Nonlinear echoes and Landau damping with insufficient regularity

We prove that the theorem of Mouhot and Villani on Landau damping near equilibrium for Vlasov-Poisson equations $\mathbb{T}_x \times \mathbb{R}_v$ cannot, in general, be extended to high Sobolev spaces case gravitational interactions. This is done by showing every space, there exists background distributions such one can construct arbitrarily small perturbations exhibit many isolated nonlinear oscillations density. These are known as plasma echoes physics community. For electrostatic interactions, we demonstrate a sequence asymptotically smaller $H^s$ which display similar echoes. shows case, any extension Villani's would have depend crucially some additional non-resonance effect coming from -- unlike Gevrey-$\nu$ with $\nu < 3$ regularity, results uniform size backgrounds. In particular, dependence obtained Gevrey class false spaces.