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 an expansion up to any order.
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…
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.
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 $L^2_x$.
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 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.
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…
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 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…
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, …
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^…
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…
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…
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…
The Vlasov-Maxwell-Boltzmann system is a fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians.
We establish the exponential time decay rate of smooth solutions of small amplitude to the Vlasov-Poisson-Fokker-Planck equations to the Maxwellian both in the whole space and in the periodic box via the uniform-in-time energy estimates and also the macroscopic equations.
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.
The hydrodynamic limit of the Vlasov-Maxwell-Boltzmann equations is considered for weak solutions. Using relative entropy estimate about an absolute Maxwellian, an incompressible Electron-Magnetohydrodynamics-Fourier limit for solutions of the Vlasov-Maxwell-Blotzmann equations over any periodic spatial domain in $\R^3$ is studied. It is shown that any properly scaled sequence of renormalized solutions of the Vlasov-Maxwell-Boltzmann equations has fluctuations that (in the weak $L^1$ topology) converge to an infinitesimal Maxwellian with fluid variables that satisfy the incompressibility and B…
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…
Following closely the classical works [8]-[12] by Glassey, Strauss, and Schaeffer, we present a version of the Glassey-Strauss representation for the Vlasov-Maxwell systems in a 3D half space when the boundary is the perfect conductor.
For the whole range of cutoff intermolecular interactions, we give a rigorous mathematical justification of the limit from the Vlasov-Maxwell-Boltzmann system to the Vlasov-Poisson-Boltzmann system as the light speed tends to infinity.Such a limit is shown to hold global-in-time in the perturbative framework.The key point in our analysis is to deduce certain a priori estimates which are independent of the light speed and the main difficulty is due to the degeneracy of the dissipative effect of the electromagnetic field for large light speed.
We consider the one and one-half dimensional multi-species relativistic Vlasov-Maxwell system with non-decaying(in space) initial data. We prove its well-posedness and nonrelativistic limit as the speed of light $c\rightarrow\infty$. These results mainly rely on a delicate analysis of energy structure and application of estimates along the characteristic lines.
We show that weak solutions of the relativistic Vlasov-Maxwell system preserve the total energy provided that the electromagnetic field is locally of bounded variation and, for any $λ$> 0, the one-particle distribution function has a square integrable $λ$-moment in the momentum variable.
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 is widely used in the literature, is not imposed on the initial data. Our proofs are based on a combination of the Klainerman vector fields method and the Fourier method, which allows us to exploit a crucial hidden null structure in the relativistic Vlasov-Maxwell system.
We consider linear stability of steady states of 1(1/2) and 3D Vlasov-Maxwell systems for collisionless plasmas. The linearized systems can be written as separable Hamiltonian systems with constraints. By using a general theory for separable Hamiltonian systems, we recover the sharp linear stability criteria obtained previously by different approaches. Moreover, we obtain the exponential trichotomy estimates for the linearized Vlasov-Maxwell systems in both relativistic and nonrelativistic cases.
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 electro-magnetic field leads to some different structure of spectrum from the classical Boltzmann equation and the closely related Vlasov-Poisson-Boltzmann 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…
It is shown that the "tangential" electric and magnetic fields, in the Glassey-Strauss representation formulas, are in fact bounded in $L^\infty_{{\rm loc},\,t} L^{2+δ}_x$ for some $δ>0$.
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.
We prove almost sharp decay estimates for the small data solutions and their derivatives of the Vlasov-Maxwell system in dimension $n \geq 4$. The smallness assumption concerns only certains weighted $L^1$ or $L^2$ norms of the initial data. In particular, no compact support assumption is required on the Vlasov or the Maxwell fields. The main ingredients of the proof are vector field methods for both the kinetic and the wave equations, null properties of the Vlasov-Maxwell system to control high velocities and a new decay estimate for the velocity average of the solution of the relativistic ma…
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 [5]. Compared with that of [5], 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 …
We prove a global-in-time limit from the two-species Vlasov-Maxwell-Boltzmann system 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, 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 [N. Jiang and Y.-L. Luo, \emph{Ann. PDE} 8 (20…
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 in…
The Cauchy problem for the Vlasov-Maxwell-Boltzmann equations (VMB) is considered. First the renormalized solution to the Vlasov equation with the Lorentz force is discussed and the difficulty on the partial differentiability of the coefficients is overcome. Then the weak stability of the renormalized solutions to the Cauchy problem of VMB is established using the compactness of velocity averages and a renormalized formulation. Furthermore, the large time behavior of the renormalized solutions to VMB is studied and it is proved that the density of particles tends to a local Maxwellian as the t…
We prove global existence of the $3D$ relativistic Vlasov-Maxwell system for a class of arbitrary large regular initial data with spherical symmetry, in which the initial distribution function of particles is assumed to decay fast but polynomially towards infinity.
In this paper, we study the controllability of the two-dimensional relativistic Vlasov-Maxwell system in a torus, by means of an interior control. We give two types of results. With the geometric control condition on the control set, we prove the local exact controllability of the system in large time. Our proof in this case is based on the return method, on some results on the control of the Maxwell equations, and on a suitable approximation scheme to solve the non-linear Vlasov-Maxwell system on the torus with an absorption procedure. Without geometric control condition, but assuming that a …
We study the "one and one-half" dimensional Vlasov-Maxwell-Fokker-Planck system and obtain the first results concerning well-posedness of solutions. Specifically, we prove the global-in-time existence and uniqueness in the large of classical solutions to the Cauchy problem and a gain in regularity of the distribution function in its momentum argument.
R. Glassey and W. Strauss have proved in [Arch. Rational Mech. Anal. 92 (1986), 59--90] that classical solutions to the relativistic Vlasov-Maxwell system in three space dimensions do not develop singularities as long as the support of the distribution function in the momentum variable remains bounded. The present paper simplifies their proof.
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 hop…
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.
