Multiphasic formulation of Vlasov equations and applications

Abstract

This work is a mathematical study of the multiphasic formulation of the Vlasov equation, which consists in recasting this collisionless kinetic equation as a system of coupled pressureless Euler equations. This framework allows to consider solutions that are only measure-valued in the velocity variable and is thus relevant to tackle physical problems where rough velocity distributions, such as Dirac masses, naturally arise. We specifically address the case of nonlinear Vlasov equations where the force field is one derivative more regular than some moments in velocity of the solution. Under this key assumption, a unified theory of local well-posedness at finite regularity (typically in Sobolev spaces) is developed from scratch for the multiphasic formulation, yielding corresponding results for the Cauchy problem of the associated Vlasov equation at low regularity. We thoroughly apply this abstract theory to two classes of equations, namely Vlasov-Poisson type systems, and Vlasov-Navier-Stokes type systems. In addition to the justification of the monokinetic limit, it also leads to specific applications for each class of equations, allowing possibly rough velocity distributions. For Vlasov-Poisson type systems, we justify the semiclassical limit from Hartree equations, and we describe the nonlinear instability of homogeneous equilibria. For Vlasov-Navier-Stokes type systems, we establish nonlinear asymptotic stability near monokinetic profiles. New results allowing a separation between the regularity in space and in velocity are proven, and along the way, we also provide new proofs of known results and generalize them to the case of rough solutions. Finally, the flexibility of the framework is exploited to obtain extensions to more sophisticated systems such as the Vlasov-Poisson equation for ions, or the Vlasov equation coupled with the compressible Navier-Stokes system.