Stable Ground States for the HMF Poisson Model

TL;DR

This paper proves the nonlinear orbital stability of steady states in the Hamiltonian Mean Field system with a Poisson potential, using novel variational techniques adapted to the problem's singularity and constraints.

Contribution

It introduces a new variational approach with specialized transformations to establish stability of steady states with Poisson interactions.

Findings

Proved nonlinear orbital stability for a broad class of steady states.

Developed new transformations to handle constraints in variational problems.

Extended stability analysis to systems with singular Poisson potentials.

Abstract

In this paper we prove the nonlinear orbital stability of a large class of steady states solutions to the Hamiltonian Mean Field (HMF) system with a Poisson interaction potential. These steady states are obtained as minimizers of an energy functional under one, two or infinitely many constraints. The singularity of the Poisson potential prevents from a direct run of the general strategy in [20, 16] which was based on generalized rearrangement techniques, and which has been recently extended to the case of the usual (smooth) cosine potential [17]. Our strategy is rather based on variational techniques. However, due to the boundedness of the space domain, our variational problems do not enjoy the usual scaling invariances which are, in general, very important in the analysis of variational problems. To replace these scaling arguments, we introduce new transformations which, although specific to our context, remain somehow in the same spirit of rearrangements tools introduced in the references above. In particular, these transformations allow for the incorporation of an arbitrary number of constraints, and yield a stability result for a large class of steady states.