# Nonlinear instability in Vlasov type equations around rough velocity profiles

### Aymeric Baradat

Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany

## Abstract

In the Vlasov-Poisson equation, every configuration which is homogeneous in space provides a stationary solution. Penrose gave in 1960 a criterion for such a configuration to be linearly unstable. While this criterion makes sense in a measure-valued setting, the existing results concerning nonlinear instability always suppose some regularity with respect to the velocity variable. Here, thanks to a multiphasic reformulation of the problem, we can prove an “almost Lyapounov instability” result for the Vlasov-Poisson equation, and an ill-posedness result for the kinetic Euler equation and the Vlasov-Benney equation (two quasineutral limits of the Vlasov-Poisson equation), both around any unstable measure.

## Cite this article

Aymeric Baradat, Nonlinear instability in Vlasov type equations around rough velocity profiles. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 3, pp. 489–547

DOI 10.1016/J.ANIHPC.2019.12.002