to the situation where the limiting density is non-homogeneous and where the Euler equation is replaced…
We study smooth, spherically-symmetric solutions to the Vlasov-Poisson system and relativistic Vlasov-Poisson system in the plasma physical case. We construct solutions that initially possess arbitrarily small C^k norms for the charge densities and the electric fields, but attain arbitrarily large L^1 norms of them at some later time.
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).
We prove the existence of axially symmetric solutions to the Vlasov--Poisson system in a rotating setting for sufficiently small angular velocity. The constructed steady states depend on Jacobi's integral and the proof relies on an implicit function theorem for operators.
We introduce a long wave scaling for the Vlasov-Poisson equation and derive, in the cold ions limit, the Korteweg-De Vries equation (in 1D) and the Zakharov-Kuznetsov equation (in higher dimensions, in the presence of an external magnetic field). The proofs are based on the relative entropy method.
Systems of Vlasov-Poisson type are kinetic models describing dilute plasma. The structure of the model differs according to whether it describes the electrons or positively charged ions in the plasma. In contrast to the electron case, where the well-posedness theory for Vlasov-Poisson systems is well established, the well-posedness theory for ion models has been investigated more recently. In this article, we prove global well-posedness for two Vlasov-Poisson systems for ions, posed on the whole three-dimensional Euclidean space $\mathbb{R}^3$, under minimal assumptions on the initial data and…
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.
For the Landau-Poisson system with Coulomb interaction in $\R^3_x$, we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.
In this paper we establish global existence and uniqueness of the solution to the three-dimensional Vlasov-Poisson system in presence of point charges in case of repulsive interaction. The present analysis extends an analogeous two-dimensional result by Caprino and Marchioro [On the plasma-charge model, to appear in Kinetic and Related Models (2010)].
In this paper, we extend to the case of initial data constituted of a Dirac mass plus a bounded density (with finite moments) the theory of Lions and Perthame [6] for the Vlasov-Poisson equation. Our techniques also provide polynomially growing in time estimates for moments of the bounded density.
We prove that the Debye screening length emerges in a spatially homogeneous plasma described by the nonlinear Vlasov-Poisson equation interacting with a point charge. While screened stationary states as well as the time-dependent problem are well-studied for the linearized equation, there are few rigorous results on screening in the nonlinear setting. As such, the results presented here cover the stationary case under the assumptions predicted by the linearized theory.
We study spherically symmetric solutions of the Vlasov-Poisson system in the context of algebras of generalized functions. This allows to model highly concentrated initial configurations and provides a consistent setting for studying singular limits of the system. The proof of unique solvability in our approach depends on new stability properties of the system with respect to perturbations.
We consider the evolution of $N$ fermions interacting through a Coulomb or gravitational potential in the mean-field limit as governed by the nonlinear Hartree equation with Coulomb or gravitational interaction. In the limit of large $N$, we study the convergence in trace norm towards the classical Vlasov-Poisson equation for a special class of mixed quasi-free states.
We consider the Vlasov-Poisson equation in $\mathbb{R}^3$ with initial data which are not $L^1$ in space and have unbounded support in the velocities. Assuming for the density a slight decay in space and a strong decay in velocities, we prove existence and uniqueness of the solution, thus generalizing the analogous result given in [5] for data compactly supported in the velocities.
The Vlasov-Poisson system with massless electrons (VPME) is widely used in plasma physics to model the evolution of ions in a plasma. It differs from the Vlasov-Poisson system (VP) for electrons in that the Poisson coupling has an exponential nonlinearity that creates several mathematical difficulties. In particular, while global well-posedness in 3D is well understood in the electron case, this problem remained completely open for the ion model with massless electrons. The aim of this paper is to fill this gap by proving uniqueness for VPME in the class of solutions with bounded density, and …
We present a probabilistic proof of the mean field limit and propagation of chaos $N$-particle systems in three dimensions with positive (Coulomb) or negative (Newton) $1/r$ potentials scaling like $1/N$ and an $N$-dependent cut-off which scales like $N^{-1/3+ ε}$. In particular, for typical initial data, we show convergence of the empirical distributions to solutions of the Vlasov-Poisson system with either repulsive electrical or attractive gravitational interactions.
In this paper, we study the quasineutral limit (in other words the limit when the Debye length tends to zero) of Vlasov-Poisson like equations describing the behaviour of ions in a plasma. We consider massless electrons, with a charge density following a Maxwell-Boltzmann law. For cold ions, using the relative entropy method, we derive the classical Isothermal Euler or the (inviscid) Shallow Water systems from fluid mechanics. In a second time, we study the combined quasineutral and strong magnetic field regime for such plasmas.
