Vlasov-Maxwell System Research Articles

The relativistic Vlasov–Maxwell system: Local smooth solvability for a weak topology

This article is devoted to the relativistic Vlasov–Maxwell system in space dimension three. We prove the local well-posedness (existence and uniqueness) for initial data (\mathrm{f}_{0}, \mathrm{E}_{0},\mathrm{B}_{0}) \in L^{\infty} \times H^{1} \times H^{1} , with \mathrm{f}_{0} compactly supported in momentum. As a byproduct, we obtain the uniqueness of weak solutions to the 3D relativistic Vlasov–Maxwell system. This result is at the interface of the classical solutions in the sense of Glassey–Strauss, and the weak solutions in the sense of DiPerna–Lions. It is the consequence of the local smooth solvability for the weak topology associated with L^{\infty} \times H^{1} \times H^{1} . We derive our result from a representation formula decoding how the momentum spreads and revealing that the domain of influence in momentum is controlled by mild information. We do so by developing a Radon Fourier analysis on the RVM system, leading to the study of a class of singular weighted integrals. In parallel, we implement our method to construct smooth solutions to the RVM system in the regime of dense, hot and strongly magnetized plasmas.

Modified scattering for the small data solutions to the Vlasov–Maxwell system

Modified scattering for the small data solutions to the Vlasov–Maxwell system

Cascade-Dissipation Balance in Astrophysical Plasmas: Insights from the Terrestrial Magnetosheath.

The differential heating of electrons and ions by turbulence in weakly collisional magnetized plasmas and the scales at which such energy dissipation is most effective are still debated. Using a large data sample measured in Earth's magnetosheath by the magnetospheric multiscale mission and the coarse-grained energy equations derived from the Vlasov-Maxwell system, we find evidence of a balance over two decades in scales between the energy cascade and dissipation rates. The decline of the cascade rate at kinetic scales (in contrast with a constant one in the inertial range), is balanced by an increasing ion and electron heating rates, estimated via the pressure strain. Ion scales are found to contribute most effectively to ion heating, while electron heating originates from both ion and electron scales. These results can potentially impact the current understanding of particle heating in turbulent magnetized plasmas as well as their theoretical and numerical modeling.

An Implicit, Asymptotic-Preserving and Energy-Charge-Conserving Method for the Vlasov-Maxwell System Near Quasi-Neutrality

An Implicit, Asymptotic-Preserving and Energy-Charge-Conserving Method for the Vlasov-Maxwell System Near Quasi-Neutrality

On Global Classical Limit of the (Relativistic) Vlasov–Maxwell System

It is shown that solutions of the (relativistic) Vlasov–Maxwell system converge pointwise to solutions of the Vlasov–Poisson system globally in time at the asymptotic rate of c^{-1}, as the light speed c tends to infinity, which extends the results of Asano and Ukai (Stud Math Appl 18:369–383, 1986), Degond (Math Methods Appl Sci 8:533–558, 1986) and Schaeffer (Commun Math Phys 104:403–421, 1986). The analysis relies on the method of Glassey and Strauss (Commun Math Phys 113:191–208, 1987) and Schaeffer (Commun Math Phys 104:403–421, 1986).

Lipschitz Continuous Solutions of the Vlasov–Maxwell Systems with a Conductor Boundary Condition

Lipschitz Continuous Solutions of the Vlasov–Maxwell Systems with a Conductor Boundary Condition

Continuous family of equilibria of the 3D axisymmetric relativistic Vlasov-Maxwell system

We consider the relativistic Vlasov-Maxwell system (RVM) on a general axisymmetric spatial domain with perfect conducting boundary which reflects particles specularly, assuming axisymmetry in the problem. We construct continuous global parametric solution sets for the time-independent RVM. The solutions in these sets have arbitrarily large electromagnetic field and the particle density functions have the form f±=μ±(e±(x,v),p±(x,v)), where e± and p± are the particle energy and angular momentum, respectively. In particular, for a certain class of examples, we show that the spectral stability changes as the parameter varies from 0 to ∞.

Energy Conservation of the DiPerna–Lions Solutions to Vlasov–Maxwell Systems

Energy Conservation of the DiPerna–Lions Solutions to Vlasov–Maxwell Systems

An asymptotic-preserving and energy-conserving particle-in-cell method for Vlasov–Maxwell equations

