Talete di Mileto

Large number asympotics for dispersed, non regular distributions of inclusions in a fluid

We shall describe the large number asymtpotics for systems consisting in a fluid surrounding a non regular distributions of inclusions. The fluid satisfies a given boundary (or initial-boundary) value problem (involving a second order operator of divergence form) in a perforated domain and the holes in the domain are occupied by the inclusions; fluid and inclusions are coupled by the boundary conditions only. These systems are meant as models at some "microscopic" scale for describing, in a suitable asymptotics, the behaviour at a macroscopic level of composite media. We shall give a general overview of the results about the homogenized limit of such a kind of systems obtained in previous w…

MR2875734 Brull, Stéphane Ghost effect for a vapor-vapor mixture. Kinet. Relat. Models 5 (2012), no. 1, 21--50. 82Cxx (76P05)

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MR2803047 (2012g:82057) 82D10 Marklund, M.; Morrison, P. J. Gauge-free Hamiltonian structure of the spin Maxwell-Vlasov equations.

Review for Mathematical Reviews

Hydrodynamic limits from multiphase Boltzmann models

We shall describe the validation of hydrodynamic models for thin sprays from a class of multiphase Boltzmann models where the collision kernels share a common structure, and include elastic collisions and some kind of inelastic collisions

Non Markovian Behavior of the Boltzmann-Grad Limit of Linear Stochastic Particle Systems

We will review some results which illustrate how the distribution of obstacles and the shape of the characteristic curves influence the convergence of the probability density of linear stochastic particle systems to the one particle probability density associated with a Markovian process in the Boltzmann-Grad asymptotics.

Zbl 1262.60093 Caputo, Pietro; Lacoin, Hubert; Martinelli, Fabio; Simenhaus, François; Toninelli, Fabio Lucio Polymer dynamics in the depinned phase: metastability with logarithmic barriers. (English) [J] Probab. Theory Relat. Fields 153, No. 3-4, 587-641 (2012).

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MR2884298 (2012k:82047) 82D10 (35B35 35Q83) Stabilité non-linéaire pour l’ équation de Vlasov-Poisson et amortissement Landau.

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MR3191476 Bonnet-Ben Dhia, A.-S.(F-ENSTA3-POE); Chesnel, L.(F-ENSTA3-POE); Ciarlet, P., Jr.(F-ENSTA3-POE) Two-dimensional Maxwell's equations with sign-changing coefficients. (English summary) Appl. Numer. Math. 79 (2014), 29–41. 78A48 (35B30 35Q60 35R25)

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Modelling of systems with dispersed phase

We shall describe some models of systems containing disperse phase and their rigorous derivation as the macroscopic limit of suitable microscopic models in which the dispersed phase is associated to a particle-like component.

MR2787978 (2012m:82050) Kimiagar, S.; Sadegh Movahed, M.; Khorram, S.; Reza Rahimi Tabar, M. Markov properties of electrical discharge current fluctuations in plasma

Review for Mathematical Reviews

Homogenization of equations describing materials interacting with clouds of particles

We shall describe the derivation of homogenized equations (in an asymp- totics of mean-field type) for systems consisting of a cloud of particles dispersed in a material enclosed in a bounded domain. We shall consider in particular the homoge- nization of the Stokes problem leading to the Brinkman force and the homogenization of a model describing the heat exchange between the material and the dispersed phase leading to a (time-dependent) two-temperature equation. In both cases, the homogenized equations can be rigorously derived using similar form for the correctors.

Non Markovianity of the Boltzmann-Grad limit of a system of random obstacles in a given force field

In this paper we consider a particle moving in a random distribution of obstacles. Each obstacle is absorbing and a fixed force field is imposed. We show rigorously that certain (very smooth) fields prevent the process obtained by the Boltzmann-Grad limit from being Markovian. Then, we propose a slightly different setting which allows this difficulty to be removed.

Zbl 1251.60070 Beltr\'an, J.; Landim, C. Metastability of reversible condensed zero range processes on a finite set Probab. Theory Relat. Fields 152, No. 3-4, 781-807 (2012).

