Particle Dynamics Subject to Impenetrable Boundaries: Existence and Uniqueness of Mild Solutions

We consider the dynamics of particle systems where the particles are confined by impenetrable barriers to a bounded, possibly non-convex domain $\Omega$. When particles hit the boundary, we consider an instant change in velocity, which turns the systems describing the particle dynamics into an ODE with discontinuous right-hand side. Other than the typical approach to analyse such a system by using weak solutions to ODEs with multi-valued right-hand sides (i.e., applying the theory introduced by Filippov in 1988), we establish the existence of mild solutions instead. This solution concept is easier to work with than weak solutions; e.g., proving uniqueness of mild solutions is straight-forward, and mild solutions provide a solid structure for proving many-particle limits. We supplement our theory of mild solutions with an application to gradient flows of interacting particle energies with a singular interaction potential, and illustrate its features by means of numerical simulations on various choices for the (non-convex) domain $\Omega$.

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