In this paper, we develop an asymptotic-preserving and energy-conserving (APEC) Particle-In-Cell (PIC) algorithm for the Vlasov–Maxwell system. This algorithm not only guarantees that the asymptotic limiting of the discrete scheme is a consistent and stable discretization of the quasi-neutral limit of the continuous model but also preserves Gauss’s law and energy conservation at the same time; therefore, it is promising to provide stable simulations of complex plasma systems even in the quasi-neutral regime. The key ingredients for achieving these properties include the generalization of Ohm’s law for electric fields such that asymptotic-preserving discretization can be achieved and a proper decomposition of the effects of the electromagnetic fields such that a Lagrange multiplier method can be appropriately employed for correcting the kinetic energy. We investigate the performance of the APEC method with three benchmark tests in one dimension, including the linear Landau damping, the bump-on-tail problem, and the two-stream instability. Detailed comparisons are conducted by including the results from the classical explicit leapfrog and the previously developed asymptotic-preserving PIC schemes. Our numerical experiments show that the proposed APEC scheme can give accurate and stable simulations of both kinetic and quasi-neutral regimes, demonstrating the attractive properties of the method across scales.

Solutions of the 2D radially symmetric Vlasov-Maxwell system with an initial focusing phase

We study radially symmetric solutions to the 2D Vlasov-Maxwell system and construct solutions that initially possess arbitrarily small $ C^k $ norms ($ k \geq 1 $) for the charge densities and the electric fields, but attain arbitrarily large $ L^\infty $ norms of them at some later time. The proof is done by carefully tracking the trajectories of the particles to observe the emergence of a concentrating behavior. It is the first result to discuss this kind of behavior happening in the Vlasov-Maxwell dynamics.

Highly efficient energy-conserving moment method for the multi-dimensional Vlasov-Maxwell system

This paper concerns an energy-conserving numerical method to solve the multi-dimensional Vlasov-Maxwell (VM) system based on the regularized moment method proposed in [7]. The globally hyperbolic moment system is deduced for the VM system under the framework of Hermite expansions, where the expansion center and the scaling factor are set as the macroscopic velocity and the local temperature, respectively. Thus, the effect of the Lorentz force term can be reduced into several ODEs regarding the macroscopic velocity and the higher-order moment coefficients, which can significantly reduce the computational cost of the whole system. An energy-conserving numerical scheme is proposed to solve the moment equations and Maxwell's equations, where only a small linear system needs to be solved implicitly. Benchmark examples such as Landau damping, two-stream instability, Weibel instability, and two-dimensional Orszag-Tang vortex problem are studied to validate the efficiency and excellent energy-preserving property of the numerical scheme.

Stable cosmologies with collisionless charged matter

It is shown that Milne models (a subclass of Friedmann–Lematre–Robertson–Walker (FLRW) spacetimes with negative spatial curvature) are nonlinearly stable in the set of solutions to the Einstein–Vlasov–Maxwell system, describing universes with ensembles of collisionless self-gravitating, charged particles. The system contains various slowly decaying borderline terms in the mutually coupled equations describing the propagation of particles and Maxwell fields. The effects of those terms are controlled using a suitable hierarchy based on the energy density of the matter fields.

Global weak solutions for the Landau–Lifshitz–Gilbert–Vlasov–Maxwell system coupled via emergent electromagnetic fields

Motivated by recent models of current driven magnetization dynamics, we examine the coupling of the Landau–Lifshitz–Gilbert equation and classical electron transport governed by the Vlasov–Maxwell system. The interaction is based on space-time gyro-coupling in the form of emergent electromagnetic fields of quantized helicity that add up to the conventional Maxwell fields. We construct global weak solutions of the coupled system in the framework of frustrated magnets with competing first- and second-order gradient interactions known to host topological solitons such as magnetic skyrmions and hopfions.

A charge-momentum-energy-conserving 1D3V hybrid Lagrangian–Eulerian method for Vlasov–Maxwell system

Here we propose hybrid Lagrangian–Eulerian (HLE) method for kinetic simulations of plasmas. The HLE method solves advection equations of shape functions unlike particle-in-cell (PIC) method. Although the PIC method cannot preserve the conservation laws of momentum and energy simultaneously, the HLE method can resolve this issue while the particle velocity is formulated by the Lagrangian description. The HLE method discretizes the advection equations and Maxwell's equations with the same scheme, so the momentum and energy exchange between charged particles and electromagnetic field exactly balances in discrete level. Numerical experiments show that linear growth rates of two-stream and Weibel instabilities are reproduced well, and global conservation is exactly preserved except for round-off errors although the velocity space is discretized by the Lagrangian description like the PIC method.

The non-relativistic limit of the Vlasov–Maxwell system with uniform macroscopic bounds

We study in this paper the non-relativistic limit from Vlasov-Maxwell to Vlasov-Poisson, which corresponds to the regime where the speed of light is large compared to the typical velocities of particles. In contrast with \cite{Asano-Ukai-86-SMA}, \cite{Degond-86-MMAS}, \cite{Schaeffer-86-CMP} which handle the case of classical solutions, we consider measure-valued solutions, whose moments and electromagnetic fields are assumed to satisfy some uniform bounds. To this end, we use a functional inspired by the one introduced by Loeper in his proof of uniqueness for the Vlasov-Poisson system \cite{Loeper-2006}. We also build a special class of measure-valued solutions, that enjoy no higher regularity with respect to the momentum variable, but whose moments and electromagnetic fields satisfy all required conditions to enter our framework.

