In this paper we prove the nonlinear orbital stability of a large class of steady state 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 [16,19] which was based on generalized rearrangement techniques, and which has been recently extended to the case of the usual (smooth) cosine potential . 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.
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Marine Fontaine, Mohammed Lemou, Florian Méhats, Stable ground states for the HMF Poisson model. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 1, pp. 217–255DOI 10.1016/J.ANIHPC.2018.05.002