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

Abstract

We consider the dynamics of interacting particle systems where the particles are confined to a bounded, possibly nonconvex domain $\Omega$. When particles hit the boundary, we consider an instant change in velocity, which turns the system describing the particle dynamics into an ODE with a discontinuous right-hand side. As an alternative to the typical approach of analyzing such a system by using weak solutions to ODEs with multivalued right-hand sides (i.e., applying the theory introduced by Filippov in 1988), we define and construct mild solutions. The merit of mild solutions is that uniqueness of such solutions follows easily from Gronwall's lemma, that certain properties of the solutions are easy to establish, and that they provide a convenient framework 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 (nonconvex) domain $\Omega$.