On the regularization of the collision solutions of the one-center problem with weak forces

TL;DR

This paper investigates regularization techniques for collision solutions in one-center problems with weak singularities, demonstrating that smoothing the potential yields a continuous extended flow despite non-differentiability.

Contribution

It introduces a method of regularization via potential smoothing for weak singularities, proving the continuity of the extended flow in collision solutions.

Findings

Regularization via smoothing makes the flow continuous.

Collision solutions can be extended as transmission trajectories.

The extended flow is not differentiable with respect to initial data.

Abstract

We study the possible regularization of collision solutions for one centre problems with a weak singularity. In the case of logarithmic singularities, we consider the method of regularization via smoothing of the potential. With this technique, we prove that the extended flow, where collision solutions are replaced with a transmission trajectory, is continuous, though not differentiable, with respect to the initial data.