Vlasov Maxwell Boltzmann System Research Articles

Green’s Function and Pointwise Behavior of the One-Dimensional Vlasov–Maxwell–Boltzmann System

Green’s Function and Pointwise Behavior of the One-Dimensional Vlasov–Maxwell–Boltzmann System

Compressible Euler–Maxwell limit for global smooth solutions to the Vlasov–Maxwell–Boltzmann system

Two fundamental models in plasma physics are given by the Vlasov–Maxwell–Boltzmann system and the compressible Euler–Maxwell system which both capture the complex dynamics of plasmas under the self-consistent electromagnetic interactions at the kinetic and fluid levels, respectively. It has remained a long-standing open problem to rigorously justify the hydrodynamic limit from the former to the latter as the Knudsen number [Formula: see text] tends to zero. In this paper, we give an affirmative answer to the problem for smooth solutions to both systems near constant equilibrium in the whole space in case when only the dynamics of electrons is taken into account. The explicit rate of convergence in [Formula: see text] over an almost global time interval is also obtained for well-prepared data. For the proof, one of main difficulties occurs to the cubic growth of large velocities due to the action of the classical transport operator on local Maxwellians and we develop new [Formula: see text]-dependent energy estimates basing on the macro–micro decomposition to characterize the asymptotic limit in the compressible setting.

Hydrodynamic Limit of the Incompressible Navier–Stokes–Fourier–Maxwell System with Ohm’s Law from the Vlasov–Maxwell–Boltzmann System: Hilbert Expansion Approach

We prove a global-in-time limit from the two-species Vlasov–Maxwell–Boltzmann system with a cutoff hard potential collision kernel to the two-fluid incompressible Navier–Stokes–Fourier–Maxwell system with Ohm’s law. Besides the techniques developed for the classical solutions to the Vlasov–Maxwell–Boltzmann equations in the past years, such as the nonlinear energy method and micro–macro decomposition are employed, in this paper, key roles are played by the decay properties of both the electric field and the wave equation with linear damping of the divergence free magnetic field. This is a companion paper of Jiang and Luo (Ann PDE, 8(1), Paper No. 4, 126, 2022) in which Hilbert expansion was not employed but only the hard sphere case was considered.

The Vlasov–Maxwell–Fokker–Planck system near Maxwellians in ℝ3

ABSTRACT Motivated by Duan and Strain [Optimal large-time behavior of the Vlasov–Maxwell–Boltzmann system in the whole space. Comm Pure Appl Math. 2011;64:1497–1546], we prove the global existence of solutions to the Vlasov–Maxwell–Fokker–Planck system near Maxwellians in the whole space by utilizing the refined energy method. In the proof of the global existence, the a priori estimates on the macroscopic component and the electromagnetic field are obtained by use of the macroscopic balance laws and the Maxwell equations, respectively. Moreover, the optimal time decay of solutions to the global Maxwellians, which can be obtained by employing the Fourier analysis.

The same class of stationary solutions to some multidimensional kinetic systems with extensive background density

In this study, we investigate the existence and uniqueness of the same class of stationary solutions for some kinetic systems comprising the Vlasov–Poisson–Fokker–Planck system, Vlasov–Poisson–Boltzmann system, and Vlasov–Maxwell–Boltzmann system in arbitrary space dimensions N(N≥3). Essentially, the problem can be reduced to solving the problem of a second order elliptic equation with exponential nonlinearity. This result had been proved in three spatial dimensions but the extension to a higher-dimensional setting makes the existence proof nontrivial. In particular, it necessary to highlight that the requirement regarding the background density function is relaxed compared with previous studies in the case of three spatial dimensions.

The Vlasov–Maxwell–Boltzmann System Near Maxwellians in the Whole Space with Very Soft Potentials

Since the work [13] by Guo [Invent. Math. 153 (2003), no. 3, 593--630], how to establish the global existence of perturbative classical solutions around a global Maxwellian to the Vlasov-Maxwell-Boltzmann system with the whole range of soft potentials has been an open problem. This is mainly due to the complex structure of the system, in particular, the degenerate dissipation at large velocity, the velocity-growth of the nonlinear term induced by the Lorentz force, and the regularity-loss of the electromagnetic fields. This paper aims to resolve this problem in the whole space provided that initial perturbation has sufficient regularity and velocity-integrability.

