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19 results on '"82D10"'

1. Discontinuous Galerkin Methods for the Vlasov–Stokes System.

Author

Hutridurga, Harsha, Kumar, Krishan, and Pani, Amiya K.

Subjects
GALERKIN methods, VLASOV equation, TWO-phase flow, ALGORITHMS, DENSITY
Abstract

This paper develops and analyses a semi-discrete numerical method for the two-dimensional Vlasov–Stokes system with periodic boundary condition. The method is based on the coupling of the semi-discrete discontinuous Galerkin method for the Vlasov equation with discontinuous Galerkin scheme for the stationary incompressible Stokes equation. The proposed method is both mass and momentum conservative. Since it is difficult to establish non-negativity of the discrete local density, the generalized discrete Stokes operator become non-coercive and indefinite, and under the smallness condition on the discretization parameter, optimal error estimates are established with help of a modified the Stokes projection to deal with the Stokes part and, with the help of a special projection, to tackle the Vlasov part. Finally, numerical experiments based on the dG method combined with a splitting algorithm are performed. [ABSTRACT FROM AUTHOR]

Published
2025
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2. Regularity of weak solutions for the relativistic Vlasov–Klein–Gordon system.

Author

Xiao, Meixia

Subjects
*VLASOV equation, *SINE-Gordon equation, *SEPARATION of variables, *DRY friction
Abstract

This article is devoted to the study of the relativistic Vlasov–Klein–Gordon system in space dimension three. The dynamical model describes a collisionless ensemble of classical particles coupled with a Klein–Gordon field. By combining the Fourier method and smoothing effect of low velocity particles, we prove a higher regularity of the field, which is widely used for the study of Vlasov equation. [ABSTRACT FROM AUTHOR]

Published
2024
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3. Effect of electron temperature on formation of travelling waves in plasma: Kinetic and hydrodynamic models.

Author

Chizhonkov, Eugene V. and Frolov, Alexander A.

Subjects
*PLASMA waves, *ELECTRON temperature, *LASER plasmas, *TEMPERATURE effect, *DISTRIBUTION (Probability theory), *VLASOV equation, *LASER pulses
Abstract

The kinetic formulation of the model problem of plasma waves excitation by a powerful short laser pulse is numerically studied for the first time. Kinetic and simplest hydrodynamic plasma models are also compared for the problem under consideration. It is shown that the considered hydrodynamic models do not provide good approximations to the solution to the Vlasov kinetic equation, namely, one leads to discontinuous solutions and the other has a significant qualitative distinction. At a low plasma temperature, the effect of non-isothermicity of the process is small, but it can lead to significant distortions of the solution during further heating. The results obtained here imply that the first two moments of the distribution function are not enough to describe the plasma hydrodynamics; higher-order moments should be used. [ABSTRACT FROM AUTHOR]

Published
2023
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5. Using linear multistep methods for the time stepping in Vlasov–Poisson simulations.

Author

Lorenzon, Denis and Elaskar, Sergio

Subjects
ORDINARY differential equations, TIME integration scheme, VLASOV equation, PLASMA astrophysics, PLASMA density
Abstract

The numerical simulation of hot and low density plasmas using the Vlasov–Poisson model is necessary for many practical applications such as the characterization of laboratory and astrophysical plasmas. The numerical treatment of the Vlasov equation is addressed using Eulerian methods when high precision and low noise are required. Among these methods, we highlight those based on finite-volumes without splitting, which have shown to be a good option for capturing small structures in phase space with low dissipation while preserving positivity. The problem is that kinetic simulations usually require the discretization of 3–6-dimensional phase-spaces which results in a huge number of ordinary differential equations (ODEs). This stands out the importance of using efficient schemes for the time integration. In this article, linear multistep methods are implemented for the time stepping of the resulting equations, and compared against traditional Runge–Kutta ones. Schemes with built-in error estimation are also implemented in an attempt to perform adaptive stepsize control. Their accuracy, stability and computational cost are compared through the simulation of classical benchmark problems for the two-dimensional Vlasov–Poisson system. [ABSTRACT FROM AUTHOR]

Published
2021
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6. Convergence analysis of asymptotic preserving schemes for strongly magnetized plasmas.

