High velocity particles in a collisionless plasma

We prove that smooth solutions of the relativistic Vlasov-Maxwell equations exist for all time provided the kinetic energy density is uniformly bounded.     

Solutions périodiques en temps des équations de Vlasov–Maxwell

We study here the existence of time periodic solution for the Vlasov–Maxwell equations in a three dimensional bounded domain. We assume that the boundary of the domain is strictly star-shaped. We give a priori estimates for the kinetic and electro-magnetic energy, and also for the normal and tangential traces of the electro-magnetic field. This method allows us to treat both classical and relativistic cases. To cite this article: M. Bostan, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Relativistic Scott correction for atoms and molecules

We prove the first correction to the leading Thomas‐Fermi energy for the ground state energy of atoms and molecules in a model where the kinetic energy of the electrons is treated relativistically. The leading Thomas‐Fermi energy, established in [25], as well as the correction given here, are of semiclassical nature. Our result on atoms and molecules is proved from a general semiclassical estimate for relativistic operators with potentials with Coulomb‐like singularities. This semiclassical estimate is obtained using the coherent state calculus introduced in [36]. The paper contains a unified treatment of the relativistic as well as the nonrelativistic case. © 2009 Wiley Periodicals, Inc.     

Kinetic Theory of Quantum Electrodynamic Plasma in a Strong Electromagnetic Field: I. The Covariant Formalism

We present a covariant approach to the kinetic theory of quantum electrodynamic plasma in a strong electromagnetic field. The method is based on the relativistic von Neumann equation for the nonequilibrium statistical operator defined on spacelike hyperplanes in Minkowski space. We use the canonical quantization of the system on hyperplanes and a covariant generalization of the Coulomb gauge. The condensate mode associated with the mean electromagnetic field is separated from the photon degrees of freedom by a time-dependent unitary transformation of the dynamic variables and the nonequilibrium statistical operator. This allows using expansions of correlation functions and of the statistical operator in powers of the fine structure constant even in the presence of a strong electromagnetic field. We present a general scheme for deriving kinetic equations in the hyperplane formalism.     

Cosmological models based on a statistical system of scalar charged degenerate fermions and an asymmetric Higgs scalar doublet

Theoretical and Mathematical Physics - On the basis of the general relativistic statistical and kinetic theory, a consistent closed cosmological model is formulated. It is based on a statistical...     

A study of current generation by electron cyclotron waves in magnetic-surface-averaged kinetic model

A two-dimensional magnetic-surface-averaged kinetic model is constructed for current generation by electron cyclotron waves in tokamak plasma. The model allows for toroidal effects, localization of the injected RF energy on a given magnetic surface, and also relativistic effects. Numerical results are reported for the dependence of current generation efficiency on a wide selection of parameters.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach Matematicheskoi Fiziki, pp. 183–200, 1993.     

Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space

We study the hypocoercivity property for some kinetic equations in the whole space and obtain the optimal convergence rates of solutions to the equilibrium state in some function spaces. The analysis relies on the basic energy method and the compensating function introduced by Kawashima to the classical Boltzmann equation and developed by Glassey and Strauss in the relativistic setting. It is also motivated by the recent work (Duan et al., 2008 [8]) on the Boltzmann equation by combining the spectrum analysis and energy method. The advantage of the method introduced in this paper is that it can be applied to some complicated system whose detailed spectrum is not known. In fact, only some estimates through the Fourier transform on the conservative transport operator and the dissipation of the linearized operator on the subspace orthogonal to the collision invariants are needed.     

Obliquely propagating shock-like structures and nonlinear solitary holes in weakly relativistic magneto-plasma

A theoretical investigation is carried out for understanding the properties of obliquely propagating shock-like structures in weakly relativistic magneto-plasma. By using the Sagdeev's pseudo-potential method, we have found the obliquely propagating shock-like solution and the relation between the amplitude, the inverse scale length, the relativistic effects and the effects of obliqueness. It is shown that the shock-like structure is nonlinear extension of the solitary hole having negative trapping parameter in weakly relativistic magneto-plasma.     

