Quasineutral limit for Vlasov-Poisson with Penrose stable data

We study the quasineutral limit of a Vlasov-Poisson system that describes the dynamics of ions in a plasma. We handle data with Sobolev regularity under the sharp assumption that the profile of the initial data in the velocity variable satisfies a Penrose stability condition. As a by-product of our analysis, we obtain a well-posedness theory for the limit equation (which is a Vlasov equation with Dirac distribution as interaction kernel) for such data

Ill-posedness of the hydrostatic Euler and singular Vlasov equations

In this paper, we develop an abstract framework to establish ill-posedness in the sense of Hadamard for some nonlocal PDEs displaying unbounded unstable spectra. We apply it to prove the ill-posedness for the hydrostatic Euler equations as well as for the kinetic incompressible Euler equations and the Vlasov-Dirac-Benney system

Instabilities in the mean field limit

Consider a system of $N$ particles interacting through Newton's second law with Coulomb interaction potential in one spatial dimension or a $\mathcal{C}^2$ smooth potential in any dimension. We prove that in the mean field limit $N \to + \infty$, the $N$ particles system displays instabilities in times of order $\log N$ for some configurations approximately distributed according to unstable homogeneous equilibria.Comment: minor typos corrected; Journal of Statistical Physics, accepte