Local existence of solutions of the vlasov-maxwell equations and convergence to the vlasov-poisson equations for infinite light velocity

We prove the local existence of smooth solutions for the Vlasov-Maxwell equations in three space variables. The existence time for such solutions is independent of the light velocity c. Then we derive regularity results for both the Vlasov-Poisson and the Vlasov-Maxwell equations. The last part of the paper is devoted to a proof of weak and strong convergence of the Vlasov-Maxwell equations towards the Vlasov-Poisson equations, when the light velocity c goes to infinity.     

Derivation and classification of Vlasov-type and magnetohydrodynamics equations: Lagrange identity and Godunov’s form

We describe the derivation of the Vlasov-Maxwell equations from the Lagrangian of classical electrodynamics, from which magnetohydrodynamic-type equations are in turn derived. We consider both the relativistic and nonrelativistic cases: with zero temperature as the exact consequence of the Vlasov-Maxwell equations and with nonzero temperature as a zeroth-order approximation of the Maxwell-Chapman-Enskog method. We obtain the Lagrangian identities and their generalizations for these cases and compare them.     

High velocity particles in a collisionless plasma

We prove that smooth solutions of the relativistic Vlasov-Maxwell equations exist for all time provided the kinetic energy density is uniformly bounded.     

Asymptotic behavior of weak solutions for the relativistic Vlasov-Maxwell equations with large light speed

We study here the behavior of time periodic weak solutions for the relativistic Vlasov-Maxwell boundary value problem in a three-dimensional bounded domain with strictly star-shaped boundary when the light speed becomes infinite. We prove the convergence toward a time periodic weak solution for the classical Vlasov-Poisson equations.     

Steady-state solutions of the Vlasov-Maxwell system and their stability

The study of the Vlasov-Maxwell system is reduced to nonlinear elliptic equations in the stationary case and hyperbolic equations in the nonstationary case. On this basis, theorems on the existence of solutions and sufficient conditions of Lyapunov stability are obtained. The cases are considered when electromagnetic fields and distribution functions can be constructed in an implicit form.     

Derivation of Orthogonality Relations in Plasma Theory using Singular Integral Equations

A mathematical technique is developed whereby weighted orthogonalityrelations can be established between the eigenfunctions of thelinear integro-differential Vlasov-Maxwell operators of thekind which occur in plasma physics, so giving to these operatorsthe property of self-adjointness. The technique involves thesolution of a singular integral equation. Previous discussionof the technique has considered the case of positive and zeroindex of this equation. The remaining case of negative indexis dealt with by considering a particular operator.     

Global weak solutions of Vlasov-Maxwell systems

We study here the Vlasov-Maxwell system in its classical and relativistic form. We prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a weak solution with large initial data. The main tools consist of a new regularity result for velocity averages of solutions of some general linear transport equations and the method of renormalization introduced by the authors in previous work.     

Stable magnetic equilibria in collisionless plasmas

There are many steady states in collisionless plasmas. Their dynamical stability has been one of the main interests in the study of plasmas. In one-dimensional space, we discover some new types of stable equilibria by constructing new invariants to the Vlasov-Maxwell system. These stable equilibria include new flat-tail solutions that connect different densities, pressures, and strengths of constant magnetic fields at x = ±∞. © 1997 John Wiley & Sons, Inc.     

Outgoing Radiation from an Isolated Collisionless Plasma

The asymptotic properties at future null infinity of the solutions of the relativistic Vlasov-Maxwell system whose global existence for small data has been established by the author in a previous work are investigated. These solutions describe a collisionless plasma isolated from incoming radiation. It is shown that a non-negative quantity associated to the plasma decreases as a consequence of the dissipation of energy in form of outgoing radiation. This quantity represents the analogue of the Bondi mass in general relativity. Communicated by Vincent Rivasseausubmitted 13/07/03, accepted 11/09/03     

Boundary value problems for the Vlasov-Maxwell equation in one dimension

Global existence in time and uniqueness of solutions are proved for the Cauchy problem for the Vlasov-Maxwell system of equations in one dimension. The limiting values of the field ±(x, t) as the space variable x → E ∞ are shown to be uniquely determined by the initial data. This result then yields existence of solutions of various boundary value problems. Solutions periodic in x are also discussed in this same framework.     

On steady states in a collisionless plasma

Motivated by the collisionless shock problem in plasma, we study the stationary Vlasov-Maxwell system in one space dimension via a dynamical system approach. We construct flat-tail and oscillatory-tail solutions, as well as solitons in the presence of a nontrivial magnetic field. Flat-tail solutions connect different densities, pressures, and strengths of constant fields at x = ±∞, while oscillatory-tail solutions oscillate as x → −∞. We also construct topological horseshoes, which imply very complicated structures of some steady states. © 1996 John Wiley & Sons, Inc.. Inc.     

High order moment closure for Vlasov-Maxwell equations

The extended magnetohydrodynamic models are derived based on the moment closure of the Vlasov-Maxwell (VM) equations. We adopt the Grad type moment expansion which was firstly proposed for the Boltzmann equation. A new regularization method for the Grad’s moment system was recently proposed to achieve the globally hyperbolicity so that the local well-posedness of the moment system is attained. For the VM equations, the moment expansion of the convection term is exactly the same as that in the Boltzmann equation, thus the new developed regularization applies. The moment expansion of the electromagnetic force term in the VM equations turns out to be a linear source term, which can preserve the conservative properties of the distribution function in the VM equations perfectly.     

