N. A. Sidorov, A. V. Sinitsyn
The theorem on the existence of bifurcation points of the stationary solutions for the Vlasov-Maxwell system with bifurcation direction is proved.
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We present a microscopic derivation of the 3-dimensional relativistic Vlasov-Maxwell system as a combined mean field and point-particle limit of an $N$-particle system of rigid charges with $N$-dependent radius. The approximation holds for typical initial particle configurations, implying in particular propagation of chaos for the respective dynamics.
The review of existence theorems of bifurcation points of solutions for nonlinear operator equation in Banach spaces is presented. The sufficient conditions of bifurcation of solutions of boundary-value problem for Vlasov-Maxwell system are considered. The analytical method of Lyapunov-Schmidt-Trenogin is employed.
We consider a modified version of the Vlasov-Maxwell system in which the usual Maxwell fields are replaced by their retarded parts. We show that solutions of this modified system exist globally for a small number of particles and that they describe a system without incoming radiation
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 $c\to\infty$ 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.
The proof of the Jacobi property of the guiding-center Vlasov-Maxwell bracket underlying the Hamiltonian structure of the guiding-center Vlasov-Maxwell equations is presented.
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.
In 1989, R. DiPerna and P.-L. Lions established the existence of global weak solutions to the Vlasov-Maxwell system. In the present notes we give a somewhat simplified proof of this result for the relativistic version of this system, the main purpose being to make this important result of kinetic theory more easily accessible to newcomers in the field. We show that the weak solutions preserve the total charge.
The Hamiltonian structure of the gauge-free gyrokinetic Vlasov-Maxwell equations is presented in terms of a Hamiltonian functional and a gyrokinetic Vlasov-Maxwell bracket. The bracket is used to show that the gyrokinetic angular-momentum conservation law can also be expressed in Hamiltonian form. The Jacobi property of the gyrokinetic Vlasov-Maxwell bracket is also demonstrated explicitly.
We treat hydrodynamic limits of the Vlasov-Maxwell-Boltzmann system for one and two species of particles in a viscous incompressible regime.
In this work, we mainly concern the limiting behavior of the electromagnetic field of two species Vlasov-Maxwell-Botlzmann system in diffusive limits. As knudsen numbers goes to zero, the electric magnetic and magnetic field may perserve or vanish. We verify rigorously Navier-Stokes, Navier-Stokes-Poisson and Navier-Stokes Maxwell limit of the two species Vlasov-Maxwell-Boltzmann system on the torus in three dimension. The justification is based on the unified and uniform estimates of solutions to the dimensionless Vlasov-Maxwell-Boltzmann. The uniform estimates of solutions are obtained by em…
Derivation of the Vlasov-Maxwell system from the Maxwell-Schrödinger equations with extended charges
该研究针对库仑规范下带扩展电荷的Maxwell-Schrödinger方程，在半经典区域证明其收敛于非相对论Vlasov-Maxwell动力学并给出近似精度估计，同时建立了两类方程的适定性与正则性理论。
In this work, the magnetohydrodynamics system is formally derived from two species Vlasov-Maxwell-Boltzmann system. By employing the hypocoercivity of the linear Boltzmann operator and overcoming the difficulties resulting from the singular Lorentz term, we first obtain the uniform estimates of solutions with respect to the Knudsen number and then derive the magnetohydrodynamics system from the dimensionless Valsov-Maxwell-Boltzmann system.
The time evolution of a collisionless plasma is modeled by the relativistic Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma consists of $N$ particle species, the particles are located in a bounded container $Ω\subset\mathbb R^3$, and are subject to boundary conditions on $\partialΩ$. Furthermore, there are external currents, typically in the exterior of the container, that may serve as a control of the plasma if adjusted suitably. We do not impose perfect conductor boundary cond…
We prove that solutions of the 3D relativistic Vlasov-Maxwell system can be extended, as long as the quantity $σ_{-1}(t, x) = \max_{|ω|=1} \,\int_{R^3} \frac{dp}{\sqrt{1+p^2}}\, \frac{1}{(1+v\cdotω)}\, f(t, x, p)$ is bounded in $L^2_x$.
