论文标题
排斥vlasov-Poisson系统的稳定性
Stability of a point charge for the repulsive Vlasov-Poisson system
论文作者
论文摘要
我们考虑了排斥性Vlasov-Poisson系统的解决方案，该系统是点电荷和小气体的组合，即$δ_ {（\ Mathcal {x}}（x}（t），\ Mathcal {v}（v}（t）（t）（t）（t）（t））}+μ^2d^2d^2d for for） $(\mathcal{X}, \mathcal{V}):\mathbb{R}\to\mathbb{R}^6$ and a small gas distribution $μ:\mathbb{R}\to L^2_{{\bf x},{\bf v}}$, and study asymptotic dynamics in the associated initial value problem.如果最初在$μ_0=μ上（t = 0）$上的合适矩很小，我们获得了上述形式的全局解决方案，并且气体分布产生的电场$ $ $ $ $ $ $ $ $几乎是最佳速率。假设除了$μ_0$的合适衍生物的界限外，电场以最佳速率衰减，并且我们为点电荷和气体分布的运动提供了修改的散射动力学。 我们的证明使哈密顿结构的重要使用至关重要。线性化系统是通过开普勒颂歌传输的，我们通过渐近动作角度转换精确地集成了该系统。由于对相关的运动学的精确了解，矩和衍生物控制的控制是通过引导分析来实现的，该分析依赖于$μ$相关的电场的衰减。然后，可以从渐近作用坐标中泊松括号的性质中推导渐近行为。
We consider solutions of the repulsive Vlasov-Poisson system which are a combination of a point charge and a small gas, i.e.\ measures of the form $δ_{(\mathcal{X}(t),\mathcal{V}(t))}+μ^2d{\bf x}d{\bf v}$ for some $(\mathcal{X}, \mathcal{V}):\mathbb{R}\to\mathbb{R}^6$ and a small gas distribution $μ:\mathbb{R}\to L^2_{{\bf x},{\bf v}}$, and study asymptotic dynamics in the associated initial value problem. If initially suitable moments on $μ_0=μ(t=0)$ are small, we obtain a global solution of the above form, and the electric field generated by the gas distribution $μ$ decays at an almost optimal rate. Assuming in addition boundedness of suitable derivatives of $μ_0$, the electric field decays at an optimal rate and we derive a modified scattering dynamics for the motion of the point charge and the gas distribution. Our proof makes crucial use of the Hamiltonian structure. The linearized system is transport by the Kepler ODE, which we integrate exactly through an asymptotic action-angle transformation. Thanks to a precise understanding of the associated kinematics, moment and derivative control is achieved via a bootstrap analysis that relies on the decay of the electric field associated to $μ$. The asymptotic behavior can then be deduced from the properties of Poisson brackets in asymptotic action coordinates.
