Large time nonlinear dynamics close to inhomogeneous stationary states of the Vlasov-HMF model
Abstract
We consider the Vlasov-HMF (Hamiltonian Mean-Field) model and symmetric perturbations of symmetric inhomogeneous stationary states that are naturally associated with a pendulum Hamiltonian flow. We first show the existence of families of such stationary states having compact support inside the separatrix region, constant value near the origin, and satisfying a linear stability (Penrose) criterion. We then describe the nonlinear dynamics of small compactly supported perturbations with finite regularity of this class of equilibria. More precisely, we prove that for large (but finite) times, in a modulated action-angle coordinate system the solution remains close to a modulated stationary state and that the magnetization of the perturbation satisfes decay estimates driven by the linear dynamics.
Related
- Weakly inhomogeneous solutions to a nonlinear Vlasov--Fokker--Planck equation in a domain
- Threshold for the existence of scattering states for inhomogeneous nonlinear Schrödinger equations without gauge invariance
- Relaxation of a Vlasov gas to an inhomogeneous state due to phase space mixing in an axisymmetric potential: A Newtonian analogy of the Kerr orbital motion
- Linear and Nonlinear Phase Mixing for the Gravitational Vlasov–Poisson System Under an External Kepler Potential
- Long-time asymptotics of nonlinear Fokker-Planck equations with inhomogeneous diffusion and natural boundary conditions
BibTeX
@misc{faou2026large,
title = {{Large time nonlinear dynamics close to inhomogeneous stationary states of the Vlasov-HMF model}},
author = {Erwan Faou and Frédéric Rousset and Tooryanand Seetohul},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16998},
eprint = {2609.16998},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16998},
}
RIS
TY - UNPB
TI - Large time nonlinear dynamics close to inhomogeneous stationary states of the Vlasov-HMF model
AU - Erwan Faou
AU - Frédéric Rousset
AU - Tooryanand Seetohul
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16998
UR - https://arxiv.org/abs/2609.16998
AB - We consider the Vlasov-HMF (Hamiltonian Mean-Field) model and symmetric perturbations of symmetric inhomogeneous stationary states that are naturally associated with a pendulum Hamiltonian flow. We first show the existence of families of such stationary states having compact support inside the separatrix region, constant value near the origin, and satisfying a linear stability (Penrose) criterion. We then describe the nonlinear dynamics of small compactly supported perturbations with finite regularity of this class of equilibria. More precisely, we prove that for large (but finite) times, in a modulated action-angle coordinate system the solution remains close to a modulated stationary state and that the magnetization of the perturbation satisfes decay estimates driven by the linear dynamics.
ER -