Landau damping in Sobolev spaces for the Vlasov-HMF model

Nonlinear echoes and Landau damping with insufficient regularity

We prove that the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations on Tx × Rv cannot, in general, be extended to Sobolev spaces. This is demonstrated by constructing a sequence of homogeneous background distributions and arbitrarily small perturbations in Hs which deviate arbitrarily far from free transport for long times (in a sense to be made p...

On Numerical Landau Damping for Splitting Methods Applied to the Vlasov-HMF Model

Landau damping in finite regularity for unconfined systems with screened interactions

We prove Landau damping for the collisionless Vlasov equation with a class of L interaction potentials (including the physical case of screened Coulomb interactions) on R x ×R v for localized disturbances of an infinite, homogeneous background. Unlike the confined case T x ×R v , results are obtained for initial data in Sobolev spaces (as well as Gevrey and analytic classes). For spatial freque...

Caldeira-Leggett Model, Landau Damping, and the Vlasov-Poisson System

The Caldeira-Leggett Hamiltonian (Eq. (1) below) describes the interaction of a discrete harmonic oscillator with a continuous bath of harmonic oscillators. This system is a standard model of dissipation in macroscopic low temperature physics, and has applications to superconductors, quantum computing, and macroscopic quantum tunneling. The similarities between the Caldeira-Leggett model and th...

A study of Landau damping with random initial inputs

For the Vlasov-Poisson equation with random uncertain initial data, we prove that the Landau damping solution given by the deterministic counterpart (Caglioti and Maffei, J. Stat. Phys., 92:301-323, 1998) depends smoothly on the random variable if the time asymptotic profile does, under the smoothness and smallness assumptions similar to the deterministic case. The main idea is to generalize th...