This paper reviews the recent mathematical progresses made on the study of the orbital stability properties for the gravitational Vlasov-Poisson system. We present in details the paper of Lemou, Méhats and Raphaël (Inventiones 2011) and we review also the previous works by Dolbeault, Guo, Hadzic, Lin, Rein, Sánchez, Soler, Wan, Wolansky. We also include a discussion of the history of this topic and the pioneering works by physicists like Antonov, Lynden-Bell and Aly. This is the text of a Bourbaki seminar given in november 2011
This work is concerned with the quasineutral limit of the one-dimensional Vlasov-Poisson equation, for initial data close to stationary homogeneous profiles. Our objective is threefold: first, we provide a proof of the fact that the formal limit does not hold for homogeneous profiles that satisfy the Penrose instability criterion. Second, we prove on the other hand that the limit is true for homogeneous profiles that satisfy some monotonicity condition, together with a symmetry condition. We handle the case of well-prepared as well as ill- prepared data. Last, we study a stationary boundary-va…
The Cauchy problem is revisited for the so-called relativistic Vlasov-Poisson system in the attractive case. Global existence and uniqueness of spherical classical solutions is proved under weaker assumptions than previously used. A new class of blowing up solutions is found when these conditions are violated. A new, non-gravitational physical vindication of the model which (unlike the gravitational one) is not restricted to weak fields, is also given.
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.
We establish the time decay rates of the solutions to the Cauchy problem for the two-species Vlasov-Poisson-Boltzmann system near Maxwellians via a refined pure energy method. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. 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 in whi…
We study the Hilbert expansion for small Knudsen number $\varepsilon$ for the Vlasov-Boltzmann-Poisson system for an electron gas. The zeroth order term takes the form of local Maxwellian: $ F_{0}(t,x,v)=\frac{ρ_{0}(t,x)}{(2πθ_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2θ_{0}(t,x)},\text{\ }θ_{0}(t,x)=Kρ_{0}^{2/3}(t,x).$ Our main result states that if the Hilbert expansion is valid at $t=0$ for well-prepared small initial data with irrotational velocity $u_0$, then it is valid for $0\leq t\leq \varepsilon ^{-{1/2}\frac{2k-3}{2k-2}},$ where $ρ_{0}(t,x)$ and $ u_{0}(t,x)$ satisfy the Euler-Poisson …
A collisionless plasma is modeled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge - dependant upon only velocity - is assumed. The situation in which mobile negative ions balance the positive charge as x tends to infinity is considered. Thus, the total positive charge and the total negative charge are both infinite. Smooth solutions with appropriate asymptotic behavior for large x, which were previously shown to exist locally in time, are continued globally. This is done by showing that the charge density decays at least as fast as x^{-6}. This art…
In this paper we consider the mean field limit and non-relativistic limit of relativistic Vlasov-Maxwell particle system to Vlasov-Poisson equation. With the relativistic Vlasov-Maxwell particle system being a starting point, we carry out the estimates (with respect to $N$ and $c$) between the characteristic equation of both Vlasov-Maxwell particle model and Vlasov-Poisson equation, where the probabilistic method is exploited. In the last step, we take both large $N$ limit and non-relativistic limit (meaning $c$ tending to infinity) to close the argument.
If the Vlasov-Poisson or Einstein-Vlasov system is linearized about an isotropic steady state, a linear operator arises the properties of which are relevant in the linear as well as nonlinear stability analysis of the given steady state. We prove that when defined on a suitable Hilbert space and equipped with the proper domain of definition this transport operator $\mathcal{T}$ is skew-adjoint, i.e., $\mathcal{T}^{\ast}=-\mathcal{T}$. In the Vlasov-Poisson case we also determine the kernel of this operator.
Dynamics of collisionless plasma described by the Poisson-Vlasov equations is connected with the Hamiltonian motions of particles and their symmetries. The Poisson equation is obtained as a constraint arising from the gauge symmetries of particle dynamics. Variational derivative constrained by the Poisson equation is used to obtain reduced dynamical equations. Lie-Poisson reduction for the group of canonical diffeomorphisms gives the momentum-Vlasov equations. Plasma density is defined as the divergence of symplectic dual of momentum variables. This definition is also given a momentum map desc…
We consider the two-dimensional Vlasov-Poisson system to model a two-component plasma whose distribution function is constant with respect to the third space dimension. First, we show how this two-dimensional Vlasov-Poisson system can be derived from the full three-dimensional system. The existence of compactly supported steady states with vanishing electric potential in a three-dimensional setting has already been investigated by A. L. Skubachevskii [15]. We show that his approach can easily be adapted to the two-dimensional system. However, our main result is to prove the existence of compac…
We consider the quantum dynamics of $N$ interacting fermions in the large $N$ limit. The particles in the system interact with each other via repulsive interaction that is regularized Coulomb potential with a polynomial cutoff with respect to $N$.From the quantum system, we derive the Vlasov-Poisson system by simultaneously estimating the semiclassical and mean-field residues in terms of the Husimi measure.
