Landau damping in dynamical Lorentz gases

We analyse the Landau damping mechanism for variants of Vlasov equations, with a time dependent linear force term and a self-consistent potential that involves an additional memory effect. This question is directly motivated by a model describing the interaction of particles with their environment, through momentum and energy exchanges with a vibrating field. We establish the stability of homogeneous states. We bring out how the coupling influences the stability criterion, in comparison to the standard Vlasov case