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Zbl 1266.35137 Jin, Shi; Wei, Dongming A particle method for the semiclassical limit of the Schr\"odinger equation and the Vlasov-Poisson equations SIAM J. Numer. Anal. 50, No. 6, 3259-3279 (2012)

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Modellistica Differenziale

Traduzione di articolo su enciclopedia dal Francese

Derivation of a Homogenized Two-Temperature Model from the Heat Equation

This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from [D. Cioranescu, F. Murat: Coll\`ege de France Seminar vol. 2. (Paris 1979-1980) Res. Notes in Math. vol. 60, pp. 98-138. Pitman, Boston, London, 1982.]

Niels H.Abel

The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow

We propose a mathematical derivation of Brinkman's force for a cloud of particles immersed in an incompressible viscous fluid. Specifically, we consider the Stokes or steady Navier-Stokes equations in a bounded domain Omega subset of R-3 for the velocity field u of an incompressible fluid with kinematic viscosity v and density 1. Brinkman's force consists of a source term 6 pi rvj where j is the current density of the particles, and of a friction term 6 pi vpu where rho is the number density of particles. These additional terms in the motion equation for the fluid are obtained from the Stokes or steady Navier-Stokes equations set in Omega minus the disjoint union of N balls of radius epsilo…

Two simple criteria to estimate an objective's performance when imaging in non design tissue clearing solutions

Tissue clearing techniques are undergoing a renaissance motivated by the need to image fluorescent neurons, and other cells, deep in the sample without physical sectioning. Optical transparency is achieved by equilibrating tissues with high refractive index (RI) solutions. When the microscope objective is not perfectly matched to the RI of the cleared sample, aberrations are introduced. We present two simple-to-calculate numerical criteria predicting: (i) the degradation in image quality (brightness and resolution) from optimal conditions of any clearing solution/objective combination; (ii) which objective, among several available, achieves the highest resolution in a given medium. We deriv…

Three-dimensional features in burning plasmas

A next major step in the research toward magnetic fusion energy production is to carry out experimental campaigns exploring regimes with relevant amount of fusion power. So far, the theoretical knowledge of the path toward a fusion burning plasma has been acquired mainly by performing numerical studies in 0 or 1-1.5 dimensions. Due to the marked anisotropy of magnetically confined plasmas, however, three-dimensional effects might play a role. In particular, the drastic change in magnetic topology associated with reconnecting modes on selected rational magnetic surfaces [1] may decrease the thermal electron conductivity parallel to the magnetic field lines, with a consequent impact on the el…

MR2864537 (Review) 82D10 Kozlov, Valery V. The Vlasov kinetic equation, dynamics of continuum and turbulence. Regul. Chaotic Dyn. 16 (2011), no. 6, 602--622. 82Dxx (37Axx)

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Zbl 1247.80009 Mahapatra, T.Ray; Nandy, S.K. Stability analysis of dual solutions in stagnation-point flow and heat transfer over a power-law shrinking surface Int. J. Nonlinear Sci. 12, No. 1, 86-94 (2011).

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Zbl 1237.52017 Artamonov, V.A.; Sánchez, S. On finite symmetry groups of some models of three-dimensional quasicrystals. (English. Russian original) [J] Sib. Math. J. 52, No. 6, 969-979 (2011); translation from Sib. Mat. Zh. 52, No. 6, 1221-1233 (2011).

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A Coherent derivation of an average ion model including the evolution of correlations between different shells

We propose in this short note a method enabling to write in a systematic way a set of refined equations for average ion models in which correlations between populations are taken into account, starting from a microscopic model for the evolution of the electronic configura- tion probabilities. Numerical simulations illustrating the improvements with respect to standard average ion models are presented at the end of the paper.

About the validation of models for multicomponent systems

We shall discuss the derivation of systems of partial differ- ential equations, where hydrodynamic equations are coupled to mean field (Vlasov type) kinetic equations, as the asymptotic limit of suitable models on a smaller scale. This kind of PDE systems can be considered as simply modelling multicomponent flows in mixtures containing a dispersed phase, such as sprays or aerosols. In this talk we shall give an overview of results of various type obtained in cooperation with E. Bernard, L. Desvillettes, F. Golse.