Spectrum Structure and Behaviors of the Vlasov--Maxwell--Boltzmann Systems

The spectrum structures and behaviors of the Vlasov--Maxwell--Boltzmann (VMB) systems for both two species and one species are studied in this paper. The analysis shows the effect of the Lorentz force induced by the electromagnetic field leads to some different structure of spectrum from the classical Boltzmann equation and the closely related VPB system. And the significant difference between the two-species VMB model and one-species VMB model are given. The structure in high frequency illustrates the hyperbolic structure of the Maxwell equation. Furthermore, the long time behaviors and the optimal convergence rates to the equilibrium of the VMB systems for both two species and one species are established based on the spectrum analysis, and in particular the phenomena of the electric field dominating and magnetic field dominating are observed for the one-species VMB system.

The Vlasov–Maxwell–Boltzmann system with a uniform ionic background near Maxwellians

The two-species Vlasov–Maxwell–Boltzmann system is an important model for plasma physics describing the time evolution of dilute charged particles consisting of electrons and ions under the influence of the self-consistent internally generated Lorentz forces. In physical situations the ion mass is usually much larger than the electron mass so that the electrons move much faster than the ions. Thus, the ions are often described by a fixed ion background and only the electrons move. For such a simple case, the two-species Vlasov–Maxwell–Boltzmann system is reduced to the one-species Vlasov–Maxwell–Boltzmann system.Although the one-species Vlasov–Maxwell–Boltzmann system is a simplified model of the two-species Vlasov–Maxwell–Boltzmann system, its global well-posedness theory near a given global Maxwellian in the perturbative framework is more difficult than the two-species case, which is partially due to the slow-decay of the electromagnetic field and up to now, the problem on the construction of global in time solutions near a given global Maxwellian in the perturbative framework for the Cauchy problem of the one-species Vlasov–Maxwell–Boltzmann system with cutoff non-hard sphere intermolecular collisions remains unsolved. It is shown in this paper that the Cauchy problem of the one-species Vlasov–Maxwell–Boltzmann system with cutoff non-hard sphere intermolecular collisions including the cutoff inverse power law potentials is globally well-posed provided that the perturbative initial data satisfies certain regularity, smallness, and integrability conditions. Our analysis is based on a new time-velocity weighted energy method with two key technical parts: one is to introduce the exponentially weighted estimates into the cutoff Boltzmann operator and the other is to design a delicate temporal energy X(t)-norm to obtain its uniform bound. As a by-product of our analysis, we can also deduce certain temporal decay estimates on the global solutions constructed above for the same range of intermolecular collisions. Notice that even for the hard sphere model, although the global solutions have been successfully constructed in [8], its large time behaviors are unknown, cf. [8].

Spectrum structure and behavior of the Vlasov–Maxwell–Boltzmann system without angular cutoff

The spectrum structure and behavior of the Vlasov–Maxwell–Boltzmann (VMB) system with physical angular non-cutoff intermolecular collisions are studied in this paper. The analysis shows the effect of the Lorentz force induced by the electro-magnetic field leads to some different spectrum structure from the non-cutoff Boltzmann equation. The spectrum structure in high frequency, quite different from the VMB system with angular cutoff assumption, also illustrates the hyperbolic structure of the Maxwell equation. Furthermore, the large time behavior and optimal convergence rates to the equilibrium of the non-cutoff VMB system are established on the spectrum analysis.

Global Solutions and Time Decay of the Non-cutoff Vlasov–Maxwell–Boltzmann System in the Whole Space

The global in time classical solutions to the Cauchy problem of the Vlasov–Maxwell–Boltzmann system near Maxwellians are obtained under the lower regularity index assumption and the weaker smallness condition on the initial perturbation in comparison with the work (Duan et al. in Kinet Relat Models 6(1):159–204, 2013). In particular, we show the relation between time decay rates and spatial derivatives of solutions.Our analysis relies on a refined energy estimates and the interpolation techniques between negative Sobolev norms and positive Sobolev norms without linear decay analysis.