Radiative Vlasov-Maxwell Equations
本文研究Radiative Vlasov-Maxwell方程，通过在Lorentz力中加入辐射反作用力修正，证明了大初始数据下解的无条件全局正则性。
The Vlasov-Maxwell equations possess a Hamiltonian structure expressed in terms of a Hamiltonian functional and a functional bracket. In the present paper, the transformation ("lift") of the Vlasov-Maxwell bracket induced by the dynamical reduction of single-particle dynamics is investigated when the reduction is carried out by Lie-transform perturbation methods. The ultimate goal of this work is to derive explicit Hamiltonian formulations for the guiding-center and gyrokinetic Vlasov-Maxwell equations that have important applications in our understanding of turbulent magnetized plasmas. Here,…
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.
We study the diffusive expansion for solutions around Maxwellian equilibrium and in a periodic box to the Vlasov-Maxwell-Boltzmann system, the most fundamental model for an ensemble of charged particles. Such an expansion yields a set of dissipative new macroscopic PDE's, the incompressible Vlasov-Navier-Stokes-Fourier system and its higher order corrections for describing a charged fluid, where the self-consistent electromagnetic field is present. The uniform estimate on the remainders is established via a unified nonlinear energy method and it guarantees the global in time validity of such a…
This paper is devoted to the study of the dynamics of charged particles in a weakly inhomogeneous dilute gas. More precisely, we consider the existence of unique global in time classical solutions for the Vlasov-MaxwellBoltzmann system and its asymptotic behavior when the particle distribution function is a small perturbation around an arbitrarily large homogenous solution of the Boltzmann equation.
We study the dynamics of many charges interacting with the Maxwell field. The particles are modeled by means of non-negative distribution functions $f^+$ and $f^-$ representing two species of charged matter with positive and negative charge, respectively. If their initial velocities are small compared to the speed of light, $c$, then in lowest order, the Newtonian or classical limit, their motion is governed by the Vlasov-Poisson system. We investigate higher order corrections with an explicit control on the error terms. The Darwin order correction, order $|v/c|^2$, has been proved previously.…
We show that nonrelativsitic scaling of the collisionless Vlasov-Maxwell system implies the existence of a formal invariant slow manifold in the infinite-dimensional Vlasov-Maxwell phase space. Vlasov-Maxwell dynamics restricted to the slow manifold recovers the Vlasov-Poisson and Vlasov-Darwin models as low-order approximations, and provides higher-order corrections to the Vlasov-Darwin model more generally. The slow manifold may be interpreted to all orders in perturbation theory as a collection of formal Vlasov-Maxwell solutions that do not excite light waves, and are therefore "dark". We p…
Algebra of Invariants for the Vlasov-Maxwell System
本文研究了相对论与非相对论多物种 Vlasov-Maxwell 系统的不变量代数，直接从系统导出不变量及其通量，证明其与 Hamiltonian 泊松对易，并揭示了单极子条件的关键作用。
Solutions of the 2D Radially Symmetric Vlasov-Maxwell System with an Initial Focusing Phase
本文研究二维径向对称Vlasov-Maxwell系统，构造了初始电荷密度和电场$C^k$范数任意小但后期$L^\infty$范数任意大的解，揭示了聚焦阶段导致的增长现象。
We consider the question of linear instability of an equilibrium of the Relativistic Vlasov-Maxwell (RVM) System that has a strong magnetic field. Standard instability results deal with systems where there are fewer particles with higher energies. In this paper we extend those results to the class of equilibria for which the number of particles does not depend monotonically on the energy. Without the standard sign assumptions, the analysis becomes significantly more involved.
