Hölder regularity for the trajectories of generalized charged particles in 1D

Thomas Geert de Jong; Patrick van Meurs; Thomas Geert de Jong; Patrick van Meurs

doi:10.3934/cam.2025028

Communications in Analysis and Mechanics

2025, Volume 17, Issue 3: 707-724. doi: 10.3934/cam.2025028

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Research article

Hölder regularity for the trajectories of generalized charged particles in 1D

Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kanazawa, Japan

Received: 05 August 2024 Revised: 26 April 2025 Accepted: 16 May 2025 Published: 09 July 2025
34E18, 74H30

We proved Hölder regularity for the particle trajectories of an interacting particle system in one dimension. The particle velocities were 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 Hölder-regularity result supplements earlier studies on the well-posedness of the particle system which implies only continuity of the trajectories. Moreover, it extends and unifies several of the previously obtained estimates on the trajectories. Our proof method relied on standard ODE techniques: we transformed the system into different variables to expose and exploit the hidden monotonicity properties.
- interacting particle systems,
- ODEs with singularities,
- regularity
Citation: Thomas Geert de Jong, Patrick van Meurs. Hölder regularity for the trajectories of generalized charged particles in 1D[J]. Communications in Analysis and Mechanics, 2025, 17(3): 707-724. doi: 10.3934/cam.2025028

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Abstract

We proved Hölder regularity for the particle trajectories of an interacting particle system in one dimension. The particle velocities were 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 Hölder-regularity result supplements earlier studies on the well-posedness of the particle system which implies only continuity of the trajectories. Moreover, it extends and unifies several of the previously obtained estimates on the trajectories. Our proof method relied on standard ODE techniques: we transformed the system into different variables to expose and exploit the hidden monotonicity properties.

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