Speaker
Description
Plasma dynamics are often modeled by Vlasov equations, where interactions are encoded by a potential kernel. A striking phenomenon in such systems is Landau damping whereby particles relax back to a natural equilibrium state when slightly perturbed. In this talk, we investigate this phenomenon around inhomogeneous (spatially-dependent) steady states. We focus on the Vlasov-HMF (Hamiltonian Mean-Field) model, a simplification of the more central Vlasov-Poisson interaction. We first present the linearized analysis, showing how Landau damping emerges via an action-angle formulation. The core of the talk then addresses the passage from this linear picture to a nonlinear result for initial data with Sobolev regularity. This step is complicated by plasma echoes: nonlinear particle interactions that can destabilize the system. Using compact support hypotheses near the origin for symmetric perturbations, we close a bootstrap argument coupling high and low regularity norms, guaranteeing that Landau damping holds for long but finite time, up to the inverse of the perturbation size.