Solutions Of Vlasov Research Articles

On Global Classical Limit of the (Relativistic) Vlasov–Maxwell System

It is shown that solutions of the (relativistic) Vlasov–Maxwell system converge pointwise to solutions of the Vlasov–Poisson system globally in time at the asymptotic rate of c^{-1}, as the light speed c tends to infinity, which extends the results of Asano and Ukai (Stud Math Appl 18:369–383, 1986), Degond (Math Methods Appl Sci 8:533–558, 1986) and Schaeffer (Commun Math Phys 104:403–421, 1986). The analysis relies on the method of Glassey and Strauss (Commun Math Phys 113:191–208, 1987) and Schaeffer (Commun Math Phys 104:403–421, 1986).

Superconvergence and accuracy enhancement of discontinuous Galerkin solutions for Vlasov–Maxwell equations

Superconvergence and accuracy enhancement of discontinuous Galerkin solutions for Vlasov–Maxwell equations

Global existence and decay rates of solutions for Vlasov–Navier–Stokes–Fokker–Planck equations with magnetic field

In this paper, we consider the two-phase flow model coupled by the incompressible magnetohydrodynamic equations and the discrete-phase (particle) dynamics equation Vlasov–Fokker–Planck in [Formula: see text]. When the initial value is a small perturbation of an equilibrium state, the global well-posedness of the strong solution in the three-dimensional global space is established by using the optimal decay estimation and energy estimation. In particular, we are able to show the algebraic convergence rate of the solution to the equilibrium state. While for periodic domain, the same result still holds, but the rate of convergence is exponential.

A low rank tensor representation of linear transport and nonlinear Vlasov solutions and their associated flow maps

We propose a low-rank tensor approach to approximate linear transport and nonlinear Vlasov solutions and their associated flow maps. The approach takes advantage of the fact that the differential operators in the Vlasov equation are tensor friendly, based on which we propose a novel way to dynamically and adaptively build up low-rank solution basis by adding new basis functions from discretization of the PDE, and removing basis from an SVD-type truncation procedure. For the discretization, we adopt a high order finite difference spatial discretization and a second order strong stability preserving multi-step time discretization. We apply the same procedure to evolve the dynamics of the flow map in a low-rank fashion, which proves to be advantageous when the flow map enjoys the low rank structure, while the solution suffers from high rank or displays filamentation structures. Hierarchical Tucker decomposition is adopted for high dimensional problems. An extensive set of linear and nonlinear Vlasov test examples are performed to show the high order spatial and temporal convergence of the algorithm with mesh refinement up to SVD-type truncation, the significant computational savings of the proposed low-rank approach especially for high dimensional problems, the improved performance of the flow map approach for solutions with filamentations.

Exact solutions for thin-walled composite open section beams using a unified state space coupled field formulation

Thin-walled open section composite beams (TWOSCB) used extensively in aerospace structures exhibit several non-classical effects such as torsional warping, warping shear, material coupling, and shear deformation. Arbitrary cross-section and material coupling make the governing equations highly coupled and difficult to solve analytically. In this paper, a unified coupled-field formulation is developed to analyze TWOSCB accommodating all non-classical effects. Using the state-space method to derive analytical expressions for displacements and forces, Vlasov solutions of isotropic beams are recovered. Results obtained for arbitrarily laminated mono-symmetric composite channel and I-section beams subjected to bending and torsion loads show excellent agreement with available results in the literature.

Temporal decays and asymptotic behaviors for a Vlasov equation with a flocking term coupled to incompressible fluid flow

We are concerned with large-time behaviors of solutions for Vlasov–Navier–Stokes equations in two dimensions and Vlasov–Stokes system in three dimensions including the effect of velocity alignment/misalignment. We first revisit the large-time behavior estimate for our main system and refine assumptions on the dimensions and a communication weight function. In particular, this allows us to take into account the effect of the misalignment interactions between particles. We then use a sharp heat kernel estimate to obtain the exponential time decay of fluid velocity to its average in L∞-norm. For the kinetic part, by employing a certain type of Sobolev norm weighted by modulations of averaged particle velocity, we prove the exponential time decay of the particle distribution, provided that local particle distribution function is uniformly bounded. Moreover, we show that the support of particle distribution function in velocity shrinks to a point, which is the mean of averaged initial particle and fluid velocities, exponentially fast as time goes to infinity. This also provides that for any p∈[1,∞], the p-Wasserstein distance between the particle distribution function and the tensor product of the local particle distributions and Dirac measure at that point in velocity converges exponentially fast to zero as time goes to infinity.

