Modèles mathématiques de type "Hamiltonian Mean-Field" ˸ stabilité et méthodes numériques autour d’états stationnaires

Abstract : In this thesis, we study the nonlinear orbital stability of steady states of "Hamiltonian mean-field" models, called HMF models. First, this study is being done theoretically by using variational methods. It is then carried out numerically by building numerical schemes wich exactly preserve steady states. Chapter 2 presents a theoretical study of the orbital stability of steady states which are solutions to the HMF Poisson system. More specifically, the orbital stability of a large class of steady states which are solutions to the HMF system with Poisson potential is proved. These steady states are obtained as minimizers of an energy functional under one, two or infinitely many constraints. The proof relies on a variational approach. However the boundedness of the space domain prevents us from using usal technics based on scale invariance. Therefore, we introduce new methods which, although specific to our context, remain somehow in the same spirit of rearrangements tools introduced for the Vlasov-Poisson system. In particular, these methods allow for the incorporation of an arbitrary number of constraints, and yield a stability result for a large class of steady states. In Chapter 3, numerical schemes exactly preserving given steady states are built. These schemes model the orbital stability property better than the classic ones. Then, a more general scheme is introduced by building a scheme wich preserves all steady states of HMF models. Lastly, by means of these schemes, we conduct a numerical study of stability of steady states solutions to HMF Poisson system. This completes the theoretical study in Chapter 2.
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Marine Fontaine. Modèles mathématiques de type "Hamiltonian Mean-Field" ˸ stabilité et méthodes numériques autour d’états stationnaires. Analyse numérique [math.NA]. École normale supérieure de Rennes, 2018. Français. ⟨NNT : 2018ENSR0013⟩. ⟨tel-01838362⟩

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