Zbl 1288.82002 Hansen, Klavs Statistical physics of nanoparticles in the gas phase. Springer Series on Atomic, Optical, and Plasma Physics 73. Dordrecht: Springer (ISBN 978-94-007-5838-4/hbk; 978-94-007-5839-1/ebook). xi, 282~p.ebook (2013).

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Zbl 1258.35214 Razafimandimby, Paul André; Sango, Mamadou Strong solution for a stochastic model of two-dimensional second grade fluids: existence, uniqueness and asymptotic behavior. (English) [J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 11, 4251-4270 (2012).

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Large Number Asymptotics for Two-Component Systems with Self-Consistent Coupling

We shall consider the large number asymptotics of particle models for partial differential equations describing two component mixtures with simplest kind of self-consistent couplings. We shall recall in particular two examples related to different classes of models, the first one having both particle-like components and the second one having only one particle-like component (the other being described as a fluid); for these examples, different techniques on the probabilistic and analytic point of view are to be used to rigorously prove the convergence to a limit of the self-consistent terms in a “mean-field”-like asymptotics. The two models were analysed resp. in Bernardin and Ricci (Kinet R…

A Formal Passage From a System of Boltzmann Equations for Mixtures Towards a Vlasov-Euler System of Compressible Fluids

A formal asymptotics leading from a system of Boltzmann equations for mixtures towards either Vlasov-Navier-Stokes or Vlasov-Stokes equations of incompressible fluids was established by the same authors and Etienne Bernard in: A Derivation of the Vlasov-Navier-Stokes Model for Aerosol Flows from Kinetic Theory Commun. Math. Sci., 15: 1703–1741 (2017) and A Derivation of the Vlasov-Stokes System for Aerosol Flows from the Kinetic Theory of Binary Gas Mixtures. KRM, 11: 43–69 (2018). With the same starting point but with a different scaling, we establish here a formal asymptotics leading to the Vlasov-Euler system of compressible fluids. Explicit formulas for the coupling terms are obtained i…

Zbl 1289.76014 Banica, Valeria; Miot, Evelyne Evolution, interaction and collisions of vortex filaments. (English) Differ. Integral Equ. 26, No. 3-4, 355-388 (2013).

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Zbl 1269.82035 Islambekov, Umar; Sims, Robert; Teschl, Gerald Lieb-Robinson bounds for the Toda lattice. (English)

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On the derivation of a linear Boltzmann equation from a periodic lattice gas

We consider the problem of deriving the linear Boltzmann equation from the Lorentz process with hard spheres obstacles. In a suitable limit (the Boltzmann-Grad limit), it has been proved that the linear Boltzmann equation can be obtained when the position of obstacles are Poisson distributed, while the validation fails, also for the "correct" ratio between obstacle size and lattice parameter, when they are distributed on a purely periodic lattice, because of the existence of very long free trajectories. Here we validate the linear Boltzmann equation, in the limit when the scatterer's radius epsilon vanishes, for a family of Lorentz processes such that the obstacles have a random distributio…

A DERIVATION OF THE VLASOV-NAVIER-STOKES MODEL FOR AEROSOL FLOWS FROM KINETIC THEORY

This article proposes a derivation of the Vlasov-Navier-Stokes system for spray/aerosol flows. The distribution function of the dispersed phase is governed by a Vlasov-equation, while the velocity field of the propellant satisfies the Navier-Stokes equations for incompressible fluids. The dynamics of the dispersed phase and of the propellant are coupled through the drag force exerted by the propellant on the dispersed phase. We present a formal derivation of this model from a multiphase Boltzmann system for a binary gaseous mixture, involving the droplets/dust particles in the dispersed phase as one species, and the gas molecules as the other species. Under suitable assumptions on the colli…

Zbl 1294.60121 Xiao, Yimin; Zheng, Xinghua Discrete fractal dimensions of the ranges of random walks in $\Bbb Z^d$ associate with random conductances Probab. Theory Relat. Fields 156, No. 1-2, 1-26 (2013).

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About the link between the detailed description of transitions in an ion and the average-ion models

We study the link which exists between microscopic (detailed) models for the evolution of the electronic configurations in a population of ions and the macroscopic (average ion) models. Rigorous asymptotics are presented in situations where they exist (large temperature; almost empty or almost full shells), and numerical simulations are presented.