Modified scattering for small data solutions to the Vlasov-Maxwell system: a short proof
本文证明，在电磁场快速衰减条件下，Vlasov-Maxwell 系统紧支撑小数据全局解的分布函数具有修正散射行为，并覆盖 Glassey-Strauss 构造的所有小数据解。
A note on the non-$L^1$ asymptotic completeness of the Vlasov-Maxwell system
本文针对Vlasov-Maxwell系统，在电荷的一般渐近条件下，利用方程的洛伦兹不变性证明其小初值解不满足线性散射，即系统不具备非L^1渐近完备性。
In this paper, we study the Vlasov-Maxwell system in the non-relativistic limit, that is in the regime where the speed of light is a very large parameter. We consider data lying in the vicinity of homogeneous equilibria that are stable in the sense of Penrose (for the Vlasov-Poisson system), and prove Sobolev stability estimates that are valid for times which are polynomial in terms of the speed of light and of the inverse of size of initial perturbations. We build a kind of higher-order Vlasov-Darwin approximation which allows us to reach arbitrarily large powers of the speed of light.
The time evolution of a collisionless plasma is modeled by the relativistic Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. 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 the plasma is located in an infinitely long cylinder and is influenced by an external magnetic field. We prove existence of stationary solutions and give conditions on the external magnetic field under wh…
We present the applications of variational-wavelet approach for computing multiresolution/multiscale representation for solution of some approximations of Vlasov-Maxwell-Poisson equations.
We present the applications of variational--wavelet approach for computing multiresolution/multiscale representation for solution of some approximations of Vlasov-Maxwell equations.
The present work establishes the mean-field limit of a N-particle system towards a regularized variant of the relativistic Vlasov-Maxwell system, following the work of Braun-Hepp [Comm. in Math. Phys. 56 (1977), 101-113] and Dobrushin [Func. Anal. Appl. 13 (1979), 115-123] for the Vlasov-Poisson system. The main ingredients in the analysis of this system are (a) a kinetic formulation of the Maxwell equations in terms of a distribution of electromagnetic potential in the momentum variable, (b) a regularization procedure for which an analogue of the total energy - i.e. the kinetic energy of the …
We consider the plasma confined in a general axisymmetric spatial domain with perfect conducting boundary which reflects particles specularly, and look at a certain class of equilibria, assuming axisymmetry in the problem. We prove a sharp criterion of spectral stability under these settings. Moreover, we provide several explicit families of stable/unstable equilibria using this criterion.
In (Arch. Rational. Mech. Anal 1986, 92:59-90), Glassey and Strauss showed that if the growth in the momentum of the particles is controlled, then the relativistic Vlasov-Maxwell system has a classical solution globally in time. Later they proved that such control is achieved if the kinetic energy density of the particles remains bounded for all time (Math. Meth. Appl. Sci. 1987, 9:46-52). Here, we show that the latter assumption can be weakened to the boundedness of the spatial density.
The uniform-in-time electrostatic limit of the linearised Vlasov-Maxwell system: the Poisson equilibrium
本文证明线性化Vlasov-Maxwell系统在光速趋于无穷时，其解一致收敛于线性化Vlasov-Poisson解，误差为c^{-1}阶，并给出纵向Landau阻尼与横向Klein-Gordon色散机制。
The global existence of solutions to the relativistic Vlasov-Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem. The main result of Glassey-Strauss (1986) shows that a solution $(f, E, B)$ remains $C^1$ as long as the momentum support of $f$ remains bounded. Alternate proofs were later given by Bouchut-Golse-Pallard (2003) and Klainerman-Staffilani (2002). We show that only the boundedness of the momentum support of $f$ after projecting to any two dimensional plane is needed for $(f, E, B)$ to remain $C^1$.
