Relativistic 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.

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 ∞.

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.

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.

Propagation of Regularity and Long Time Behavior of the 3D Massive Relativistic Transport Equation II: Vlasov–Maxwell System

Given any smooth, suitably small initial data, which decays polynomially at infinity, we prove global regularity for the 3D relativistic massive Vlasov–Maxwell system. In particular, the compact support assumption, which was widely used in the literature, is not imposed on the initial data. Our proofs are based on a combination of the Klainerman vector field method and the Fourier method, which allows us to exploit a crucial hidden null structure in the relativistic Vlasov–Maxwell system.

A toy model for the relativistic Vlasov-Maxwell system

<p style='text-indent:20px;'>The global-in-time existence of classical solutions to the relativistic Vlasov-Maxwell (RVM) system in three space dimensions remains elusive after nearly four decades of mathematical research. In this note, a simplified "toy model" is presented and studied. This toy model retains one crucial aspect of the RVM system: the phase-space evolution of the distribution function is governed by a transport equation whose forcing term satisfies a wave equation with finite speed of propagation.</p>

Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus

<p style='text-indent:20px;'>Although the nuclear fusion process has received a great deal of attention in recent years, the amount of mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the <i>Vlasov-Maxwell</i> system in a two-dimensional annulus when a huge (<i>but finite-in-time</i>) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The authors hope that this work is a step towards a more generalized work on the three-dimensional Tokamak structure. The highlight of this work is the physical assumptions on the external magnetic potential well which remains finite <i>within a finite time interval</i> and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the <i>Vlasov-Maxwell</i> system.</p>

On the 3D Relativistic Vlasov-Maxwell System with Large Maxwell Field

This paper is devoted to the study of relativistic Vlasov-Maxwell system in three space dimension. For a class of large initial data, we prove the global existence of classical solution with sharp decay estimate. The initial Maxwell field is allowed to be arbitrarily large and the initial density distribution is assumed to be small and decay with rate $(1+|x|+|v|)^{-9-}$. In particular, there is no restriction on the support of the initial data.

Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system

<p style='text-indent:20px;'>This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function <inline-formula><tex-math id="M1">\begin{document}$ f(t,\cdot) $\end{document}</tex-math></inline-formula>. Magnetically confined plasmas are characterized by the presence of a strong <i>external</i> magnetic field <inline-formula><tex-math id="M2">\begin{document}$ x \mapsto \epsilon^{-1} \mathbf{B}_e(x) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent <i>internal</i> electromagnetic fields <inline-formula><tex-math id="M4">\begin{document}$ (E,B) $\end{document}</tex-math></inline-formula> are supposed to be small. In the non-magnetized setting, local <inline-formula><tex-math id="M5">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since <inline-formula><tex-math id="M6">\begin{document}$ \epsilon^{-1} $\end{document}</tex-math></inline-formula> is large, standard results predict that the lifetime <inline-formula><tex-math id="M7">\begin{document}$ T_\epsilon $\end{document}</tex-math></inline-formula> of solutions may shrink to zero when <inline-formula><tex-math id="M8">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> goes to <inline-formula><tex-math id="M9">\begin{document}$ 0 $\end{document}</tex-math></inline-formula>. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound (<inline-formula><tex-math id="M10">\begin{document}$ 0<T<T_\epsilon $\end{document}</tex-math></inline-formula>) on the lifetime of solutions and uniform Sup-Norm estimates. Furthermore, a bootstrap argument shows <inline-formula><tex-math id="M11">\begin{document}$ f $\end{document}</tex-math></inline-formula> remains at a distance <inline-formula><tex-math id="M12">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.</p>

Weak solutions of the relativistic Vlasov–Maxwell system with external currents

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 , and are subject to boundary conditions on ∂Ω. 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 conditions for the electromagnetic fields but consider the fields as functions on whole space and model objects, that are placed in space, via given matrix‐valued functions ε (the permittivity) and μ (the permeability). A weak solution concept is introduced and existence of global‐in‐time solutions is proved, as well as the redundancy of the divergence part of the Maxwell equations in this weak solution concept.