Malliavin, Paul

Articolo su Enciclopedia

Large number asymptotics of two-component systems

We shall analyze the asymptotics of two-component systems with at least one particle component when the number of particles becomes large; choosing suitable scalings for the parameters, we find the set of coupled partial differential equations modeling those systems in the limit.

Zbl 1246.60036 Kabluchko, Zakhar Distribution of levels in high-dimensional random landscapes Ann. Appl. Probab. 22, No. 1, 337-362 (2012).

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crittografia

MR2909539 Pankavich, Stephen A particle method for a collisionless plasma with infinite mass. Math. Comput. Simulation 82 (2012), no. 7, 1278--1286. 82D10 (65M75 76M28 76X05)

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Empirical measures and Vlasov hierarchies

The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the N-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which implies a stability estimate in the sense of the Monge-Kantorovich distance with exponent 1 on the i…

Mode-particle Interactions as Sources of Gamma-ray Bubbles in the Galaxy

A plasma outflow coming from the center of Our Galaxy is simulated by an ion beam reaching a nearly stationary magnetic field that can excite lower hybrid waves efficiently in the rarefied plasma (10-2-10-4 cm-3) surrounding the whole galaxy. The electrostatic lower hybrid waves are driven to instability via Cerenkov interaction. In this work we derive (by using a fluid model) the relevant dispersion relation and the expression for the growth rate by solving the dispersion relation both analytically and numerically. Parametric studies show that the growth rate of the instability increases with the beam density. The instability is found to have the maximum growth rate when the perpendicular …

Thermostats: Modeling non-equilibrium dynamics

The Vlasov Limit for a System of Particles which Interact with a Wave Field

In two recent publications [Commun. PDE, vol.22, p.307--335 (1997), Commun. Math. Phys., vol.203, p.1--19 (1999)], A. Komech, M. Kunze and H. Spohn studied the joint dynamics of a classical point particle and a wave type generalization of the Newtonian gravity potential, coupled in a regularized way. In the present paper the many-body dynamics of this model is studied. The Vlasov continuum limit is obtained in form equivalent to a weak law of large numbers. We also establish a central limit theorem for the fluctuations around this limit.

Rigorous analysis of a model of spray in quasi-static condition

MR2988961 (Review) Barbier, Matthieu; Trizac, Emmanuel Field induced stationary state for an accelerated tracer in a bath. J. Stat. Phys. 149 (2012), no. 2, 317--341. 82Cxx

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A Derivation of the Vlasov-Stokes System for Aerosol Flows from the Kinetic Theory of Binary Gas Mixtures

In this short paper, we formally derive the thin spray equation for a steady Stokes gas, i.e. the equation consists in a coupling between a kinetic (Vlasov type) equation for the dispersed phase and a (steady) Stokes equation for the gas. Our starting point is a system of Boltzmann equations for a binary gas mixture. The derivation follows the procedure already outlined in [Bernard-Desvillettes-Golse-Ricci, arXiv:1608.00422 [math.AP]] where the evolution of the gas is governed by the Navier-Stokes equation.

MR2875740 (Review) 82C40 (76P05) Torrilhon, Manuel H-theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinet. Relat. Models 5 (2012), no. 1, 185–201.

Review for Mathematical Reviews

Zbl 1284.82002 El-Batanouny, Michael; Wooten, Frederick Symmetry and condensed matter physics. A computational approach. (English) Cambridge: Cambridge University Press (ISBN 978-0-521-82845-1/hbk)(2008).