大型不正定陀螺系统本征值问题

陀螺动力系统可以导入哈密顿辛几何体系,在哈密顿陀螺系统的辛子空间迭代法的基础上提出了一种能够有效计算大型不正定哈密顿函数的陀螺系统本征值问题的算法．利用陀螺矩阵既为哈密顿矩阵而本征值又是纯虚数或零的特点,将对应哈密顿函数为负的本征值分离开来,构造出对应哈密顿函数全为正的本征值问题,利用陀螺系统的辛子空间迭代法计算出正定哈密顿矩阵的本征值,从而解决了大型不正定陀螺系统的本征值问题,算例证明,本征解收敛得很快．     

On a Characteristic Initial Value Problem in Plasma Physics

The relativistic Vlasov-Maxwell system of plasma physics is considered with initial data on a past light cone. This characteristic initial value problem arises in a natural way as a mathematical framework to study the existence of solutions isolated from incoming radiation. Various consequences of the mass-energy conservation and of the absence of incoming radiation condition are first derived assuming the existence of global smooth solutions. In the spherically symmetric case, the existence of a unique classical solution in the future of the initial cone follows by arguments similar to the case of initial data at time t=0. The total mass-energy of spherically symmetric solutions equals the (properly defined) mass-energy on backward and forward light cones. Communicated by Sergiu Klainerman submitted 8/03/05, accepted 26/05/05     

The Darwin Approximation of the Relativistic Vlasov-Maxwell System

We study the relativistic Vlasov-Maxwell system which describes large systems of particles interacting by means of their collectively generated forces. If the speed of light c is considered as a parameter then it is known that in the Newtonian limit c the Vlasov-Poisson system is obtained. In this paper we determine the next order approximate system, which in the case of individual particles usually is called the Darwin approximation.submitted 07/01/04, accepted 30/07/04     

Finite dimensional lie-poisson approximations to vlasov-poisson equations

Many of the basic equations of conservative continuum mechanics (Euler, Vlasov-Poisson, Vlasov-Maxwell, MHD, etc.) are Hamiltonian systems with respect to Lie-Poisson brackets on dual spaces of infinite dimensional Lie algebras. The development of Lie-Poisson integrators for finite dimensional Lie-Poisson systems has shown that they are superior in the numerical simulations of these systems, especially with regard to long term phenomena. This paper shows how to truncate one of these systems, the Vlasov-Poisson equation of plasma physics, to a finite dimensional Lie-Poisson system. This requires replacing the functions on single-particle phase space, with their Poisson bracket Lie algebra structure, by a finite dimensional Lie algebra. Replacing the densities by their moments of order up to k about a fixed reference point corresponds to replacing the functions by their Taylor expansions up to order k. Unfortunately, these truncated Taylor expansions do not form a Lie algebra, since the functions which vanish through order k do not form an ideal under Poisson bracket. Geometrically, this corresponds to the fact that canonical transformations which fix the reference point do not form a normal subgroup. Introducing the location of a reference point in phase space as an extra variable and truncating with respect to this moving point turns out to decouple the “location” from “shape” coordinates of a lump of density in phase space, as far as the Poisson bracket is concerned. One can then replace the shape coordinates by a finite number of moments. The central result of the paper is a construction of the decoupling map described above in the general context of the decomposition of a Lie group as a product of subgroups. The main theorem is first proved by the general theory of Poisson reduction, then by explicit calculation, and lastly by showing that the Poisson isomorphism follows from the lift of a natural groupoid isomorphism. The groupoid aspect of the theory also provides natural Poisson maps, useful in the application of Ruth-type integration techniques, which do not seem easily derivable from the general theory of Poisson reduction. © 1994 John Wiley & Sons, Inc.     

Pairs of edge disjoint Hamiltonian circuits in 5-connected planar graphs

Summary A variety of examples of 4-connected 4-regular graphs with no pair of disjoint Hamiltonian circuits were constructed in response to Nash-Williams conjecture that every 4-connected 4-regular graph is Hamiltonian and also admits a pair of edge-disjoint Hamiltonian circuits. Nash-Williams's problem is especially interesting for planar graphs since 4-connected planar graphs are Hamiltonian. Examples of 4-connected 4-regular planar graphs in which every pair of Hamiltonian circuits have edges in common are included in the above mentioned examples.B. Grünbaum asked whether 5-connected planar graphs always admit a pair of disjoint Hamiltonian circuits. In this paper we introduce a technique that enables us to construct infinitely many examples of 5-connected planar graphs, 5-regular and non regular, in which every pair of Hamiltonian circuits have edges in common.     

Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference

Summary The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is, motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian. The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws) is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for capillary-gravity waves, showing how new waves can be found in this framework.     

Symplectic Transformations and a Reduction Method for Asymptotically Linear Hamiltonian Systems

In many fields of applications, especially in applications from mechanics, many equations of motion can be written as Hamiltonian systems. In this paper, we study a class of asymptotically linear Hamiltonian systems. We construct a symplectic transformation which reduces the linear systems of the Hamiltonian systems. This reduction method can be applied to study the existence of periodic solutions for a class of asymptotically linear Hamiltonian systems under weaker conditions on the linear systems of the Hamiltonian systems.