6D phase space collective modes in Vlasov-Maxwell system

Plasma electromagnetic (EM) kinetic simulation faces two unavoidable difficulties. One arises from the fact that Maxwell equations determine a simultaneity relation between transverse electric field and local growth rates of probability distribution function (PDF) in Vlasov-Maxwell (V-M) simulation in Eulerian approach, so does that between and Lagrangian particles’ time-varying rates which refers to displacement of -element in phase space in V-M simulation in Lagrangian approach, and macroparticles’ accelerating rates in Particle-in-Cell (PIC) simulation. These simultaneous with are termed as bottom objects’ growth rates (BOGRs) in this work. Directly solving the BOGRs needs to diagonalize a large full matrix, and hence often be approximated. The other arises from the fact that Lagrangian particles’ time-histories should uniformly converge with respect to the time-step. This severe requirement is difficult to be satisfied and hence some of Lagrangian particles’ time-histories lose fidelity. We propose a strict alternative method free from two difficulties. The initial-value problem of V-M system can be interpreted by phase space allowed deformation of an initial -profile. By virtue of a more compact description of in which a conditional probability density function (C-PDF) well reflects some macroscopic conservation laws behind the V-M system, we can interpret the initial-value problem of in terms of standing-wave oscillation in the C-PDF. A key subtle difference between microscopic scalar field and microscopic vector field can also lead to a similar scheme of particle simulation.PACS: 52.65.-y.

Higher-Order Accurate and Conservative Hybrid Numerical Scheme for Relativistic Time-Fractional Vlasov-Maxwell System

The historical analysis demonstrates that plasma scientists produced a variety of numerical methods for solving “kinetic” models, i.e., the Vlasov-Maxwell (VM) system. Still, on the other hand, a significant fact or drawback of most algorithms is that they do not preserve conservation philosophies. This is a crucial fact that cannot be disregarded since the Vlasov Maxwell system is associated with conservation rules and is capable of assessing after the accomplishment of certain helpful mathematical actions. To examine the fractional-order routine of charged particles, we constructed a fractional-order plasma model and proposed a higher-order conservative numerical approach based on operational matrices theory and Shifted Gegenbauer estimations. Numerical convergence is investigated to confirm its competence and compatibility. This concept may be used in problems involving variable order and multidimensionality, such as those involving Vlasov and Boltzmann systems.

Asymptotic properties of small data solutions of the Vlasov-Maxwell system in high dimensions

Asymptotic properties of small data solutions of the Vlasov-Maxwell system in high dimensions

Atangana-Baleanu Caputo fractional-order modeling of plasma particles with circular polarization of LASER light: An extended version of Vlasov-Maxwell system

In this study, we design a specific geometry of plasma particles and further translate it into mathematical form, i.e., formulated time-fractional semi relativistic Vlasov Maxwell system. This extended version of model is very significant and ground-breaking because it has ability to study the behavior of plasma particles at macroscopic and microscopic time-evaluation scales, which is not studied yet. This model is further tackled with numerical strategy, projected in accordance with spectral and finite difference approximations. The numerical results demonstrate that there are certain variations of the plasma particles that were out of sight at fractional scale and dynamically, we have entered into the real profundity of the problem. Numerical convergence of the projected technique is also inspected. These designed tools, such as the plasma and numerical methods, are very significant and can say that it is a remarkable contribution in this field. In addition, the established numerical structure can utilize to scrutinize the required numerical solution of highly non-linear and fractional problems.

A semi‐relativistic time‐fractional Vlasov–Maxwell code for numerical simulation based on circular polarization and Landau damping instability

AbstractThe Vlasov–Maxwell system is a revolutionary tool to model and analyze the dynamic behavior of the collisionless plasma in the existence of the electromagnetic field. In this perspective, the investigation of this system with the concept of the time‐fractional derivative is the new benchmark and also one of the main objectives of this article. In order to construct the numerical solution, we propose and implement a novel scheme which grounded on finite‐difference and spectral approximations. The temporal variable is tackled by using finite‐difference approximation, whereas spatial variables approximated via sifted Gegenbauer polynomials. Various simulations are performed to explore new phenomena and demonstrate the accuracy and reliability of the proposed method. Convergence and error bound of the suggested methods are investigated theoretically, whereas stability analysis performed numerically. Moreover, the developed method can be used conveniently to examine the numerical solution of other multidimensional highly nonlinear fractional or variable order problems of physical nature.