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A SIMPLE PARTICLE MODEL FOR A SYSTEM OF COUPLED EQUATIONS WITH ABSORBING COLLISION TERM

We study a particle model for a simple system of partial differential equations describing, in dimension $d\geq 2$, a two component mixture where light particles move in a medium of absorbing, fixed obstacles; the system consists in a transport and a reaction equation coupled through pure absorption collision terms. We consider a particle system where the obstacles, of radius $\var$, become inactive at a rate related to the number of light particles travelling in their range of influence at a given time and the light particles are instantaneously absorbed at the first time they meet the physical boundary of an obstacle; elements belonging to the same species do not interact among themselves…

Modelling of Systems with a Dispersed Phase: “Measuring” Small Sets in the Presence of Elliptic Operators

When modelling systems with a dispersed phase involving elliptic operators, as is the case of the Stokes or Navier-Stokes problem or the heat equation in a bounded domain, the geometrical structure of the space occupied by the dispersed phase enters in the homogenization process through its capacity, a quantity which can be used to define the equivalence classes in \(H^1\). We shall review the relationship between capacity and homogenization terms in the limit when the number of inclusions becomes large, focusing in particular on the situation where the distribution of inclusions is not necessarily too regular (i.e. it is not periodic).

Nonlocal Second Order Vehicular Traffic Flow Models And Lagrange-Remap Finite Volumes

In this paper a second order vehicular macroscopic model is derived from a microscopic car–following type model and it is analyzed. The source term includes nonlocal anticipation terms. A Finite Volume Lagrange–remap scheme is proposed.

Meccanica Statistica

Articolo Enciclopedia

Validation of models for sprays

We consider complex fluids consisting of a dispersed phase (solid particles or liquid droplets) immersed in a gas. A class of models describing the dynamics of such a kind of systems is given by a system of partial differential equations where a kinetic equation, describing the dispersed phase, is coupled to a fluid equation for the background gas. The coupling is given by the drag force exerted by the gas on the dispersed phase. Within this class, we shall analyse the case where the kinetic equation is a Vlasov-type equation and the fluid equation are of Stokes or Navier-Stokes type. We shall discuss the validation problem for this class of models, i.e. the derivation of the equations of t…

stocastico

Particle models for kinetic equations: an introduction and some rigorous results

We shall give an introduction to the validity problem for kinetic equations and we shall review some convergence theorems concerning the derivation from a microscopic dynamics of systems of partial differential equations describing, at the mesoscopic scale, collections of particles interacting through various (collisions and mean field) type of interaction.

Castelnuovo, Emma

Articolo su Enciclopedia

Alan Mathison Turing

Analisi Matematica

Traduzione dal Francese Articolo di Enciclopedia

Modellizzazione e calcolo

Traduzione dal Francese di articolo su enciclopedia

The Boltzmann-Grad limit of a stochastic Lorentz gas in a force field

MR2993918 (Review) 82C40 (35A35 35Q20) Fournier, Nicolas (F-UPEC-LAM); Godinho, David (F-UPEC-LAM) Asymptotic of grazing collisions and particle approximation for the Kac equation without cutoff. (English summary) Comm. Math. Phys. 316 (2012), no. 2, 307–344.1432-0916

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Lax, Peter David

Articolo su Enciclopedia

Gottfried WIlhem Leibniz

Zbl 1251.78010 Pelinovsky, Dmitry E.; Simpson, Gideon; Weinstein, Michael I. Polychromatic solitary waves in a periodic and nonlinear Maxwell system SIAM J. Appl. Dyn. Syst. 11, No. 1, 478-506, electronic only (2012).

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Zbl 1259.60054 Lachièze-Rey, Raphaël Concave majorant of stochastic processes and Burgers turbulence. (English) [J] J. Theor. Probab. 25, No. 2, 313-332 (2012).

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Ladyzhenskaya, Olga Alexandrovna

Articolo in Enciclopedia

Derivation of Models for Thin Sprays from a Multiphase Boltzmann Model

We shall review the validation of a class of models for thin sprays where a Vlasov type equation is coupled to an hydrodynamic equation of Navier–Stokes or Stokes type. We present a formal derivation of these models from a multiphase Boltzmann system for a binary mixture: under suitable assumptions on the collision kernels and in appropriate asymptotics (resp. for the two different limit models), we prove the convergence of solutions to the multiphase Boltzmann model to distributional solutions to the Vlasov–Navier–Stokes or Vlasov–Stokes system. The proofs are based on the procedure followed in Bardos et al. (J Stat Phys 63:323–344 (1991), [2]) and explicit evaluations of the coupling term…

MR2736188 (Review) 82D10 (35Q40 82D37) Chen, Li; Chen, Xiuqing; Zhang, Chunlei Vanishing electron mass limit in the bipolar Euler-Poisson system. Nonlinear Anal. Real World Appl. 12 (2011), no. 2, 1002–1012

Review for Mathematical Reviews

About the link between the detailed description of transitions in a ion and the average-ion models.