Author

Filbet, Francis, Rodrigues, L. Miguel, and Zakerzadeh, Hamed

Subjects
VLASOV equation, LARMOR radius, PLASMA frequencies, MAGNETIC fields
Abstract

The present paper is devoted to the convergence analysis of a class of asymptotic preserving particle schemes (Filbet and Rodrigues in SIAM J. Numer. Anal. 54(2):1120–1146, 2016) for the Vlasov equation with a strong external magnetic field. In this regime, classical Particle-in-Cell methods are subject to quite restrictive stability constraints on the time and space steps, due to the small Larmor radius and plasma frequency. The asymptotic preserving discretization that we are going to study removes such a constraint while capturing the large-scale dynamics, even when the discretization (in time and space) is too coarse to capture fastest scales. Our error bounds are explicit regarding the discretization, stiffness parameter, initial data and time. [ABSTRACT FROM AUTHOR]

Published
2021
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7. Scattering Structure and Landau Damping for Linearized Vlasov Equations with Inhomogeneous Boltzmannian States.

Author

Després, Bruno

Subjects
*LANDAU damping, *VLASOV equation, *SCATTERING (Mathematics)
Abstract

We study the linearized Vlasov–Poisson–Ampère equation for non-constant Boltzmannian states with one region of trapped particles in dimension one and construct the eigenstructure in the context of the scattering theory. This is based on the use of semi-discrete variables (moments in velocity), and it leads to a new Lippmann–Schwinger variational equation. The continuity in quadratic norm of the operator is proved, and the well posedness is proved for a small value of the scaling parameter. It gives a proof of Linear Landau damping for inhomogeneous Boltzmannian states. The linear HMF model is an example. [ABSTRACT FROM AUTHOR]

Published
2019
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8. Existence, uniqueness and asymptotic behavior for the Vlasov-Poisson system with radiation damping.

Author

Chen, Jing, Zhang, Xian, and Gao, Ran

Subjects
*NUMERICAL solutions to the Cauchy problem, *VLASOV equation, *RADIATION damping, *VELOCITY, *LAGRANGIAN functions, *PRINCIPLE of virtual work, *GRONWALL inequalities
Abstract

We investigate the Cauchy problem for the Vlasov-Poisson system with radiation damping. By virtue of energy estimate and a refined velocity average lemma, we establish the global existence of nonnegative weak solution and asymptotic behavior under the condition that initial data have finite mass and energy. Furthermore, by building a Gronwall inequality about the distance between the Lagrangian flows associated to the weak solutions, we can prove the uniqueness of weak solution when the initial data have a higher order velocity moment. [ABSTRACT FROM AUTHOR]

Published
2017
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9. Convergence of the Two-Species Vlasov-Poisson System to the Pressureless Euler Equations.

Author

Qiu, Jianlong and Yang, Xiuhui

Subjects
*STOCHASTIC convergence, *VLASOV equation, *EULER equations (Rigid dynamics), *DEBYE length, *MAXIMUM entropy method
Abstract

In this paper we study the quasineutral limit of the two-species Vlasov-Poisson system. For well-prepared initial data, we establish the convergence of the Vlasov-Poisson system to the pressureless Euler system as the Debye length tends to zero. [ABSTRACT FROM AUTHOR]

Published
2016
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10. Convergence Analysis of Asymptotic Preserving Schemes for Strongly Magnetized plasmas

