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Particle Dynamics with Elastic Collision at the Boundary: Existence and Partial Uniqueness of Solutions

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Abstract

We consider the dynamics of point particles which are confined to a bounded, possibly nonconvex domain \(\Omega \). Collisions with the boundary are described as purely elastic collisions. This turns the description of the particle dynamics into a coupled system of second order ODEs with discontinuous right-hand side. The main contribution of this paper is to develop a precise solution concept for this particle system, and to prove existence of solutions. In this proof we construct a solution by passing to the limit in an auxiliary problem based on the Yosida approximation. In addition to existence of solutions, we establish a partial uniqueness theorem, and show by means of a counterexample that uniqueness of solutions cannot hold in general.

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Acknowledgements

MK and PvM gratefully acknowledge support from JSPS KAKENHI Grant Number 20KK0058. PvM gratefully acknowledges support from JSPS KAKENHI Grant Number 20K14358.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Kanazawa University, Kanazawa, Japan

    M. Kimura & P. van Meurs

  2. School of Mathematics, Chengdu Normal University, Chengdu, China

    Z. X. Yang

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  1. M. Kimura
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  2. P. van Meurs
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  3. Z. X. Yang
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Correspondence to P. van Meurs.

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Kimura, M., van Meurs, P. & Yang, Z.X. Particle Dynamics with Elastic Collision at the Boundary: Existence and Partial Uniqueness of Solutions. Acta Appl Math 174, 5 (2021). https://doi.org/10.1007/s10440-021-00423-4

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