The generalized Friedmann model as a self-similar solution of Vlasov–Poisson equation system

The generalized Friedmann model as a self-similar solution of Vlasov–Poisson equation system

A statistical description for the Quasi-Stationary-States of the dipole-type Hamiltonian Mean Field Model based on a family of Vlasov solutions

For non-equilibrium quasi-stationary-states (QSS), we present a theoretical background based on a family of Vlasov equation solutions constructed by non-Gaussian distributions. Proposing a transformation, we connect the Vlasov stationary solutions to a non-standard theoretical perspective. Such one is suitable to describe the QSS involved in the d-HMF model, which occur while the system evolves towards equilibrium. Our results complement the notion of Tsallis formalism and represent input on a theoretical description of the systems behavior with long-range interactions out of equilibrium.

Sharp Asymptotics for the Solutions of the Three-Dimensional Massless Vlasov\u2013Maxwell System with Small Data

This paper is concerned with the asymptotic properties of the small data solutions to the massless Vlasov–Maxwell system in 3d. We use vector field methods to derive almost optimal decay estimates in null directions for the electromagnetic field, the particle density and their derivatives. No compact support assumption in x or v is required on the initial data, and the decay in v is in particular initially optimal. Consistently with Proposition 8.1 of Bigorgne (Asymptotic properties of small data solutions of the Vlasov–Maxwell system in high dimensions. arXiv:1712.09698, 2017), the Vlasov field is supposed to vanish initially for small velocities. In order to deal with the slow decay rate of the solutions near the light cone and to prove that the velocity support of the particle density remains bounded away from 0, we make crucial use of the null properties of the system.

Sharp Asymptotic Behavior of Solutions of the 3d Vlasov\u2013Maxwell System with Small Data

We study the asymptotic properties of the small data solutions of the Vlasov–Maxwell system in dimension three. No neutral hypothesis nor compact support assumptions are made on the data. In particular, the initial decay in the velocity variable is optimal. We use vector field methods to obtain sharp pointwise decay estimates in null directions on the electromagnetic field and its derivatives. For the Vlasov field and its derivatives, we obtain, as in Fajman et al. (The Stability of the Minkowski space for the Einstein-Vlasov system, 2017. arXiv:1707.06141), optimal pointwise decay estimates by a vector field method where the commutators are modification of those of the free relativistic transport equation. In order to control high velocities and to deal with non integrable source terms, we make fundamental use of the null structure of the system and of several hierarchies in the commuted equations.

Uniformly accurate methods for Vlasov equations with non-homogeneous strong magnetic field

In this paper, we consider the numerical solution of highly-oscillatory Vlasov and Vlasov-Poisson equations with non-homogeneous magnetic field. Designed in the spirit of recent uniformly accurate methods, our schemes remain insensitive to the stiffness of the problem, in terms of both accuracy and computational cost. The specific difficulty (and the resulting novelty of our approach) stems from the presence of a non-periodic oscillation, which necessitates a careful ad-hoc reformulation of the equations. Our results are illustrated numerically on several examples.

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].

Kinetics of the Raman scattering in a laser corona using a transform method

AbstractThis paper is an extension of our previous paper (Mašek and Rohlena, 2015), where we applied a transform method for the solution of Vlasov–Maxwell set of equations in a one-dimensional geometry to describe the Raman backscattering of the heating ns laser wave in the external corona of the generated laser plasma in a strongly non-linear regime. The method is stabilized by a simplified Fokker–Planck collision term, which, in turn, is used for a study of the influence of collisional and collisionless damping mechanisms of the daughter electron plasma wave (EPW) on the instability development and their competition resulting in a different instability behavior in various plasma configurations. The physics of trapped electrons is studied in detail and compared to the resulting Raman reflectivity. The Raman reflectivity was found to depend strongly on the intensity of laser irradiation in the different regions of the plasma corona. This is discussed in detail from the point of view of trapped electrons behavior in the EPW. Moreover, a study of the Raman reflectivity dependence on the electron–ion collision frequency (average plasma ionization) is presented, too. The results supplement the physical picture of the collision and collisionless processes influencing the Raman instability non-linear development.

