Convergence des solutions faibles du système de Vlasov–Maxwell stationnaire vers des solutions faibles du système de Vlasov–Poisson stationnaire quand la vitesse de la lumière tend vers l'infini

We study here the behavior of weak solutions for the relativistic stationary Vlasov–Maxwell system with boundary conditions in a three-dimensional bounded domain with strictly star-shaped boundary, when the light speed becomes infinite. We prove the convergence toward a weak solution for the stationary Vlasov–Poisson system. The time periodic problem and the problem with initial-boundary conditions can be treated by the same method. To cite this article: M. Bostan, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Boundary value problem for the N‐dimensional time periodic Vlasov–Poisson system

In this work, we study the existence of time periodic weak solution for the N‐dimensional Vlasov–Poisson system with boundary conditions. We start by constructing time periodic solutions with compact support in momentum and bounded electric field for a regularized system. Then, the a priori estimates follow by computations involving the conservation laws of mass, momentum and energy. One of the key point is to impose a geometric hypothesis on the domain: we suppose that its boundary is strictly star‐shaped with respect to some point of the domain. These results apply for both classical or relativistic case and for systems with several species of particles. Copyright © 2006 John Wiley & Sons, Ltd.     

A two-species plasma in the limit of large ion mass

Classical solutions of the relativistic Vlasov–Maxwell system are considered, describing a collisionless plasma with two species of particles. ions and electrons. It is shown that as the ion mass m tends to infinity, the corresponding solution of the relativistic Vlasov–Maxwell system tends to the solution of a system, in which the ions are given by a fixed ion background and only the electrons move. The convergence is uniform on compact time intervals, with an asymptotic convergence rate of m^?1.     

The classical limit of the relativistic Vlasov–Maxwell system in two space dimensions

The motion of a collisionless plasma is modelled by the Vlasov–Maxwell system. In this paper, solutions of the relativistic Vlasov–Maxwell system are considered in two space dimensions. The speed of light, c, appears as a parameter in the system. With representations of the electric and magnetic fields, conditions are established under which solutions of the relativistic Vlasov–Maxwell system converge pointwise to solutions of the non‐relativistic Vlasov–Poisson system as c tends to infinity, at the asymptotic rate of 1/c. Copyright © 2004 John Wiley & Sons, Ltd.     

Periodic solutions of the 1D Vlasov–Maxwell system with boundary conditions

We study the 1D Vlasov–Maxwell system with time‐periodic boundary conditions in its classical and relativistic form. We are mainly concerned with existence of periodic weak solutions. We shall begin with the definitions of weak and mild solutions in the periodic case. The main mathematical difficulty in dealing with the Vlasov–Maxwell system consist of establishing L^∞ estimates for the charge and current densities. In order to obtain this kind of estimates, we impose non‐vanishing conditions for the incoming velocities, which assure a finite lifetime of all particles in the computational domain ]0,L[. The definition of the mild solution requires Lipschitz regularity for the electro‐magnetic field. It would be enough to have a generalized flow but the result of DiPerna Lions (Invent. Math. 1989; 98 : 511–547) does not hold for our problems because of boundary conditions. Thus, in the first time, the Vlasov equation has to be regularized. This procedure leads to the study of a sequence of approximate solutions. In the same time, an absorption term is introduced in the Vlasov equation, which guarantees the uniqueness of the mild solution of the regularized problem. In order to preserve the periodicity of the solution, a time‐averaging vanishing condition of the incoming current is imposed: \def\d{{\rm d}}\def\incdist#1#2{\int_{0}^{T}\d t\int_{v_{x}#10}\int_{v_{y}}v_xg_{#2}(t,v_x,v_y)\,\d v}$$\incdist{>}{0}+\incdist{<}{L}=0$$\nopagenumbers\end (1) where g₀, g_L are incoming distributors (2) (3) The existence proof uses the Schauder fixed point theorem and also the velocity averaging lemma of DiPerna and Lions (Comm. Pure Appl. Math. 1989; XVII : 729–757). In the last section we treat the relativistic case. Copyright © 2000 John Wiley & Sons, Ltd.     

Solutions périodiques en temps des équations de Vlasov–Maxwell

We study here the existence of time periodic solution for the Vlasov–Maxwell equations in a three dimensional bounded domain. We assume that the boundary of the domain is strictly star-shaped. We give a priori estimates for the kinetic and electro-magnetic energy, and also for the normal and tangential traces of the electro-magnetic field. This method allows us to treat both classical and relativistic cases. To cite this article: M. Bostan, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On global symmetric solutions to the relativistic Vlasov–Poisson equation in three space dimensions

A collisionless plasma is modelled by the Vlasov–Maxwell system. In the presence of very large velocities, relativistic corrections are meaningful. When magnetic effects are ignored this formally becomes the relativistic Vlasov–Poisson equation. The initial datum for the phase space density ƒ₀(x, v) is assumed to be sufficiently smooth, non‐negative and cylindrically symmetric. If the (two‐dimensional) angular momentum is bounded away from zero on the support of ƒ₀(x, v), it is shown that a smooth solution to the Cauchy problem exists for all times. Copyright © 2001 John Wiley & Sons, Ltd.     

