Nonlinear stability criteria for the HMF Model

TL;DR

This paper establishes nonlinear stability criteria for a broad class of inhomogeneous steady states in the HMF model, extending rearrangement techniques from gravitational systems to this context.

Contribution

It introduces a simple stability criterion for steady states decreasing in energy and extends rearrangement methods to non-compact steady states in the HMF model.

Findings

Proves nonlinear stability under the energy-decreasing condition.

Develops explicit stability inequalities for the HMF model.

Handles non-compact steady states relevant to physical applications.

Abstract

We study the nonlinear stability of a large class of inhomogeneous steady state solutions to the Hamiltonian Mean Field (HMF) model. Under a simple criterion, we prove the nonlinear stability of steady states which are decreasing functions of the microscopic energy. To achieve this task, we extend to this context the strategy based on generalized rearrangement techniques which was developed recently for the gravitational Vlasov-Poisson equation. Explicit stability inequalities are established and our analysis is able to treat non compactly supported steady states to HMF, which are physically relevant in this context but induces additional difficulties, compared to the Vlasov-Poisson system.