In the paper "Optimal control of a Vlasov-Poisson plasma by an external magnetic field - The basics for variational calculus" [arXiv:1708.02464] we have already introduced a set of admissible magnetic fields and we have proved that each of those fields induces a unique strong solution of the Vlasov-Poisson system. We have also established that the field-state operator that maps any admissible field onto its corresponding solution is continuous and weakly compact. In this paper we will show that this operator is also Fréchet differentiable and we will continue to analyze the optimal control pro…
In this work we recast the collisional Vlasov-Maxwell and Vlasov-Poisson equations as systems of coupled stochastic and partial differential equations, and we derive stochastic variational principles which underlie such reformulations. We also propose a stochastic particle method for the collisional Vlasov-Maxwell equations and provide a variational characterization of it, which can be used as a basis for a further development of stochastic structure-preserving particle-in-cell integrators.
The Nordström-Vlasov system provides an interesting relativistic generalization of the Vlasov-Poisson system in the gravitational case, even though there is no direct physical application. The study of this model will probably lead to a better mathematical understanding of the class of non-linear systems consisting of hyperbolic and transport equations. In this paper it is shown that solutions of the Nordström-Vlasov system converge to solutions of the Vlasov-Poisson system in a pointwise sense as the speed of light tends to infinity, providing a further and rigorous justification of this mode…
As is well known from the work of R. Glassey} and J. Schaeffer, the main energy estimates which are used in global existence results for the gravitational Vlasov-Poisson system do not apply to the relativistic version of this system, and smooth solutions to the initial value problem with spherically symmetric initial data of negative energy blow up in finite time. For similar reasons the variational techniques by which Y. Guo and G. Rein obtained nonlinear stability results for the Vlasov-Poisson system do not apply in the relativistic situation. In the present paper a direct, non-variational …
We consider the three dimensional Vlasov-Poisson system that is equipped with an external magnetic field to describe a plasma. The aim of various concrete applications is to control a plasma in a desired fashion. This can be modeled by an optimal control problem. For that reason the basics for calculus of variations will be introduced in this paper. We have to find a suitable class of fields that are admissible for this procedure as they provide unique global solutions of the Vlasov-Poisson system. Then we can define a field-state operator that maps any admissible field onto its corresponding …
We present the applications of variational-wavelet approach for computing multiresolution/multiscale representation for solution of some approximations of Vlasov-Maxwell-Poisson equations.
We consider spherically symmetric steady states of the Vlasov-Poisson system, which describe equilibrium configurations of galaxies or globular clusters. If the microscopic equation of state, i.e., the dependence of the steady state on the particle energy (and angular momentum) is fixed, a one-parameter family of such states is obtained. In the polytropic case the mass of the state along such a one-parameter family is a monotone function of its radius. We prove that for the King, Woolley-Dickens, and related models this mass-radius relation takes the form of a spiral.
We consider the Vlasov-Poisson system with spherical symmetry and an exterior potential which is induced by a point mass in the center. This system can be used as a simple model for a newtonian galaxy surrounding a black hole. For this system, we establish a global existence result for classical solutions with shell-like initial data, i.e. the support of the density is bounded away from the point mass singularity. We also prove existence and stability of stationary solutions which describe static shells, where we use a variational approach which was established by Y. Guo and G. Rein.
We study existence and uniqueness of the solution to the Vlasov-Poisson system describing a plasma constituted by different species evolving in $\mathbb{R}^3$, whose particles interact via the Coulomb potential. The species can have both positive or negative charge. It is assumed that initially the particles are distributed according to a spatial density with a power-law decay in space, allowing for unbounded mass, and an exponential decay in velocities given by a Maxwell-Boltzmann law, extending a result obtained by the same authors, which was restricted to finite total mass.
Structure preservation using discrete gradients in the Vlasov-Poisson-Landau system
National Research Council, Naval Research Laboratory（国家研究委员会，海军研究实验室）；University at Buffalo, The State University of New York（纽约州立大学布法罗分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
The Caldeira-Leggett Hamiltonian (Eq. (1) below) describes the interaction of a discrete harmonic oscillator with a continuous bath of harmonic oscillators. This system is a standard model of dissipation in macroscopic low temperature physics, and has applications to superconductors, quantum computing, and macroscopic quantum tunneling. The similarities between the Caldeira-Leggett model and the linearized Vlasov-Poisson equation are analyzed, and it is shown that the damping in the Caldeira-Leggett model is analogous to that of Landau damping in plasmas [1]. An invertible linear transformatio…
Numerical evidence is given that spherically symmetric perturbations of stable spherically symmetric steady states of the gravitational Vlasov-Poisson system lead to solutions which oscillate in time. The oscillations can be periodic in time or damped. Along one-parameter families of polytropic steady states we establish an Eddington-Ritter type relation which relates the period of the oscillation to the central density of the steady state. The numerically obtained periods are used to estimate possible periods for typical elliptical galaxies.
