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About a Variant of the 1 d Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation"

Claude Bardos (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report) . It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full non linear problem and also on natural connections with several other equations of mathematical physic.

Effect of the polarization drift in a strongly magnetized plasma

Daniel Han-Kwan (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a strongly magnetized plasma described by a Vlasov-Poisson system with a large external magnetic field. The finite Larmor radius scaling allows to describe its behaviour at very fine scales. We give a new interpretation of the asymptotic equations obtained by Frénod and Sonnendrücker [SIAM J. Math. Anal. 32 (2001) 1227–1247] when the intensity of the magnetic field goes to infinity. We introduce the so-called polarization drift and show that its contribution is not negligible in the...

Effect of the polarization drift in a strongly magnetized plasma

Daniel Han-Kwan (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a strongly magnetized plasma described by a Vlasov-Poisson system with a large external magnetic field. The finite Larmor radius scaling allows to describe its behaviour at very fine scales. We give a new interpretation of the asymptotic equations obtained by Frénod and Sonnendrücker [SIAM J. Math. Anal. 32 (2001) 1227–1247] when the intensity of the magnetic field goes to infinity. We introduce the so-called polarization drift and show that its contribution is not negligible in the...

Équations de transport à coefficient dont le gradient est donné par une intégrale singulière

François Bouchut, Gianluca Crippa (2007/2008)

Séminaire Équations aux dérivées partielles

Nous rappelons tout d’abord l’approche maintenant classique de renormalisation pour établir l’unicité des solutions faibles des équations de transport linéaires, en mentionnant les résultats récents qui s’y rattachent. Ensuite, nous montrons comment l’approche alternative introduite par Crippa et DeLellis estimant directement le flot lagrangien permet d’obtenir des résultats nouveaux. Nous établissons l’existence et l’unicité du flot associé à une équation de transport dont le coefficient a un gradient...

Mean-Field Optimal Control

Massimo Fornasier, Francesco Solombrino (2014)

ESAIM: Control, Optimisation and Calculus of Variations

We introduce the concept of mean-field optimal control which is the rigorous limit process connecting finite dimensional optimal control problems with ODE constraints modeling multi-agent interactions to an infinite dimensional optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of interacting agents. While in the classical mean-field theory one studies the behavior of a large number of small individuals freely interacting...

New Results in Velocity Averaging

François Golse (2001/2002)

Séminaire Équations aux dérivées partielles

This paper discusses two new directions in velocity averaging. One is an improvement of the known velocity averaging results for L 1 functions. The other shows how to adapt some of the ideas of velocity averaging to a situation that is essentially a new formulation of the Vlasov-Maxwell system.

On the Plasma-Charge problem

Mario Pulvirenti (2009/2010)

Séminaire Équations aux dérivées partielles

This short report is a review on recent results of S. Caprino, C. Marchioro, E. Miot and the author on the initial value problem associated to the evolution of a continuous distribution of charges (plasma) in presence of a finite number of point charges.

Propriétés dispersives pour des équations cinétiques et applications à l’équation de Vlasov-Poisson

Delphine Salort (2008/2009)

Séminaire Équations aux dérivées partielles

On considère l’équation de Vlasov-Poisson en dimension 3. On montre des résultats d’existence et d’unicité de solutions faibles de l’équation de Vlasov-Poisson avec densité bornée pour des données initiales ayant strictement moins de six moments dans L x , ξ 1 . La preuve est basée sur une nouvelle approche qui consiste à établir des effets de moments a priori pour des équations de transport avec des termes de force peu réguliers.

Solving the Vlasov equation in complex geometries

J. Abiteboul, G. Latu, V. Grandgirard, A. Ratnani, E. Sonnendrücker, A. Strugarek (2011)

ESAIM: Proceedings

This paper introduces an isoparametric analysis to solve the Vlasov equation with a semi-Lagrangian scheme. A Vlasov-Poisson problem modeling a heavy ion beam in an axisymmetric configuration is considered. Numerical experiments are conducted on computational meshes targeting different geometries. The impact of the computational grid on the accuracy and the computational cost are shown. The use of analytical mapping or Bézier patches does not induce...

The mean-field limit for the dynamics of large particle systems

François Golse (2003)

Journées équations aux dérivées partielles

This short course explains how the usual mean-field evolution PDEs in Statistical Physics - such as the Vlasov-Poisson, Schrödinger-Poisson or time-dependent Hartree-Fock equations - are rigorously derived from first principles, i.e. from the fundamental microscopic models that govern the evolution of large, interacting particle systems.

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