Objectifs :
Objectifs: The aim of this lecture is to present the state of art of the
Cauchy problem for the collisionless kinetic equations such as the
Vlasov-Poisson and Vlasov-Maxwell systems. Collisionless kinetic
equations appear, among others, in plasma physics and astrophysics. In
plasma physics, these models describe accurately the wave-particle
interaction which plays a crucial role in turbulent plasmas such as
magnetic fusion plasmas (ITER project). In astrophysics these models
allow to describe the large scale structure of the universe such as
clusters of galaxies and the dark matter.
Here, we present the theory of weak solutions and classical regular
solutions. Existence theory of classical solutions is based on natural a
priori estimates like among others conservation of energy, on the theory
of characteristics and on the control of the velocity support of the
distribution function. Uniqueness follows from regularity properties of
classical solutions. Existence theory of weak solutions relies on a priori
functional estimates and on compactness results such as standard
compact embeddings in Sobolev spaces for the Vlasov-Poisson
equations, or averaging lemmas for the Vlasov-Maxwell system.
Uniqueness of weak solutions is a trickier task and sometimes an open
problem.

Programme :
1) Introduction: theory of characteristics; formal properties of Vlasov-
Poisson & Vlasov-Maxwell systems; conservation laws; basic a priori
estimates.
2) Weak and classical solutions for the Vlasov-Poisson system.
3) Weak and classical solutions for the Vlasov-Maxwell system.

Pré-requis :
functional analysis, partial differential equations

Equipe pédagogique :
Nicolas Besse (OCA, LAGRANGE)

Les Compétences et capacités visées à l'issue de la formation (fiches RNCP)

Arrêté du 22 février 2019 définissant les compétences des diplômés du doctorat et inscrivant le doctorat au répertoire national de la certification professionnelle. https://www.legifrance.gouv.fr/loda/id/JORFTEXT000038200990/

Bloc 1 : Conception et élaboration d’une démarche de recherche et développement, d’études et prospective

- Disposer d'une expertise scientifique tant générale que spécifique d'un domaine de recherche et de travail déterminé

La formation participe à l'objectif suivant :être directement utile pour la réalisation des travaux personnels de recherche