Modified scattering dynamics in the Vlasov-Poisson equation near an attractive point mass

TL;DR

The paper analyzes the Vlasov-Poisson system near an attractive point mass under radial symmetry and small, localized perturbations. By exploiting the Hamiltonian structure via the method of asymptotic actions and action-angle variables, it derives a global well-posedness theory for Lagrangian and, under stronger regularity, strong solutions. The nonlinear long-time behavior is shown to follow a modified scattering regime with a logarithmic phase correction governed by an asymptotic effective field $\mathcal{F}_{\infty}$, linking the asymptotics to a final profile $\gamma_{\infty}$. This work provides the first global-in-time stability and detailed asymptotics for dynamics near an attractive point mass in the Vlasov-Poisson system, with potential extensions to near-vacuum and broader settings.

Abstract

We study the long-time behavior of radially symmetric solutions to the Vlasov-Poisson equation consisting of an attractive point mass and a small, suitably localized and absolutely continuous distribution of particles: if the latter is initially localized on hyperbolic trajectories for the associated Kepler problem, we obtain global in time, unique Lagrangian solutions that asymptotically undergo a modified scattering dynamics (in the sense of distributions). A key feature of this result is its low regularity regime, which does not make use of derivative control, but can be upgraded to strong solutions and strong convergence by propagation of regularity.

Figures (4)

• Figure 1: Plots of the radial function $t\mapsto r(t)$ of the solution to \ref{['eq:Sec2:CharSys']} for $t_0=0$ and different values of actions $a$ and angular momentum $\ell$. Observe that for small values of $\ell$ the function behaves more singular at time $t=0$. Compare this with Remark \ref{['rem:ZeroEll']}.
• Figure 2: Plots of the velocity $t\mapsto v(t)$ of the solution to \ref{['eq:Sec2:CharSys']} for $t_0=0$ and different values of actions $a$ and angular momentum $\ell$. Note that the velocity approaches the asymptotic velocity $a$ when $t\to \infty$. Furthermore, observe that for small values of $\ell$ the velocity becomes singular close to $t=0$. Compare this with Remark \ref{['rem:ZeroEll']}.
• Figure 3: Plot of the radial function $t\mapsto r(t)$ (left) and velocity $t\mapsto v(t)$ (right) of the solution to \ref{['eq:Sec2:CharSys']} when $t_0 = \ell=0$ and different values of actions $a$. At time $t=0$ the radial function has a singular behavior. Accordingly, the velocity goes to infinity as predicted in Remark \ref{['rem:ZeroEll']}.
• Figure 4: Plots of the function $H_\kappa(x)$ for small and large values of $x$ in the cases $\kappa=0.1, \, 0.5, \, 0.9$.

Theorems & Definitions (75)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Lemma 2.1: Hyperbolic trajectories
• Remark 2.2
• Remark 2.3: Rescaling the angular momentum
• proof : Proof of Lemma \ref{['lem:HypTraj']}
• Remark 2.4: Trajectories with $\ell=0$
• Remark 2.5: Parabolic trajectories