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

Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium

This work is devoted to the analysis of the linear Boltzmann equation in a bounded domain, in the presence of a force deriving from a potential. The collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity. We study how the association of transport and collision phenomena can lead to convergence to equilibrium, using concepts and ideas from control theory. We prove two main classes of results. On the one hand, we show that convergence towards an equilibrium is equivalent to an almost everywhere geometric control condition. The equilibria (which are not necessarily Maxwellians with our general assumptions on the collision kernel) are described in terms of the equivalence classes of an appropriate equivalence relation. On the other hand, we characterize the exponential convergence to equilibrium in terms of the Lebeau constant, which involves some averages of the collision frequency along the flow of the transport. We handle several cases of phase spaces, including those associated to specular reflection in a bounded domain, or to a compact Riemannian manifold

On the controllability of the Vlasov-Poisson system in the presence of external force fields

In this work, we are interested in the controllability of Vlasov-Poisson systems in the presence of an external force field (namely a bounded force field or a magnetic field), by means of a local interior control. We are able to extend the results of [7], where the only present force was the self-consistent electric field

Stability issues in the quasineutral limit of the one-dimensional Vlasov-Poisson equation

This work is concerned with the quasineutral limit of the one-dimensional Vlasov-Poisson equation, for initial data close to stationary homogeneous profiles. Our objective is threefold: first, we provide a proof of the fact that the formal limit does not hold for homogeneous profiles that satisfy the Penrose instability criterion. Second, we prove on the other hand that the limit is true for homogeneous profiles that satisfy some monotonicity condition, together with a symmetry condition. We handle the case of well-prepared as well as ill- prepared data. Last, we study a stationary boundary-value problem for the formal limit, the so-called quasineutral Vlasov equation. We show the existence of numerous stationary states, with a lot of freedom in the construction (compared to that of BGK waves for Vlasov-Poisson): this illustrates the degeneracy of the limit equation.Comment: 50 page