On the one- and one-half-dimensional relativistic Vlasov-Fokker-Planck-Maxwell system

In this paper, we study the relativistic Vlasov-Fokker-Planck-Maxwell system in one space variable and two momentum variables. This non-linear system of equations consists of a transport equation for the phase space distribution function combined with Maxwell's equations for the electric and magnetic fields. It is important in modelling distribution of charged particles in the kinetic theory of plasma. 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.     

Kinetic schemes for the relativistic gas dynamics

Summary. A kinetic solution for the relativistic Euler equations is presented. This solution describes the flow of a perfect gas in terms of the particle density n, the spatial part of the four-velocity u and the inverse temperature . In this paper we present a general framework for the kinetic scheme of relativistic Euler equations which covers the whole range from the non-relativistic limit to the ultra-relativistic limit. The main components of the kinetic scheme are described now. (i) There are periods of free flight of duration _M, where the gas particles move according to the free kinetic transport equation. (ii) At the maximization times t_n=n_M, the beginning of each of these free-flight periods, the gas particles are in local equilibrium, which is described by Jüttners relativistic generalization of the classical Maxwellian phase density. (iii) At each new maximization time t_n>0 we evaluate the so called continuity conditions, which guarantee that the kinetic scheme satisfies the conservation laws and the entropy inequality. These continuity conditions determine the new initial data at t_n. iv If in addition adiabatic boundary conditions are prescribed, we can incorporate a natural reflection method into the kinetic scheme in order to solve the initial and boundary value problem. In the limit _M0 we obtain the weak solutions of Eulers equations including arbitrary shock interactions. We also present a numerical shock reflection test which confirms the validity of our kinetic approach. Mathematics Subject Classification (1991):65M99, 76Y05This work is supported by the project Long-time behaviour of nonlinear hyperbolic systems of conservation laws and their numerical approximation, contract # DFG WA 633/7-2.     

Evidence for the charged θ-meson

A K-meson is found to decay at rest into a nearly relativistic secondary particle. The secondary particle produces a nuclear disintegration in flight and is identified as a π-meson of kinetic energy ～ 110 MeV. The event is interpreted as the decay of a θ^±-meson according to the scheme:

$$\theta ^ \pm \to \pi ^ \pm + \pi ^0 + (222 \pm 12 MeV.)$$     

Quasi-exact solution of the problem of relativistic bound states in the (1+1)-dimensional case

We investigate the problem of bound states for bosons and fermions in the framework of the relativistic configurational representation with the kinetic part of the Hamiltonian containing purely imaginary finite shift operators e^±ihd/dx instead of differential operators. For local (quasi)potentials of the type of a rectangular potential well in the (1+1)-dimensional case, we elaborate effective methods for solving the problem analytically that allow finding the spectrum and investigating the properties of wave functions in a wide parameter range. We show that the properties of these relativistic bound states differ essentially from those of the corresponding solutions of the Schrödinger and Dirac equations in a static external potential of the same form in a number of fundamental aspects both at the level of wave functions and of the energy spectrum structure. In particular, competition between ? and the potential parameters arises, as a result of which these distinctions are retained at low-lying levels in a sufficiently deep potential well for ? ? 1 and the boson and fermion energy spectra become identical.     

相对论性Birkhoff系统动力学方程的代数结构与Poisson积分方法

定义相对论性Ｐｆａｆｆ作用量,得到相对论性Ｐｆａｆｆ Ｂｉｒｋｈｏｆｆ原理和相对论性Ｂｉｒｋｈｏｆｆ方程．证明了自治形式和半自治形式的相对论性Ｂｉｒｋｈｏｆｆ方程具有相容代数结构和Ｌｉｅ代数结构;一般非 自治形式的相对论性Ｂｉｒｋｈｏｆｆ方程没有代数结构．研究一种特殊的非自治形式的相对论性Ｂｉｒｋｈｏｆｆ方程,它具有相容代数结构和Ｌｉｅ容许代数结构．给出相对论性Ｂｉｒｋｈｏｆｆ方程的Ｐｏｉｓｓｏｎ积分 方法．最后给出应用性实例．     

