On the neighborhood of an inhomogeneous stable stationary solution of the Vlasov equation -Case of the Hamiltonian mean-field model
Résumé
We consider the one-dimensional Vlasov equation with an attractive cosine potential, and its non homogeneous stationary states that are decreasing functions of the energy. We show that in the Sobolev space W 1,p (p > 2) neighborhood of such a state, all stationary states that are decreasing functions of the energy are stable. This is in sharp contrast with the situation for homogeneous stationary states of a Vlasov equation, where a control over strictly more than one derivative is needed to ensure the absence of unstable stationary states in a neighborhood of a reference stationary state
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