H\"{o}lder regularity for the trajectories of generalized charged particles in 1D

Abstract

We prove H\"{o}lder regularity for the trajectories of an interacting particle system. The particle velocities are given by the nonlocal and singular interactions with the other particles. Particle collisions occur in finite time. Prior to collisions the particle velocities become unbounded, and thus the trajectories fail to be of class $C^1$. Our H\"older-regularity result supplements earlier studies on the well-posedness of the particle system which imply only continuity of the trajectories. Moreover, it extends and unifies several of the previously obtained estimates on the trajectories. Our proof method relies on standard ODE techniques: we transform the system into different variables to expose and exploit the hidden monotonicity properties.