Stability of the relativistic Vlasov–Maxwell–Boltzmann system for short range interaction

The Cauchy problem of the relativistic Vlasov–Maxwell–Boltzmann system for short range interaction is investigated. For perturbative initial data with suitable regularity and integrability, we prove the large time stability of solutions to the relativistic Vlasov–Maxwell–Boltzmann system, and also obtain as a byproduct the convergence rates of solutions. For the proof, a new interactive instant energy functional is introduced to capture the the macroscopic dissipation and the very weak electro-magnetic dissipation of the linearized system. A refined time–velocity weighted energy method is also applied to compensate the weaker dissipation of the linearized collision operator in the case of non-hard potential models. The results also extend the case of “hard ball” model considered by Guo and Strain (2012) to the short range interactions.

The Boltzmann equation with frictional force for soft potentials in the whole space

This paper is concerned with the Cauchy problem of the Boltzmann equation with frictional force for both cutoff and non-cutoff soft potentials in the whole space and our main purpose is to establish its global solvability result near a given global Maxwellian and to deduce the temporal convergence rates of such a global solution toward the global Maxwellian when initial perturbation is sufficiently small. The analysis is based on the time-weighted energy method building also upon the recent studies of the cutoff Vlasov–Poisson–Boltzmann system [14,15], the non-cutoff Vlasov–Poisson–Boltzmann system [8], and non-cutoff Vlasov–Maxwell–Boltzmann system [9].

Negative Sobolev spaces and the two-species Vlasov–Maxwell–Landau system in the whole space

A global solvability result of the Cauchy problem of the two-species Vlasov–Maxwell–Landau system near a given global Maxwellian is established by employing an approach different than that of [2]. Compared with that of [2], the minimal regularity index and the smallness assumptions we imposed on the initial data are weaker. Our analysis does not rely on the decay of the corresponding linearized system and the Duhamel principle and thus it can be used to treat the one-species Vlasov–Maxwell–Landau system for the case of γ>−3 and the one-species Vlasov–Maxwell–Boltzmann system for the case of −1<γ≤1 to deduce the global existence results together with the corresponding temporal decay estimates.

Dissipation of one-species VMB system with non-uniform ionic background

The Vlasov–Maxwell–Boltzmann system is a system that describes the dynamics of charged particles when there is electromagnetic field. There have been extensive studies on the energy estimates and global existence of solutions to the one- and two-species Vlasov–Maxwell–Boltzmann system when the background density is constant. In this work, we first deal with the energy estimates of the one-species Vlasov–Maxwell–Boltzmann system when the background density is a small perturbation from the stationary state. Then, we handle the dissipation rate for the linearized one-species Vlasov–Maxwell–Boltzmann system with the background density having a small perturbation around the stationary state.

Solutions of the Vlasov–Maxwell–Boltzmann system with long-range interactions

We establish the existence of renormalized solutions of the Vlasov–Maxwell–Boltzmann system with a defect measure in the presence of long-range interactions. We also present a control of the defect measure by the entropy dissipation only, which turns out to be crucial in the study of hydrodynamic limits.

Dissipative Property of the Vlasov–Maxwell–Boltzmann System with a Uniform Ionic Background

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 naturally existing ones are designed to capture the optimal dissipation rate of the system, which in turn yields the optimal $L^p$-$L^q$ type time-decay estimates of the corresponding linearized solution operator. These results show a special feature of the one-species Vlasov-Maxwell-Boltzmann system different from the case of two-species, that is, the dissipation of the magnetic field in one-species is strictly weaker than the one in two-species. As a by-product, the global existence of solutions to the nonlinear Cauchy problem is also proved by constructing some similar energy functionals but the time-decay rates of the obtained solution still remain open.

Vlasov–Maxwell–Boltzmann Diffusive Limit

Inspired by the work (Bastea et al. in J Stat Phys 1011087–1136, 2000) for binary fluids, 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 PDEs, 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 an expansion up to any order.

The Vlasov–Maxwell–Boltzmann System in the Whole Space

The Vlasov–Maxwell–Boltzmann system is one of the most fundamental models to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove existence of global in time classical solutions to the Cauchy problem near Maxwellians.