On the neighborhood of an inhomogeneous stable stationary solution of the Vlasov equation—Case of an attractive cosine potential

Abstract

We consider the one-dimensional Vlasov equation with an attractive cosine potential, and its non-homogeneous stable stationary states that are decreasing functions of the energy. We show that in the Sobolev space W1,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 [Z. Lin and C. Zeng, Commun. Math. Phys. 306, 291-331 (2011)].