We study smooth, global-in-time solutions of the relativistic Vlasov-Maxwell system that possess arbitrarily large charge densities and electric fields. In particular, we construct spherically symmetric solutions that describe a thin shell of equally charged particles concentrating arbitrarily close to the origin and which give rise to charge densities and electric fields as large as one desires at some finite time. We show that these solutions exist even for arbitrarily small initial data or any desired mass. In the latter case, the time at which solutions concentrate can also be made arbitra…
In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space $\R^3$. The existence of global in time nearby Maxwellian solutions is known from [34] in 2006. However the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work [10] on time decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of $O(t^{-3/2+\frac{3}{2r}})$ in $L^2_ξ(L^r_x)$-norm for any $2\leq r\leq \infty$ if i…
We construct a one-parameter family of stationary axisymmetric and asymptotically flat spacetimes solutions to the Einstein-Vlasov system bifurcating from the Kerr spacetime. The constructed solutions have the property that the spatial support of the matter is a finite, axisymmetric shell located away from the black hole. Our proof is mostly based on the analysis of the set of trapped timelike geodesics for stationary axisymmetric spacetimes close to Kerr, where the geodesic flow is not necessarily integrable. Moreover, the analysis of the Einstein field equations relies on the modified Carter…
Global mild solutions to the Vlasov-Maxwell-Fokker-Planck system
本文研究 Vlasov–Maxwell–Fokker–Planck 系统，在小振幅初值和近 Maxwell 分布条件下，于 $L^1_kL^\infty_TL^2_v$ 空间证明温和解全局存在，并建立其指数时间衰减。
In the study of solutions to the relativistic Boltzmann equation, their regularity with respect to the momentum variables has been an outstanding question, even local in time, due to the initially unexpected growth in the post-collisional momentum variables which was discovered in 1991 by Glassey & Strauss \cite{MR1105532}. We establish momentum regularity within energy spaces via a new splitting technique and interplay between the Glassey-Strauss frame and the center of mass frame of the relativistic collision operator. In a periodic box, these new momentum regularity estimates lead to a proo…
When particle speeds are large the motion of a collisionless plasma is modeled by the relativistic Vlasov Maxwell system. Large time behavior of solutions which depend on one position variable and two momentum variables is considered. In the case of a single species of charge it is shown that there are solutions for which the charge density does not decay in time. This is in marked contrast to results for the non-relativistic Vlasov Poisson system in one space dimension. The case when two oppositely charged species are present and the net total charge is zero is also considered. In this case, …
Techniques for coordinate changes that depend on both dependent and independent variables are developed and applied to the Maxwell-Vlasov Hamiltonian theory. Particle coordinate changes with a new velocity variable dependent on the magnetic field, with spatial coordinates unchanged, are lifted to transform the noncanonical Poisson bracket and, thus, the field Hamiltonian structure of the Vlasov-Maxwell equation. Several examples are given including magnetic coordinates, where the velocity is decomposed into components parallel and perpendicular to the local magnetic field, and the case of sphe…
We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincaré theory. Possible applications of these formulations…
It is argued that the relativistic Vlasov--Maxwell equations of the kinetic theory of plasma approximately describe a relativistic system of $N$ charged point particles interacting with the electromagnetic Maxwell fields in a Bopp--Landé--Thomas--Podolsky (BLTP) vacuum, provided the microscopic dynamics lasts long enough.The purpose of this work is not to supply an entirely rigorous vindication, but to lay down a conceptual road map for the microscopic foundations of the kinetic theory of special-relativistic plasma, and to emphasize that a rigorous derivation seems feasible. Rather than worki…
We consider the $3$-dimensional relativistic Vlasov-Maxwell system with data without compact support in momentum space. We prove two continuation criteria for solutions to this system. First, we show that a regular solution can be continued if the integral of the electromagnetic field along any characteristic is assumed to be bounded. This can be viewed as a generalization of the classical result of Glassey-Strauss (1986) to data with non-compact momentum support. Second, we extend the methods in our companion paper (Part I) to show that a regular solution can be extended as long as $\|\pel_0^…
