Thesis title: Stability of a point charge for the repulsive Vlasov–Poisson system

Authors：Benoît Pausader，Klaus Widmayer，Jiaqi Yang

Publication：Journal of the European Mathematical Society

Abstract:

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 $\delta_{(X(t),V(t))}+\mu^2d{\bf x}d{\bf v}$ for some $(X,V):\mathbb{R}\to\mathbb{R}^6$ and a small gas distribution $\mu:\mathbb{R}\to L^2_{{\bf x},{\bf v}}$, and study asymptotic dynamics in the associated initial value problem. If initially suitable moments on $\mu_0=\mu(t=0)$ are small, we obtain a global solution of the above form, and the electric field generated by the gas distribution $\mu$ decays at an almost optimal rate. Assuming in addition boundedness of suitable derivatives of $\mu_0$, the electric field decays at an optimal rate, and we derive 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 $\mu$. The asymptotic behavior can then be deduced from the properties of Poisson brackets in asymptotic action coordinates.

我们考虑三维空间中互斥的单个带电粒子和一个无碰撞等离子体之间的相互作用对应的Vlasov–Poisson系统，研究了相应初值问题的适定性和渐近行为。在等离子体分布的初值足够小时，我们得到了全局强解的存在唯一性，以及在长时间下，单个带电粒子的运动近似于匀速运动，等离子体中的电荷分布沿修饰过的轨道收敛。

Discipline: Mathematical physics

Paper address:

https://ems.press/journals/jems/articles/14298087