Vlasov-HMF模型非齐次稳态附近的大时间非线性动力学
Large time nonlinear dynamics close to inhomogeneous stationary states of the Vlasov-HMF model
AI总结：
本文研究Vlasov-HMF模型对称非齐次稳态的扰动，证明存在满足Penrose判据的稳态族，并展示在调制作用-角坐标下，大时间尺度内解接近调制稳态且扰动磁化强度按线性动力学衰减。
AI中文摘要：
我们考虑Vlasov-HMF（哈密顿平均场）模型，以及对称非齐次稳态的对称扰动，这些稳态自然与摆哈密顿流相关联。我们首先证明存在这样的稳态族，其支撑紧致于分界线区域内，在原点附近取常数值，并满足线性稳定性（Penrose）判据。然后，我们描述这类平衡态的小型紧支撑且具有有限正则性的扰动的非线性动力学。更精确地，我们证明在大的（但有限的）时间内，在调制的作用-角坐标系中，解仍然接近调制的稳态，并且扰动的磁化强度满足由线性动力学驱动的衰减估计。
英文摘要：
We consider the Vlasov-HMF (Hamiltonian Mean-Field) model and symmetric perturbations of symmetric inhomogeneous stationary states that are naturally associated with a pendulum Hamiltonian flow. We first show the existence of families of such stationary states having compact support inside the separatrix region, constant value near the origin, and satisfying a linear stability (Penrose) criterion. We then describe the nonlinear dynamics of small compactly supported perturbations with finite regularity of this class of equilibria. More precisely, we prove that for large (but finite) times, in a modulated action-angle coordinate system the solution remains close to a modulated stationary state and that the magnetization of the perturbation satisfes decay estimates driven by the linear dynamics.