The Vlasov–Maxwell–Fokker–Planck system in two space dimensions

We consider the Cauchy problem for the Vlasov–Maxwell–Fokker–Planck system in the plane. It is shown that for smooth initial data, as long as the electromagnetic fields remain bounded, then their derivatives do also. Glassey and Strauss have shown this to hold for the relativistic Vlasov–Maxwell system in three dimensions, but the method here is totally different. In the work of Glassey and Strauss, the relativistic nature of the particle transport played an essential role. In this work, the transport is nonrelativistic, and smoothing from the Fokker–Planck operator is exploited. Copyright © 2015 John Wiley & Sons, Ltd.     

On Derivation and Classification of Vlasov Type Equations and Equations of Magnetohydrodynamics. The Lagrange Identity,the Godunov Form,and Critical Mass

The derivation of the Vlasov–Maxwell and the Vlasov–Poisson–Poisson equations from Lagrangians of classical electrodynamics is described. The equations of electromagnetohydrodynamics (EMHD) type and electrostatics with gravitation are obtained. We obtain and compare the Lagrange equalities and their generalizations for different types of the Vlasov and EMHD equations. The conveniences of writing the EMHD equations in twice divergent form are discussed. We analyze exact solutions to the Vlasov–Poisson–Poisson equations with the presence of gravitation where we have different types of nonlinear elliptic equations for trajectories of particles with critical mass m ² = e ²/G, which has an obvious physical sense, where G denotes the gravitation constant and e is the electron charge. As a consequence we have different behaviors of particles: divergence or collapse of their trajectories.     

On Stationary Solutions of the Relativistic Vlasov–Maxwell System

We study stationary solutions of the relativistic Vlasov–Maxwell system of plasma physics which have a special form introduced (in the classical setting) by Rudykh, Sidorov and Sinitsy and establish their existence under suitable assumptions on the ansatz functions. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. Math. Meth. Appl. Sci., Vol. 20, 667–677 (1997).     

Sur quelques limites de la physique des particules chargées vers la (magnéto)hydrodynamiqueOn some limits in charged particle physics towards (magneto)hydrodynamic equations

We discuss the connection between different scalings limits of the quantum-relativistic Dirac–Maxwell system. In particular we give rigorous results for the quasi-neutral/non-relativistic limit of the Vlasov–Maxwell system: we obtain a magneto-hydro-dynamic system when we consider the magnetic field as a non-relativistic effect and we obtain the Euler equation when we see it as a relativistic effect. A mathematical key is the modulated energy method. To cite this article: Y. Brenier et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 239–244.     

Decay in time for a one‐dimensional two‐component plasma

The motion of a collisionless plasma is described by the Vlasov–Poisson (VP) system, or in the presence of large velocities, the relativistic VP system. Both systems are considered in one space and one momentum dimension, with two species of oppositely charged particles. A new identity is derived for both systems and is used to study the behavior of solutions for large times. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Stochastic differential equations for multi-dimensional domain with reflecting boundary

Summary In this paper we prove that there exists a unique solution of the Skorohod equation for a domain inR ^d with a reflecting boundary condition. We remove the admissibility condition of the domain which is assumed in the work [4] of Lions and Sznitman. We first consider a deterministic case and then discuss a stochastic case.     

On a Vlasov–Schrödinger–Poisson Model

Abstract

A Vlasov–Schrödinger–Poisson system is studied, modeling the transport and interactions of electrons in a bidimensional electron gas. The particles are assumed to have a wave behaviour in the confinement direction (z) and to behave like point particles in the directions parallel to the electron gas (x). For each fixed x and at each time t, the eigenfunctions and the eigenenergies of the Schrödinger operator in the z are computed. The occupation number of each eigenfunction is computed through the resolution of a Vlasov equation in the x direction, the force field being the gradient of the eigenenergy. The whole system is coupled to the Poisson equation for the electrostatic interaction. Existence of weak solutions is shown for boundary value problems in the stationary and time-dependent regimes.     

Finite speed propagation of the solutions for the relativistic Vlasov–Maxwell system

In this paper, we investigate the continuous dependence with respect to the initial data of the solutions for the 1D and 1.5D relativistic Vlasov–Maxwell system. More precisely, we prove that these solutions propagate with finite speed. We formulate our results in the framework of mild solutions, i.e., the particle densities are solutions by characteristics and the electro-magnetic fields are Lipschitz continuous functions.     

Nonlinear stability and instability of relativistic Vlasov‐Maxwell systems

We prove the nonlinear stability or instability of certain periodic equilibria of the 1½D relativistic Vlasov‐Maxwell system. In particular, for a purely magnetic equilibrium with vanishing electric field, we prove its nonlinear stability under a sharp criterion by extending the usual Casimir‐energy method in several new ways. For a general electromagnetic equilibrium we prove that nonlinear instability follows from linear instability. The nonlinear instability is macroscopic, involving only the L¹‐norms of the electromagnetic fields. © 2006 Wiley Periodicals, Inc.     

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.     

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.