Recent developments on quasineutral limits for Vlasov-type equations

Abstract

Kinetic equations of Vlasov type are in widespread use as models in plasma physics. A well known example is the Vlasov-Poisson system for collisionless, unmagnetised plasma. In these notes, we discuss recent progress on the quasineutral limit in which the Debye length of the plasma tends to zero, an approximation widely assumed in applications. The models formally obtained from Vlasov-Poisson systems in this limit can be seen as kinetic formulations of the Euler equations. However, rigorous results on this limit typically require a structural or strong regularity condition. Here we present recent results for a variant of the Vlasov-Poisson system, modelling ions in a regime of massless electrons. We discuss the quasineutral limit from this system to the kinetic isothermal Euler system, in a setting with rough initial data. Then, we consider the connection between the quasineutral limit and the problem of deriving these models from particle systems. We begin by presenting a recent result on the derivation of the Vlasov-Poisson system with massless electrons from a system of extended charges. Finally, we discuss a combined limit in which the kinetic isothermal Euler system is derived.

On the Rigorous Derivation of the Incompressible Euler Equation from Newton's Second Law

“…The mean-field limit N → ∞ of the system (1.6) formally leads to the Vlasov-Poisson equation, as discussed above. Several authors [4,8,9,3,23,13,17,14,15,12,11] have considered the quasineutral limit of Vlasov-Poisson with varying assumptions on the initial datum. But in the monokinetic regime of present interest, Brenier [3] has rigorously shown that the quasineutral limit leads to the incompressible Euler equation (1.1).…”