An endpoint case of the renormalization property for the relativistic Vlasov–Maxwell system

The aim of this paper is to improve the previous work on the relativistic Vlasov–Maxwell system, one of the most important equations in plasma physics. Recently, Bardos et al. [Q. Appl. Math. 78, 193–217 (2020)] presented a proof of an Onsager type conjecture on the renormalization property and the entropy conservation laws for the relativistic Vlasov–Maxwell system. Particularly, the authors proved that if the distribution function u∈L∞(0,T;Wθ,p(R6)) and the electromagnetic field E,B∈L∞(0,T;Wκ,q(R3)) with θ, κ ∈ (0, 1) such that θκ + κ + 3θ − 1 > 0 and 1/p + 1/q ≤ 1, then the renormalization property and entropy conservation laws hold. To determine a complete proof of this work, in this paper, we improve their results under weaker regularity assumptions for a weak solution to the relativistic Vlasov–Maxwell equations. More precisely, we show that under similar hypotheses, the renormalization property and entropy conservation laws for the weak solution to the relativistic Vlasov–Maxwell system even hold for the endpoint case θκ + κ + 3θ − 1 = 0. Our proof is based on better estimations on regularization operators.

A relativistic canonical symplectic particle-in-cell method for energetic plasma analysis

A relativistic canonical symplectic particle-in-cell (RCSPIC) method for simulating energetic plasma processes is established. By use of the Hamiltonian for the relativistic Vlasov–Maxwell system, we obtain a discrete relativistic canonical Hamiltonian dynamical system, based on which the RCSPIC method is constructed by applying the symplectic temporal discrete method. Through a 106-step numerical test, the RCSPIC method is proven to possess long-term energy stability. The ability to calculate energetic plasma processes is shown by simulations of the reflection processes of a high-energy laser (1 × 1020 W cm−2) on the plasma edge.

Semi-Lagrangian Vlasov simulation of the interaction between femtosecond chirped and double laser pulses with a thin plasma slab

The wakefield generation and electrons acceleration in the interaction of femtosecond chirped and double laser pulses with an underdense thin plasma slab is examined by numerical solution of the relativistic Vlasov-Maxwell system of equations using the backward semi-Lagrangian method (BSL). Unlike particle-in-cell methods, the BSL method is free of numerical noise and provides an accurate representation of the distribution function, even in very low density regions of phase space where acceleration and trapping occur. The effect of density fluctuations in low density regions of phase space are important, since they affect the acceleration processes over the following interactions. Therefore, a detailed structure of the distribution function is needed to investigate this process in these regions, which only can be obtained by Vlasov codes. Excitation of wakefields by chirped, chirp free, and double pulse lasers in an underdense plasma slab are simulated using the BSL code and the results are compared. It is shown that by making use of chirped laser pulses, the mean kinetic energy of plasma electrons can be enhanced up to 40% compared with the chirp free case. The investigation of wakefield generation by double laser pulses with a time repetition equal to one plasma period, exhibits amplification of wakefield amplitude after the entrance of the second laser pulse into plasma slab.

Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder

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 which the plasma is confined inside the cylinder, i.e., it stays away from the boundary of the cylinder.

Hot Plasma in a Container---an Optimal Control Problem

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 is located in a bounded container $\Omega\subset\mathbb R^3$, for example a fusion reactor. 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 model objects, that are placed in space, via given matrix-valued functions $\varepsilon$ (the permittivity) and $\mu$ (the permeability). A typical aim in fusion plasma physics is to keep the amount of particles hitting $\partial\Omega$ as small as possible (since they damage the reactor wall), while the control costs should not be too exhaustive (to ensure efficiency). This leads to a minimizing problem with a PDE constraint. This problem is analyzed in detail. In particular, we prove existence of minimizers and establish an approach to derive first order optimality conditions.

Effect of external magnetic field on wakefield generation in underdense plasma slab using backward semi-Lagrangian Vlasov code

Laser wakefield excitation in the interaction of femtosecond laser pulse with an underdense thin magnetized plasma slab is studied by one dimensional relativistic Vlasov-Maxwell system of equations. The interaction is modeled by relativistic Vlasov equation in propagation direction of laser pulse and fluid equations in transverse direction. Backward semi-Lagrangian method (BSL) is used for numerical solution of Vlasov equation. Generation of wakefields by a right and left handed circularly polarized (RCP) and (LCP) laser pulses in the interaction with nonmagnetized and magnetized plasma slab are examined and compared. Two directions are considered for external magnetic field, parallel and antiparallel to direction of laser pulse propagation. The results show that applying magnetic field along (opposite to) the propagation direction of RCP (LCP) laser pulse increases amplitude of density steepening and consequently wakefields amplitude, and mean kinetic energy of electrons compared with nonmagnetized plasma. The results are in complete agreement with previous analytical and numerical reports.