Author

Francis Filbet, Hamed Zakerzadeh, Luis Miguel Rodrigues, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées, Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Discretization, Gyroradius, 65L04, Particle methods 2010 MSC: 35Q83, 010103 numerical & computational mathematics, Plasma oscillation, 01 natural sciences, Convergence (routing), FOS: Mathematics, Applied mathematics, Vlasov-Poisson system, Asymptotic preserving schemes, Mathematics - Numerical Analysis, 0101 mathematics, 82D10, Mathematics, Spacetime, 65M75, Homogeneous magnetic field, Applied Mathematics, Numerical analysis, Vlasov equation, 65M15, Numerical Analysis (math.NA), 010101 applied mathematics, Constraint (information theory), Computational Mathematics, High-order time discretization, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; The present paper is devoted to the convergence analysis of a class of asymptotic preserving particle schemes [Filbet & Rodrigues, SIAM J. Numer. Anal., 54 (2) (2016)] for the Vlasov equation with a strong external magnetic field. In this regime, classical Particle-In-Cell (PIC) methods are subject to quite restrictive stability constraints on the time and space steps, due to the small Larmor radius and plasma frequency. The asymptotic preserving discretization that we are going to study removes such a constraint while capturing the large-scale dynamics, even when the discretization (in time and space) is too coarse to capture fastest scales. Our error bounds are explicit regarding the discretization, stiffness parameter, initial data and time.

Published
2021
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11. Phase-space moment-equation model of highly relativistic electron-beams in plasma-wakefield accelerators.

Author

Robson, R.E., Mehrling, T., and Osterhoff, J.

Subjects
*ELECTRON beams, *PARTICLE accelerators, *NUMERICAL analysis, *VLASOV equation, *SIMULATION methods & models
Abstract

We formulate a new procedure for modelling the transverse dynamics of relativistic electron beams with significant energy spread when injected into plasma-based accelerators operated in the blow-out regime. Quantities of physical interest, such as the emittance, are furnished directly from solution of phase space moment equations formed from the relativistic Vlasov equation. The moment equations are closed by an Ansatz, and solved analytically for prescribed wakefields. The accuracy of the analytic formulas is established by benchmarking against the results of a semi-analytic/numerical procedure which is described within the scope of this work, and results from a simulation with the 3D quasi-static PIC code HiPACE. [ABSTRACT FROM AUTHOR]

Published
2015
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12. On the controllability of the relativistic Vlasov–Maxwell system.

Author

Glass, Olivier and Han-Kwan, Daniel

Subjects
*VLASOV equation, *MAXWELL equations, *TWO-dimensional models, *TORUS, *MATHEMATICAL proofs, *APPROXIMATION theory
Abstract

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 strip of the torus is contained in the control set and under certain additional conditions on the initial data, we establish a controllability result on the distribution function only, also in large time. Here, we need some additional arguments based on the asymptotics of the Vlasov–Maxwell system with large speed of light and on our previous results concerning the controllability of the Vlasov–Poisson system with an external magnetic field [14] . [ABSTRACT FROM AUTHOR]

Published
2015
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13. ASYMPTOTICS OF THE THREE DIMENSIONAL VLASOV EQUATION IN THE LARGE MAGNETIC FIELD LIMIT

Author

Francis Filbet, L. Miguel Rodrigues, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UNIV-RENNES), Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UPS), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-PRES Université de Toulouse-Université Toulouse III - Paul Sabatier (UPS), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1)

Subjects
asymptotic analysis 2010 MSC: 35Q83, guiding center approximation, General Mathematics, Structure (category theory), FOS: Physical sciences, Curvature, 01 natural sciences, 010305 fluids & plasmas, Mathematics - Analysis of PDEs, Vlasov equation, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Gyrokinetics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Algebraic number, 82D10, Mathematical Physics, Physics, 35B40, Mathematical Physics (math-ph), Action (physics), Magnetic field, 010101 applied mathematics, Classical mechanics, 78A35, gyrokinetics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

International audience; We study the asymptotic behavior of solutions to the Vlasov equation in the presence of a strong external magnetic field. In particular we provide a mathematically rigorous derivation of the guiding-center approximation in the general three dimensional setting under the action of large inhomogeneous magnetic fields. First order corrections are computed and justified as well, including electric cross field, magnetic gradient and magnetic curvature drifts. We also treat long time behaviors on two specific examples, the two dimensional case in carte-sian coordinates and a poloidal axi-symmetric geometry, the former for expository purposes. Algebraic manipulations that underlie concrete computations make the most of the linearity of the stiffest part of the system of characteristics instead of relying on any particular variational structure.

