Penrose condition around measure-valued velocity profiles

In 1960, Penrose studied a condition (now called the Penrose condition) for a homogeneous solution to the Vlasov-Poisson equations to be linearly stable. This condition turned out to be relevant when studying the convergence of the Vlasov-Poisson equations in various regimes. Despite the fact that it makes sense for measured-valued profiles, there is no general result in this setting, neither in the stable case (nonlinear stability, well-posedness for the asymptotic equations), nor in the unstable one (nonlinear instability, ill-posedness for the asymptotic equations). In this presentation, I will show how a multi-fluid formulation of the equations makes it possible to deal with such questions and I will present a first result in the unstable case.