We study the link which exists between microscopic (detailed) models for the evolution of the electronic configurations in a population of ions and the macroscopic (average ion) models. Rigorous asymptotics are presented in situations where they exist (large temperature; almost empty or almost full shells), and numerical simulations are presented.

MR2811471 (2012i:82048) 82C40 Silva, C. A. B.; Vasconcellos, Aurea R.; Galvao Ramos, J.; Luzzi, Roberto Generalized kinetic equation for far-from-equilibrium many-body systems.

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Alessio Figalli

Articolo Celebrativo Celebratory Article

MR2598885 (Review) 35Q83 (35B40 82D10) Glassey, Robert; Schaeffer, Jack; Pankavich, Stephen. Time decay for solutions to one-dimensional two component plasma equations. Quart. Appl. Math. 68 (2010), no. 1, 135--141

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MR2353181 (Review) 82C40 (35F20 76X05 82-02 85A30) Kiessling, Michael K.-H. (1-RTG) Microscopic derivations of Vlasov equations. (English summary) Commun. Nonlinear Sci. Numer. Simul. 13 (2008), no. 1, 106–113.

recensione per Mathematical Reviews

MR2569625 (Review) 82B23 (82-02 82B05 82B20 82C05 82C23) Campa, Alessandro; Dauxois, Thierry (F-ENSLY-LP); Ruffo, Stefano (I-FRNZ-EG) Statistical mechanics and dynamics of solvable models with long-range interactions. (English summary) Phys. Rep. 480 (2009), no. 3-6, 57–159. MATHEMATICAL REVIEWS, p. MR2569625, ISSN: 0025-5629

Review

MR2109474 (2005m:82133a) 82C40 (81Q05) Pesci, Adriana I.; Goldstein, Raymond E. Mapping of the classical kinetic balance equations onto the Schrödinger equation. (English summary) Nonlinearity 18 (2005), no. 1, 211–226.

recensione per Mathematical Reviews AMS

MR2119935 (2005k:35396) 35Q60 (35D05 76X05 82D10) Rein, Gerhard (D-BAYR) Global weak solutions to the relativistic Vlasov-Maxwell system revisited. (English summary) Commun. Math. Sci. 2 (2004), no. 2, 145–158.

recensione per Mathematical Reviews

MR2425604 (2009f:82050) 82C70 (35Q72 60D05 65C30 82D10) Clouet, Jean-Francois; Golse, Francois; Puel, Marjolaine; Sentis, Rémi On the slowing down of charged particles in a binary stochastic mixture. (English summary) Kinet. Relat. Models 1 (2008), no. 3, 387–404.

recensione per Mathematical Reviews AMS

MR3434006 Arkeryd, Leif [Arkeryd, Leif O.] (S-CHAL); Nouri, Anne (F-AMU-IM) Well-posedness of the Cauchy problem for a space-dependent anyon Boltzmann equation. (English summary) SIAM J. Math. Anal. 47 (2015), no. 6, 4720{4742.

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The dissipative linear Boltzmann equation for hard spheres

The authors prove existence and uniqueness of a Maxwellian, normalized equilibrium state for a dissipative linear Boltzmann equation with hard-sphere collision kernel modeling a granular gas and, for initial data with finite temperature and entropy, strong L 1 convergence (obtained through compactness arguments) toward the equilibrium of the solutions in the space-homogeneous case. The form of the equilibrium state, which is universal for the family of collision operators including hard, soft and Maxwellian interactions, is guessed through a grazing collision asymptotics and then proved to be the equilibrium state through Fourier analysis. Uniqueness and strong L 1 convergence proofs follow…

MR2239361 (Review) 82C40 Yang, Tong (PRC-CHK); Zhao, Hui-Jiang (PRC-WUHAN-MS) A new energy method for the Boltzmann equation. (English summary) J. Math. Phys. 47 (2006), no. 5, 053301, 19 pp.

recensione per Mathematical Reviews