非弹性边界条件下动力学Fokker-Planck方程的非唯一性
Nonuniqueness for the kinetic Fokker-Planck equation with inelastic boundary conditions
摘要：
我们描述了一维空间中边界靠近奇异集的区域中动力学Fokker-Planck方程解的结构。我们特别研究的非弹性边界条件，其特征在于通过一个系数$r$描述的能量损失量的粒子与域的边界的碰撞的方程的解决方案的行为。这个问题的一个特殊特征是临界指数rc的开始，这是从McKean的分析中得出的（参见。[26])的性质的随机过程相关的福克-普朗克方程的考虑。本文严格地证明了所考虑问题的解是非唯一的，如果$r < r_c$和唯一的，如果$r_{c}<r\leq 1$。特别是，这种非唯一性解释了在物理文献中发现的与福克-普朗克方程相关的随机微分方程的数值模拟的不同行为。在本文结果的证明中，我们使用了相应文献[16]中的几个渐近公式和计算。
英文摘要：
We describe the structure of solutions of the kinetic Fokker-Planck equations in domains with boundaries near the singular set in one-space dimension. We study in particular the behaviour of the solutions of this equation for inelastic boundary conditions which are characterized by means of a coefficient $r$ describing the amount of energy lost in the collisions of the particles with the boundaries of the domain. A peculiar feature of this problem is the onset of a critical exponent rc which follows from the analysis of McKean (cf. [26]) of the properties of the stochastic process associated to the Fokker-Planck equation under consideration. In this paper, we prove rigorously that the solutions of the considered problem are nonunique if $r < r_c$ and unique if $r_{c}<r\leq 1$. In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the Fokker-Planck equation. In the proof of the results of this paper we use several asymptotic formulas and computations in the companion paper [16].
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