The action principle by Low [Proc. R. Soc. Lond. A 248, 282--287] for the classic Vlasov-Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler-Poincaré formulation of Vlasov-Maxwell-type systems, effectively starting with Low's action and using constrained variations for the Eulerian description of particle motion, has been known for a while [J. Math. …
We consider the 1/2-dimensional relativistic Vlasov-Maxwell system that describes the time-evolution of a plasma. We find a relatively simple criterion for spectral instability of a wide class of equilibria. This class includes non-homogeneous equilibria that need not satisfy any additional symmetry properties (as was the case in previous results), nor should they be monotone in the particle energy. The criterion is given in terms of the spectral properties of two Schrödinger operators that arise naturally from Maxwell's equations. The spectral analysis of these operators is quite delicate, an…
Continuous Family of Equilibria of the 3D Axisymmetric Relativistic Vlasov-Maxwell System
本文在轴对称有界域上构造相对论性Vlasov-Maxwell系统的连续参数化平衡族，解具有任意大电磁场，并发现谱稳定性随参数变化。
Derivation of the Maxwell-Schrödinger and Vlasov-Maxwell Equations from Non-Relativistic QED
该论文从非相对论量子电动力学出发，在平均场极限下推导出麦克斯韦-薛定谔方程和弗拉索夫-麦克斯韦方程，证明了多体量子态收敛于半经典Slater行列式与相干光子态的乘积。
Motivated by recent discussions on the possible role of quantum computation in plasma simulations, here we present different approaches to Koopman's Hilbert-space formulation of classical mechanics in the context of Vlasov-Maxwell kinetic theory. The celebrated Koopman-von Neumann construction is provided with two different Hamiltonian structures: one is canonical and recovers the usual Clebsch representation of the Vlasov density, the other is noncanonical and appears to overcome certain issues emerging in the canonical formalism. Furthermore, the canonical structure is restored for a variant…
In this paper we discuss the dissipative property of near-equilibrium classical solutions to the Cauchy problem of the Vlasov-Maxwell-Boltzmann System in the whole space $\R^3$ when the positive charged ion flow provides a spatially uniform background. The most key point of studying this coupled degenerately dissipative system here is to establish the dissipation of the electromagnetic field which turns out to be of the regularity-loss type. Precisely, for the linearized non-homogeneous system, some $L^2$ energy functionals and $L^2$ time-frequency functionals which are equivalent with the nat…
Consider the relativistic Vlasov-Maxwell system with initial data of unrestricted size. In the two dimensional and the two and a half dimensional cases, Glassey-Schaeffer (1997, 1998, 1998) proved that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the 2D and the $2\frac 12$D cases, we prove the global existence, uniqueness and regularity for solutions arising from this c…
Linear Landau damping for the Vlasov-Maxwell system in $\mathbb{R}^3$
本文研究了整体空间中线性化相对论 Vlasov-Maxwell 系统，在径向光滑快速衰减平衡态假设下，证明了电磁场的代数色散衰减，并刻画了电标量势与磁矢势由不同色散波包驱动的主导行为。
Stability of Big Bang singularity for the Einstein-Maxwell-scalar field-Vlasov system in the full strong sub-critical regime
针对3+1维爱因斯坦-麦克斯韦-标量场-弗拉索夫系统，通过分析数学结构与设计精细论证，确定新强亚临界区域并证明该区域内卡斯纳解的非线性稳定性，扩展了相关系统的研究结果。
The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We only consider a 'two-dimensional' version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma in a proper way. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applic…
A Presymplectic and Symmetry Reduced Formulation of the Maxwell Vlasov System
本文提出了一种基于偏辛和对称约简的麦克斯韦-维拉索夫系统统一几何 formulation，结合变分方法和控制理论，统一了拉格朗日、哈密顿、规范、约简和控制理论的多个方面。
We present the applications of methods from nonlinear local harmonic analysis for calculations in nonlinear collective dynamics described by different forms of Vlasov-Maxwell-Poisson equations. Our approach is based on methods provided the possibility to work with well-localized in phase space bases, which gives the most sparse representation for the general type of operators and good convergence properties. The consideration is based on a number of anzatzes, which reduce initial problems to a number of dynamical systems and on variational-wavelet approach to polynomial approximations for nonl…
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