Smooth solutions for one‐dimensional relativistic radiation hydrodynamic equations

In this research article, we investigated the existence of local smooth solutions for relativistic radiation hydrodynamic equations in one spatial variable. The proof is based on a classical iteration method and the Banach contraction mapping principle. However, because of the complexity of relativistic radiation hydrodynamics equations, we first rewrite this system into a semilinear form to construct the iteration scheme and then use left eigenvectors to decouple the system instead of applying standard energy method on symmetric hyperbolic systems. Different from multidimensional case, we just use the characteristic method, which can keep the properties of the initial data. Copyright © 2015 John Wiley & Sons, Ltd.     

The <Emphasis Type="Italic">k</Emphasis>-Essence in the Relativistic Theory of Gravitation and General Relativity

We consider a model of a scalar field with a nontrivial kinetic part (k-essence) on the background of a flat homogeneous isotropic universe in the framework of the relativistic theory of gravitation and general relativity. Such a scalar field simulates the substance of an ideal fluid and serves as a model of dark energy because it leads to cosmological acceleration at later times. For finding a suitable cosmological scenario, it is more convenient to determine the dependence of the energy density of such a field on the scale factor and only then find the corresponding Lagrangian. Based on the solution of such an inverse problem, we show that in the relativistic theory of gravitation, either any scalar field of this type leads to instabilities, or the compression stage ends at an unacceptably early stage. We note that a consistent model of dark energy in the relativistic theory of gravitation can be a scalar field with a negative potential (ekpyrosis) of Steinhardt–Turok. In general relativity, the k-essence model is viable and can represent both dark energy and dark matter. We consider several specific k-essence models.     

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.     

On the determination of approximate periodic solutions of some non-linear ODE

We propose a numerical-symbolic method for the approximation of periodic solutions of a type of non-linear ODE. The efficiency of our method is contrasted with the harmonic balance method and with another one which combines the differential transformation method with Padé approximants on a non trivial example: the relativistic oscillator. It is shown that our method is computationally more reliable.     

Characteristic decompositions and boundary value problems for two‐dimensional steady relativistic Euler equations

In this paper, we investigate Goursat problems, and mixed initial and boundary value problems for the two‐dimensional steady relativistic Euler equations. The global existence of classical solutions to these problems are obtained by using the characteristic decomposition method. Some applications of these results in supersonic flow in two‐dimensional ducts and the two‐dimensional relativistic jet are discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

The Vlasov–Maxwell–Fokker–Planck system in two space dimensions

We consider the Cauchy problem for the Vlasov–Maxwell–Fokker–Planck system in the plane. It is shown that for smooth initial data, as long as the electromagnetic fields remain bounded, then their derivatives do also. Glassey and Strauss have shown this to hold for the relativistic Vlasov–Maxwell system in three dimensions, but the method here is totally different. In the work of Glassey and Strauss, the relativistic nature of the particle transport played an essential role. In this work, the transport is nonrelativistic, and smoothing from the Fokker–Planck operator is exploited. Copyright © 2015 John Wiley & Sons, Ltd.     

A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation

The numerical approximation of one-dimensional relativistic Dirac wave equations is considered within the recent framework consisting in deriving local scattering matrices at each interface of the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously shown that such a discretization preserves exactly the \(L^2\) norm despite being explicit in time. This construction is well-suited for particles for which the reference velocity is of the order of \(c\), the speed of light. Moreover, when \(c\) diverges, that is to say, for slow particles (the characteristic scale of the motion is non-relativistic), Dirac equations are naturally written so as to let a “diffusive limit” emerge numerically, like for discrete 2-velocity kinetic models. It is shown that an asymptotic-preserving scheme can be deduced from the aforementioned well-balanced one, with the following properties: it yields unconditionally a classical Schrödinger equation for free particles, but it handles the more intricate case with an external potential only conditionally (the grid should be such that \(c \Delta x\rightarrow 0\)). Such a stringent restriction on the computational grid can be circumvented easily in order to derive a seemingly original Schrödinger scheme still containing tiny relativistic features. Numerical tests (on both linear and nonlinear equations) are displayed.