Abstract:
We prove global existence of strong solutions for the Vlasov-Poisson system in a convex bounded domain in the plasma physics case assuming homogeneous Dirichlet boundary conditions for the electric potential and the specular reflection boundary conditions for the distribution density.

Abstract:
We prove the existence of least energy nodal solution for a class of Schr\"odinger-Poisson system in a bounded domain $\Omega \subset \mathbb{R}^3$ with nonlinearity having a subcritical growth.

Abstract:
We consider the Vlasov-Poisson system with spherical symmetry and an exterior potential which is induced by a point mass in the center. This system can be used as a simple model for a newtonian galaxy surrounding a black hole. For this system, we establish a global existence result for classical solutions with shell-like initial data, i.e. the support of the density is bounded away from the point mass singularity. We also prove existence and stability of stationary solutions which describe static shells, where we use a variational approach which was established by Y. Guo and G. Rein.

Abstract:
In this note, we show uniqueness of weak solutions to the Vlasov-Poisson system on the only condition that the macroscopic density $\rho$ defined by $\rho(t,x) = \int_{\Rd} f(t,x,\xi)d\xi$ is bounded in $\Linf$. Our proof is based on optimal transportation.

Abstract:
In this paper, we extend to the case of initial data constituted of a Dirac mass plus a bounded density (with finite moments) the theory of Lions and Perthame [6] for the Vlasov-Poisson equation. Our techniques also provide polynomially growing in time estimates for moments of the bounded density.

Abstract:
We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in \cite{Tro86}. Here we shall give a simpler proof of this result, and explain why it implies the so-called "Propagation of molecular chaos". More precisely, both results will be a direct consequence of a weak-strong stability result on the one dimensional Vlasov-Poisson equation that is interesting by it own. We also prove the existence of global solutions to the $N$ particles dynamic starting from any initial positions and velocities, and the existence of global solutions to the Vlasov-Poisson equation starting from any measures with bounded first moment in velocity.

Abstract:
We characterize the $L^p$-range, $1 < p < +\infty$, of the Poisson transform on the Shilov boundary for non-tube bounded symmetric domains. We prove that this range is a Hua-Hardy type space for harmonic functions satisfying a Hua system.

Abstract:
We prove the existence of axially symmetric solutions to the Vlasov--Poisson system in a rotating setting for sufficiently small angular velocity. The constructed steady states depend on Jacobi's integral and the proof relies on an implicit function theorem for operators.

Abstract:
As is well known from the work of R. Glassey} and J. Schaeffer, the main energy estimates which are used in global existence results for the gravitational Vlasov-Poisson system do not apply to the relativistic version of this system, and smooth solutions to the initial value problem with spherically symmetric initial data of negative energy blow up in finite time. For similar reasons the variational techniques by which Y. Guo and G. Rein obtained nonlinear stability results for the Vlasov-Poisson system do not apply in the relativistic situation. In the present paper a direct, non-variational approach is used to prove nonlinear stability of certain steady states of the relativistic Vlasov-Poisson system against spherically symmetric, dynamically accessible perturbations. The resulting stability estimates imply that smooth solutions with spherically symmetric initial data which are sufficiently close to the stable steady states exist globally in time.

Abstract:
A collisionless plasma is modeled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge - dependant upon only velocity - is assumed. The situation in which mobile negative ions balance the positive charge as x tends to infinity is considered. Thus, the total positive charge and the total negative charge are both infinite. Smooth solutions with appropriate asymptotic behavior for large x, which were previously shown to exist locally in time, are continued globally. This is done by showing that the charge density decays at least as fast as x^{-6}. This article also establishes spatial decay estimates for the electrostatic field and its derivatives.