Published
2020
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14. L2-Hypocoercivity and large time asymptotics of the linearized Vlasov-Poisson-Fokker-Planck system

Author

Jean Dolbeault, M. Lazhar Tayeb, Xingyu Li, Lanoir Addala, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Fokker-Planck operator, drift-diffusion equations, hypocoercivity, diffusion limit, Poisson distribution, 01 natural sciences, convergence to equilibrium, 010305 fluids & plasmas, symbols.namesake, Mathematics - Analysis of PDEs, Vlasov equation, Vlasov–Poisson-Fokker–Planck system, Simple (abstract algebra), 0103 physical sciences, large-time behavior, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 82C40, 35H10, 35P15, 35Q84, 35R09, 47G20, 82C21, 82D10, 82D37, 010306 general physics, Mathematical Physics, 2010 Mathematics Subject Classification . 82C40, 35H10, 35P15, 35Q84, 35R09, 47G20, 82C21, 82D10, 82D37, Physics, Coupling, Scalar (physics), 82C40, Statistical and Nonlinear Physics, Nonlinear system, electrostatic forces, Rate of convergence, confinement, symbols, Fokker–Planck equation, Poisson coupling, rate of convergence, Analysis of PDEs (math.AP)
Abstract

International audience; This paper is devoted to the linearized Vlasov-Poisson-Fokker-Planck system in presence of an external potential of confinement. We investigate the large time behaviour of the solutions using hypocoercivity methods and a notion of scalar product adapted to the presence of a Poisson coupling. Our framework provides estimates which are uniform in the diffusion limit. As an application in a simple case, we study the one-dimensional case and prove the exponential convergence of the nonlinear Vlasov-Poisson-Fokker-Planck system without any small mass assumption.

Published
2019
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15. Expansion of a singularly perturbed equation with a two-scale converging convection term

Author

Alexandre Mouton, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), and Laboratoire Paul Painlevé - UMR 8524 (LPP)

Subjects
Convection, 01 natural sciences, Two-scale convergence, Vlasov equation, FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics - Numerical Analysis, 35Q83, 76M40, 78A35, 82D10, 0101 mathematics, convection equation, Physics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Magnetic confinement fusion, Numerical Analysis (math.NA), Plasma, Charged particle, 010101 applied mathematics, Amplitude, Distribution function, Convection–diffusion equation, Gyrokinetic approximations, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In many physical contexts, evolution convection equations may present some very large amplitude convective terms. As an example, in the context of magnetic confinement fusion, the distribution function that describes the plasma satisfies the Vlasov equation in which some terms are of the same order as $\epsilon^{-1}$, $\epsilon \ll 1$ being the characteristic gyrokinetic period of the particles around the magnetic lines. In this paper, we aim to present a model hierarchy for modeling the distribution function for any value of $\epsilon$ by using some two-scale convergence tools. Following Fr\'enod \\& Sonnendr\"ucker's recent work, we choose the framework of a singularly perturbed convection equation where the convective terms admit either a high amplitude part or a an oscillating part with high frequency $\epsilon^{-1} \gg 1$. In this abstract framework, we derive an expansion with respect to the small parameter $\epsilon$ and we recursively identify each term of this expansion. Finally, we apply this new model hierarchy to the context of a linear Vlasov equation in three physical contexts linked to the magnetic confinement fusion and the evolution of charged particle beams.