A new class of exact solutions for Vlasov–Poisson plasmas

A new class of exact solutions of the Vlasov–Poisson equations based on the Jeans’ theorem is considered. The self-consistent solutions for the evolution of the system from given initial particle distributions make use of particular invariants of particle motion with predetermined spatial structure. The relation between the time-dependent coefficients of the latter and the macroscopic plasma quantities, or moments of the distribution function, are obtained. As an illustration, the evolution of an expanding plasma with oscillations of the electron and ion expansion fronts is presented.

APPLICATION OF THE GALERKIN-VLASOV METHOD TO THE FLEXURAL ANALYSIS OF SIMPLY SUPPORTED RECTANGULAR KIRCHHOFF PLATES UNDER UNIFORM LOADS

Plates are important structural elements used to model bridge decks, retaining walls, floor slabs, spacecraft panels, aerospace structures, and ship hulls amongst. Plates have been modelled using three dimensional elasticity theory, Reissner’s theory, Kirchhoff theory, Shimpi’s theory, Von Karman’s theory, etc. The resulting plate equations have also been solved using classical and numerical techniques.In this research, the Galerkin-Vlasov variational method was used to present a general formulation of the Kirchhoff plate problem with simply supported edges and under distributed loads. The problem was then solved to obtain the displacements, and the bending moments in a Kirchhoff plate with simply supported edges and under uniform load. Maximum values of the displacement and the bending moments were found to occur at the plate center. The Galerkin Vlasov solutions for a rectangular simply supported Kirchhoff plate carrying uniform load was found to be exactly identical with the Navier double trigonometric series solution. http://dx.doi.org/10.4314/njt.v35i4.7

Collisionless self-gravitating statistical systems of scalarly interacting particles

We consider the theory of collisionless statistical systems with interparticle scalar interaction. A mathematical model of such systems is constructed, and an exact solution of the Vlasov equation for the isotropic homogenous model of the Universe is found. Asymptotic solutions of the self-consistent Vlasov–Einstein model for conformally invariant scalar interactions are found. We obtain and study an exact cosmological solution of a self-consistent set of equations consisting of a collisionless kinetic equation for a plasma with interparticle scalar interaction and the scalar field equation with a source. It is shown that in the ultrarelativistic limit the scalar interaction of particles leads to generation of an effective mass of scalar bosons.

Mathematical and numerical methods for Vlasov–Maxwell equations: The contribution of data mining

This paper deals with the applications of data mining techniques in the evaluation of numerical solutions of Vlasov–Maxwell models. This is part of the topic of characterizing the model and approximation errors via learning techniques. We give two examples of application. The first one aims at comparing two Vlasov–Maxwell approximate models. In the second one, a scheme based on data mining techniques is proposed to characterize the errors between a P1 and a P2 finite element Particle-In-Cell approach. Beyond these examples, this original approach should operate in all cases where intricate numerical simulations like for the Vlasov–Maxwell equations take a central part.

First order Two-Scale Particle-in-Cell numerical method for the Vlasov equation

The aim of this work is to build an accurate numerical method for the simulation of the long time evolution of the Vlasov solution $\feps$ with an electric field $\Eeps = E_0+ \eps E_1$ for small $\eps$. To this purpose, we use the Two-Scale Convergence theory to determine a first order approximation $F+\eps F_1$ of $\feps$\!, then particle approximations to build an algorithm to obtain a numerical approximation of $F+\eps F_1$.

Sectionally analytic solutions of the Vlasov–Poisson equations

We investigate the solutions of the kinetic Vlasov–Poisson equations, which govern a plasma made of electrons and one species of mobile ions, in one rectangular dimension. We present a new formulation of Poisson’s equation as an integral inverse problem. We prove inversion formulas which allow us to write the solution of this problem in such a way that the energy distribution of either of the particle species is related, in a straightforward way, to the energy distribution of the other species. We show that these distributions are retrieved from the boundary values of suitable sectionally analytic functions. These latter functions are the extension of the particle distributions into their respective complex energy domains.

Nonlinear Stage of Alfvén Ion-Cyclotron Instability

A model describing nonlinear saturated state of Alfvén ion-cyclotron instability is presented. Spirally symmetric exact solutions of Vlasov – Maxwell equations describing nonlinear Alfvén waves in collisionless plasmas are presented. On their basis the equation describing ion distribution function in non-linear equilibrium of a weak-collision plasma with the wave is obtained, taking into account an uniform fast ion injection, ion drag caused by electrons and neglecting ion angle scattering. A solution of this equation and relations of the wave parameters with the injection parameters are obtained analytically in the case of the infinitely narrow velocity distribution of the injected ions