Onsager-type conjecture and renormalized solutions for the relativistic Vlasov–Maxwell system

In this paper we give a proof of an Onsager type conjecture on conservation of energy and entropies of weak solutions to the relativistic Vlasov--Maxwell equations. As concerns the regularity of weak solutions, say in Sobolev spaces $W^{\alpha,p}$, we determine Onsager type exponents $\alpha$ that guarantee the conservation of all entropies. In particular, the Onsager exponent $\alpha$ is smaller than $\alpha = 1/3$ established for fluid models. Entropies conservation is equivalent to the renormalization property, which have been introduced by DiPerna--Lions for studying well-posedness of passive transport equations and collisionless kinetic equations. For smooth solutions renormalization property or entropies conservation are simply the consequence of the chain rule. For weak solutions the use of the chain rule is not always justified. Then arises the question about the minimal regularity needed for weak solutions to guarantee such properties. In the DiPerna--Lions and Bouchut--Ambrosio theories, renormalization property holds under sufficient conditions in terms of the regularity of the advection field, which are roughly speaking an entire derivative in some Lebesgue spaces (DiPerna--Lions) or an entire derivative in the space of measures with finite total variation (Bouchut--Ambrosio). In return there is no smoothness requirement for the advected density, except some natural a priori bounds. Here we show that the renormalization property holds for an electromagnetic field with only a fractional space derivative in some Lebesgue spaces. To compensate this loss of derivative for the electromagnetic field, the distribution function requires an additional smoothness, typically fractional Sobolev differentiability in phase-space. As concerns the conservation of total energy, if the macroscopic kinetic energy is in $L^2$, then total energy is preserved.

The relativistic Vlasov–Maxwell equations for strongly magnetized plasmas

An important challenge in plasma physics is to determine whether ionized gases can be confined by strong magnetic fields. After properly formulating the model, this question leads to a penalized version of the Relativistic Vlasov Maxwell system, marked by the role of a singular factor e −1 corresponding to the inverse of a cyclotron frequency. In this paper, we prove in this context the existence of classical C 1-solutions for a time independent of e. We also investigate the stability of these smooth solutions.

Regularity of weak solutions for the relativistic Vlasov–Maxwell system

We investigate the regularity of weak solutions of the relativistic Vlasov–Maxwell system by using Fourier analysis and the smoothing effect of low velocity particles. This smoothing effect has been used by several authors (see Glassey and Strauss 1986; Klainerman and Staffilani, 2002) for proving existence and uniqueness of [Formula: see text]-regular solutions of the Vlasov–Maxwell system. This smoothing mechanism has also been used to study the regularity of solutions for a kinetic transport equation coupled with a wave equation (see Bouchut, Golse and Pallard 2004). Under the same assumptions as in the paper “Nonresonant smoothing for coupled wave[Formula: see text]+[Formula: see text]transport equations and the Vlasov–Maxwell system”, Rev. Mat. Iberoamericana 20 (2004) 865–892, by Bouchut, Golse and Pallard, we prove a slightly better regularity for the electromagnetic field than the one showed in the latter paper. Namely, we prove that the electromagnetic field belongs to [Formula: see text], with [Formula: see text].

Quadratic conservative scheme for relativistic Vlasov–Maxwell system

For more than half a century, most of the plasma scientists have encountered a violation of the conservation laws of charge, momentum, and energy whenever they have numerically solved the first-principle equations of kinetic plasmas, such as the relativistic Vlasov–Maxwell system. This fatal problem is brought by the fact that both the Vlasov and Maxwell equations are indirectly associated with the conservation laws by means of some mathematical manipulations. Here we propose a quadratic conservative scheme, which can strictly maintain the conservation laws by discretizing the relativistic Vlasov–Maxwell system. A discrete product rule and summation-by-parts are the key players in the construction of the quadratic conservative scheme. Numerical experiments of the relativistic two-stream instability and relativistic Weibel instability prove the validity of our computational theory, and the proposed strategy will open the doors to the first-principle studies of mesoscopic and macroscopic plasma physics.