Published
2015

16. Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics

Author

Mihai Bostan, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Scientific computation and visualization (CALVI), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection (LSIIT), Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection (LSIIT)

Subjects
Physics, Asymptotic analysis, Advection, Applied Mathematics, 010102 general mathematics, Magnetic confinement fusion, Context (language use), Plasma modeling, 01 natural sciences, Action (physics), Magnetic field, 010101 applied mathematics, Slow motion, Classical mechanics, Vlasov equation, Transport equations, Physics::Plasma Physics, [PHYS.PHYS.PHYS-PLASM-PH]Physics [physics]/Physics [physics]/Plasma Physics [physics.plasm-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, Gyrokinetic models, Analysis, 35Q75, 78A35, 82D10
Abstract

International audience; The subject matter of this paper concerns the asymptotic regimes for transport equations with advection fields having components of very disparate orders of magnitude. Such models arise in the magnetic confinement context, where charged particles move under the action of strong magnetic fields. According to the different possible orderings between the typical physical scales (Larmor radius, Debye length, cyclotronic frequency, plasma frequency) we distinguish several regimes: guiding-center approximation, finite Larmor radius regime, etc. The main purpose is to derive the limit models: we justify rigorously the convergence towards these limit models and we investigate the well-posedness of them.

Published
2010
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17. Decay in Time for a One-Dimensional Two-Component Plasma

Author

Robert T. Glassey, Stephen Pankavich, and Jack Schaeffer

Subjects
82C21, General Mathematics, FOS: Physical sciences, Space (mathematics), Momentum, Mathematics - Analysis of PDEs, Physics::Plasma Physics, FOS: Mathematics, Initial value problem, Boundary value problem, 82D10, Mathematical Physics, Mathematics, General Engineering, Vlasov equation, Plasma, Mathematical Physics (math-ph), 35L60, Charged particle, 35Q99, Quantum electrodynamics, Physics::Space Physics, Kinetic theory of gases, 82C22, Analysis of PDEs (math.AP)
Abstract

The motion of a collisionless plasma is described by the Vlasov-Poisson system, or in the presence of large velocities, the relativistic Vlasov-Poisson system. Both systems are considered in one space and one momentum dimension, with two species of oppositely charged particles. A new identity is derived for both systems and is used to study the behavior of solutions for large times., Comment: 17 pages, no figures

Published
2010
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18. Transport equations with singular coefficients. Application to the gyro-kinetic models in plasma physics

Author

Bostan, Mihai, Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS), Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC), Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Scientific computation and visualization (CALVI), Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection (LSIIT), Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), INRIA, projet CALVI INRIA, Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection (LSIIT)

Subjects
Vlasov equation, Transport equations, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35Q75, 78A35, 82D10, Gyro-kinetic models
Abstract

The subject matter of this paper concerns the asymptotic regimes for transport equations with singular coefficients. Such models arise for example in plasma physics, when dealing with charged particles moving under the action of strong magnetic fields. These regimes are motivated by the magnetic confinement fusion. The stiffness of the coefficients comes from the multi-scale character of the problem. According to the different possible orderings between the typical physical scales (Larmor radius, Debye length, cyclotronic frequency, plasma frequency) we distinguish several regimes. From the mathematical point of view the analysis of such regimes reduces to stability properties for transport equations whose coefficients have different magnitude orders, depending on some small parameter. The main purpose is to derive limit models by letting the small parameter vanish. In the magnetic confinement context these asymptotics can be assimilated to homogenization procedures with respect to the fast cyclotronic movement of particles around the magnetic lines. We justify rigorously the convergence towards these limit models and we investigate the well-posedness of them.

Published
2008

19. Smooth Solutions to the Relativistic Vlasov-Maxwell System

Author

Christophe Pallard

Subjects
Physics, Applied Mathematics, General Mathematics, Vlasov equation, Inhomogeneous electromagnetic wave equation, Plasma modeling, System of linear equations, Momentum, Classical mechanics, 35Q75, Vlasov-Maxwell system, Bounded function, Kinetic equations, Gravitational singularity, 76X05, 82D10, Focus (optics)
Abstract

We focus on the existence of classical solutions to the relativistic Vlasov-Maxwell system of equations. We discuss an alternative proof of the result by Glassey and Strauss showing that smooth solutions do not develop singularities as long as the momentum support remains bounded.

Published
2007

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