Introducing a Dirac Potential instead of the standard Poisson potential in the Vlasov equation change it into a very singular problem at a cross road of very subjects. In particular Penrose's method differenciates between linearly well posed and ill posed problems. It leads to a reuslt of ill posedness for the full non linear problem. On the other hand when restricted to kinetic hydrodynamic it gives rises to a well posed problem which locally in times is related to the semi classical limit of the Non Linear Schrodinger Equation. Moreover since the Non linear Schrodinger equation is integrable by inverse scattering some type of integrability with an infinite set of conserved quantities shows up in the present equation. In fact this equation can also be used as a model for water waves (in some convenient scaling). Then it carries the name of Benney equation and it is in this case that properties related to integrability were observed by Benney, Miura Novikov and others.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

In the first part of the talk we focus on the dynamics of a many-particle self-gravitating system, described with either classical or relativistic kinetic theory. As a way to gain insight into the relativistic models, we will review what is known about dispersive behavior in the classical setting. Then we shall present some preliminary results concerning dispersive behavior for the relativistic models, together with necessary conditions for the existence of steady states as well. The second part of the talk is devoted to discuss several macroscopic limits of kinetic models, paying special attention to the relativistic BGK equation.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

A solution of the relativistic Vlasov-Maxwell system is said to be isolated from incoming radiation if it is not hit by electromagnetic energy coming from the pass null infinity of Minkowski space. In this talk I will review the results available concerning the existence and the properties of isolated solutions to the relativistic Vlasov-Maxwell system.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

The Nordström-Vlasov system describes the evolution of a population of self-gravitating collisionless particles. We study the existence and uniqueness of mild solution for the Cauchy problem in one dimension. This approach does not require any derivative for the initial particle density. For any initial particle density uniformly bounded with respect to the space variable by some function with finite kinetic energy and any initial smooth data for the field equation we construct a global solution, preserving the total energy. Moreover the solution propagates with finite speed, which coincides with the light speed.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

In this talk, I will discuss two recent results on the Cauchy problem for the relativistic Vlasov-Darwin (RVD) system: the existence and uniqueness of global in time classical solutions with a small initial datum, and the uniqueness of weak solutions with compact support. These results rely on the formulation of the RVD system in terms of the generalized phase space variables, and the scalar and vector potentials. Existence of classical solutions is proved by constructive arguments. Improvements are made on the time decay estimates previously known for the electromagnetic field and its space derivatives. Uniqueness of weak solutions is proved with techniques derived from optimal transportation. This work is in collaboration with Martial Agueh and Reinhard Illner.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

I will present the theorem obtained with C. Villani proving the Landau damping in the nonlinear perturbative regime for the classical Vlasov-Poisson equation, and discuss the method and some further developements in progress.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

The Vlasov-Maxwell system is a fundamental kinetic model of plasma dynamics. When one considers relativistic velocities and includes effects due to collisions with a fixed background of particles, the result is the relativistic Vlasov-Maxwell-Fokker-Planck system. The first Lorentz-invariant model of this type was recently derived by Calogero and Felix in 2010. Hence, we discuss the first well-posedness results for global-in-time classical solutions of this system posed in a lower-dimensional setting. Our methods utilize a gain in regularity stemming from the Fokker-Planck term to arrive at smooth solutions even from weak initial data.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

This talk will deal with non-linear stability of spherical steady states to Vlasov systems. In the case of the classical gravitational Vlasov-Poisson system, we have proved recently that all spherical stationary states which are deceasing functions of their microscopic energy are non linearly stable under general perturbations. The proof is based on two main steps: A quantitative control of the potential field via a Poincaré-Antonov like inequality, and a compactness argument leading to the stability of the whole distribution function. The first aim of this talk is to present a new functional inequality which makes fully quantitative the compactness part of this proof as well. Applications of this inequality to other contexts will also be discussed. The second aim of the talk will be to explore the extension of this strategy to relativistic contexts: the relativistic Vlasov-Poisson (RVP) and the Vlasov-Manev (VM) systems. We show that the strategy extends to the RVP although the potential field is less regular than in the classical case, while the theory for VM faces a serious difficulty : A Poincaré-Antonov inequality with a fractional Laplacian is not available. Nevertheless, we show how a standard variational approach in the case of the VM system may be used to prove the non-linear stability of the minimizers of the Hamiltonian, although the uniqueness of these minimizers is not guaranteed (here, the Euler-Lagrange equation is a fractional Laplacian and not a Poisson equation). Finally, in the case of the so-called "Pure Manev system", we exhibit a continuous family of self-similar blow-up solutions whose profiles are close to the steady states.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)