On the one and one-half dimensional relativistic Vlasov–Maxwell–Fokker–Planck system with non-vanishing viscosity

The relativistic Vlasov–Maxwell–Fokker–Planck system is used in modelling distribution of charged particles in plasma. It consists of a transport equation coupled with the Maxwell system. The diffusion term in the equation models the collisions among particles, whereas the viscosity term signifies the dynamical frictional forces between the particles and the background reservoir. In the case of one space variable and two momentum variables, we prove the existence of a classical solution when the initial density decays fast enough with respect to the momentum variables. The solution which shares this same decay condition along with its first derivatives in the momentum variables is unique. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Classical solvability of the relativistic Vlasov–Maxwell system with bounded spatial density

In (Arch. Rat. Mech. Anal. 1986; 92:59–90), Glassey and Strauss showed that if the growth in the momentum of the particles is controlled, the relativistic Vlasov–Maxwell system has 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. Copyright © 2009 John Wiley & Sons, Ltd.     

Linearized quantum and relativistic Fokker–Planck–Landau equations

After a recent work on spectral properties and dispersion relations of the linearized classical Fokker–Planck–Landau operator [8], we establish in this paper analogous results for two more realistic collision operators: The first one is the Fokker–Planck–Landau collision operator obtained by relativistic calculations of binary interactions, and the second is a collision operator (of Fokker–Planck–Landau type) derived from the Boltzmann operator in which quantum effects have been taken into account. We apply Sobolev–Poincaré inequalities to establish the spectral gap of the linearized operators. Furthermore, the present study permits the precise knowledge of the behaviour of these linear Fokker–Planck–Landau operators including the transport part. Relations between the eigenvalues of these operators and the Fourier‐space variable in a neighbourhood of 0 are then investigated. This study is a first natural step when one looks for solutions near equilibrium and their hydrodynamic limit for the full non‐linear problem in all space in the spirit of several works [3, 6, 20, 2] on the non‐linear Boltzmann equation. Copyright © 2000 John Wiley & Sons, Ltd.     

The classical limit of the relativistic Vlasov–Maxwell system in two space dimensions

The motion of a collisionless plasma is modelled by the Vlasov–Maxwell system. In this paper, solutions of the relativistic Vlasov–Maxwell system are considered in two space dimensions. The speed of light, c, appears as a parameter in the system. With representations of the electric and magnetic fields, conditions are established under which solutions of the relativistic Vlasov–Maxwell system converge pointwise to solutions of the non‐relativistic Vlasov–Poisson system as c tends to infinity, at the asymptotic rate of 1/c. Copyright © 2004 John Wiley & Sons, Ltd.     

Steady states in plasma physics—the Vlasov–Fokker–Planck equation

In this paper we investigate the non-linear Vlasov–Fokker–Planck (VFP) equation, a both physically and mathematically interesting modification of Vlasov's equation, which describes a plasma in a thermal bath. We prove existence, uniqueness and representation results for steady states of the VFP equation both in the case of a mollified interaction potential and for the VFP–Poisson system. The uniqueness and representation results are of special interest since they distinguish special solutions of the Vlasov equation.     

Global weak solutions to the Cauchy problem of compressible Navier–Stokes–Vlasov–Fokker–Planck equations

A fluid–particles system of the compressible Navier‐Stokes equations and Vlasov‐Fokker‐Planck equation (including the case of Vlasov equation) in three‐dimensional space is considered in this paper. The coupling arises from a drag force exerted by the fluid onto the particles. We study a Cauchy problem with large data, and establish the existence of global weak solutions through an approximation scheme, energy estimates, and weak convergence. Copyright © 2016 John Wiley & Sons, Ltd.     

L∞‐estimates for the Vlasov–Poisson–Fokker–Planck equation

We consider the Vlasov–Poisson–Fokker–Planck equation in three dimensions as the backward Kolmogorov equation associated to a non‐linear diffusion process. In this way we derive new L^∞‐estimates on the spatial density which are uniform in the diffusion parameters. Copyright © 2000 John Wiley & Sons, Ltd.     

Variational formulations for Vlasov–Poisson–Fokker–Planck systems

Time‐discrete variational schemes are introduced for both the Vlasov–Poisson–Fokker–Planck (VPFP) system and a natural regularization of the VPFP system. The time step in these variational schemes is governed by a certain Kantorovich functional (or scaled Wasserstein metric). The discrete variational schemes may be regarded as discretized versions of a gradient flow, or steepest descent, of the underlying free energy functionals for these systems. For the regularized VPFP system, convergence of the variational scheme is rigorously established. Copyright © 2000 John Wiley & Sons, Ltd.     

Global weak solutions for the initial–boundary-value problems Vlasov–Poisson–Fokker–Planck System

This work is devoted to prove the existence of weak solutions of the kinetic Vlasov–Poisson–Fokker–Planck system in bounded domains for attractive or repulsive forces. Absorbing and reflection-type boundary conditions are considered for the kinetic equation and zero values for the potential on the boundary. The existence of weak solutions is proved for bounded and integrable initial and boundary data with finite energy. The main difficulty of this problem is to obtain an existence theory for the linear equation. This fact is analysed using a variational technique and the theory of elliptic–parabolic equations of second order. The proof of existence for the initial–boundary value problems is carried out following a procedure of regularization and linearization of the problem. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Long-time behaviour for solutions of the Vlasov–Poisson–Fokker–Planck equation

