Nonlinear Stability Criteria for the HMF Model

Abstract

Abstract. We study the nonlinear stability of a large class of inhomogeneous steady state solutions to the Hamiltonian Mean Field (HMF) model. Under a simple criterion, we prove the nonlinear stability of steady states which are decreasing functions of the microscopic energy. To achieve this task, we extend to this context the strategy based on generalized rearrangement techniques which was developed recently for the gravitational Vlasov-Poisson equation. Explicit stability inequalities are established and our analysis is able to treat non compactly supported steady states to HMF, which are physically relevant in this context but induces additional difficulties, compared to the Vlasov-Poisson system.

Stable ground states for the HMF Poisson model

“…This model is closer to the Vlasov-Poisson system than the HMF model with a cosine potential. The Poisson interaction potential is however more singular, which induces serious technical difficulties and prevent from a complete application of the strategy introduced in [19] for the Vlasov-Poisson system or in [17] for the HMF model with a cosine potential. For this reason, our analysis is based on variational methods.…”

“…We get the compactness of (f n ) n via the compactness of (f * φn ) n thanks to monotonicity properties of H with respect to the transformation f * φ which will be detailed in Lemma 4.3. To define this new function, we use the generalization of symmetric rearrangement with respect to the microscopic energy e = v 2 2 + φ(θ) introduced in [17]. For more generalized results, see also [16].…”

On Linear Damping Around Inhomogeneous Stationary States of the Vlasov-HMF Model

“…We then conclude by showing the existence of stationary states η i.e. of couple (M 0 , G) satisfying (1.5)) fulfilling the stability condition and relate it with more classical stability condition from the physics literature [3] that was also used in [31] conditioning the nonlinear orbital stability of the inhomogeneous steady states of Vlasov-HMF. Strikingly enough, the key argument relies on explicit formulae in the action-angle change of variable and the direct verification that some terms do not vanish from known Fourier expansions of elliptic functions that can be found in [12].…”

“…Written in this form, the Penrose criterion (2.13) is difficult to check, but we can relate it to a more classical condition that was found in [31] or [3] to ensure orbital stability of inhomogeneous stationary states in the nonlinear equation.…”

Nonlinear Instability of Inhomogeneous Steady States Solutions to the HMF Model

“…Here m 0 is the magnetization of the stationary state f 0 defined by m 0 " ş ρ f 0 cos θdθ. It is shown in [22] that (essentially) if F is decreasing then f 0 is nonlinearly stable by the HMF flow (1.1) provided that the criterion κ 0 ă 1 is satisfied, where κ 0 is given by…”