We study the long-time behaviour of solutions of the Vlasov–Poisson–Fokker–Planck equation for initial data small enough and satisfying some suitable integrability conditions. Our analysis relies on the study of the linearized problems with bounded potentials decaying fast enough for large times. We obtain global bounds in time for the fundamental solutions of such problems and their derivatives. This allows to get sharp bounds for the decay of the difference between the solutions of the Vlasov–Poisson–Fokker–Planck equation and the solution of the free equation with the same initial data. Thanks to these bounds, we get an explicit form for the second term in the asymptotic expansion of the solutions for large times. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

On global symmetric solutions to the relativistic Vlasov–Poisson equation in three space dimensions

A collisionless plasma is modelled by the Vlasov–Maxwell system. In the presence of very large velocities, relativistic corrections are meaningful. When magnetic effects are ignored this formally becomes the relativistic Vlasov–Poisson equation. The initial datum for the phase space density ƒ₀(x, v) is assumed to be sufficiently smooth, non‐negative and cylindrically symmetric. If the (two‐dimensional) angular momentum is bounded away from zero on the support of ƒ₀(x, v), it is shown that a smooth solution to the Cauchy problem exists for all times. Copyright © 2001 John Wiley & Sons, Ltd.     

A low velocity approximation for the relativistic Vlasov-Maxwell system

An approximation for the relativistic Vlasov-Maxwell (RVM) system of partial differential equations in the one-space, two-momenta case is proposed. The speed of light, c, appears as a parameter in this system. The approximation is obtained by modifying certain integral operators appearing in integral representations, due to Glassey and Strauss, of the electric and magnetic fields, and replaces the hyperbolic Maxwell system with one that is elliptic in nature (for each fixed t). Solutions of the modified problem are shown to converge in a pointwise sense to solutions of (RVM) at the asymptotic rate of 1/c² as c tends to infinity.     

A two-species plasma in the limit of large ion mass

Classical solutions of the relativistic Vlasov–Maxwell system are considered, describing a collisionless plasma with two species of particles. ions and electrons. It is shown that as the ion mass m tends to infinity, the corresponding solution of the relativistic Vlasov–Maxwell system tends to the solution of a system, in which the ions are given by a fixed ion background and only the electrons move. The convergence is uniform on compact time intervals, with an asymptotic convergence rate of m^?1.     

Homogenized Diffusion Limit of a Vlasov–Poisson–Fokker–Planck Model

The approximation by diffusion and homogenization of the initial-boundary value problem of the Vlasov–Poisson–Fokker–Planck model is studied for a given velocity field with spatial macroscopic and microscopic variations. The L¹-contraction property of the Fokker–Planck operator and a two-scale Hybrid-Hilbert expansion are used to prove the convergence towards a homogenized Drift–Diffusion equation and to exhibit a rate of convergence.     

Exponential convergence toward equilibrium for homogeneous Fokker–Planck-type equations

We consider homogeneous solutions of the Vlasov–Fokker–Planck equation in plasma theory proving that they reach the equilibrium with a time exponential rate in various norms. By Csiszar–Kullback inequality, strong L¹-convergence is a consequence of the ‘sharp’ exponential decay of relative entropy and relative Fisher information. To prove exponential strong decay in Sobolev spaces H^k, k ⩾ 0, we take into account the smoothing effect of the Fokker–Planck kernel. Finally, we prove that in a metric for probability distributions recently introduced in [9] and studied in [4, 14] the decay towards equilibrium is exponential at a rate depending on the number of moments bounded initially. Uniform bounds on the solution in various norms are then combined, by interpolation inequalities, with the convergence in this weak metric, to recover the optimal rate of decay in Sobolev spaces. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons, Ltd.     

The Vlasov–Poisson–Fokker–Planck equation in an interval with kinetic absorbing boundary conditions

We study the initial–boundary value problem for the Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions. We first prove the existence of weak solutions of the linearized equation in an interval with absorbing boundary conditions. Moreover, the weak solution converges to zero exponentially in time. Then we extend the above results to the fully nonlinear Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions; the existence and the longtime behavior of weak solutions. Finally, we prove that the weak solution is actually a classical solution by showing the hypoellipticity of the solution away from the grazing set and the Hölder continuity of the solution up to the grazing set.     

Global existence for the Vlasov–Darwin system in ℝ3 for small initial data

We prove the global existence of weak solutions to the Vlasov–Darwin system in R3 for small initial data. The Vlasov–Darwin system is an approximation of the Vlasov–Maxwell model which is valid when the characteristic speed of the particles is smaller than the light velocity, but not too small. In contrast to the Vlasov–Maxwell system, the total energy conservation does not provide an L2‐bound on the transverse part of the electric field. This difficulty may be overcome by exploiting the underlying elliptic structure of the Darwin equations under a smallness assumption on the initial data. We finally investigate the convergence of the Vlasov–Darwin system towards the Vlasov–Poisson system. Copyright © 2003 John Wiley & Sons, Ltd.     

Instability in a Vlasov–Fokker–Planck binary mixture

This paper is concerned with a kinetic model of a Vlasov–Fokker–Planck system used to describe the evolution of two species of particles interacting through a potential and a thermal reservoir at given temperature. We prove that at low temperature, the homogeneous equilibrium is dynamically unstable under certain perturbations. Our work is motivated by a problem arising in [4].     

On Stationary Solutions of the Relativistic Vlasov–Maxwell System

We study stationary solutions of the relativistic Vlasov–Maxwell system of plasma physics which have a special form introduced (in the classical setting) by Rudykh, Sidorov and Sinitsy and establish their existence under suitable assumptions on the ansatz functions. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. Math. Meth. Appl. Sci., Vol. 20, 667–677 (1997).     

Anomalous transport of particles in plasma physics

We investigate the long time/small mean-free-path asymptotic behavior of the solutions of a Vlasov–Lévy–Fokker–Planck equation and show that the asymptotic dynamics for the VLFP are described by an anomalous diffusion equation.