{
  "id": "1812.08969",
  "version": "v1",
  "published": "2018-12-21T06:37:14.000Z",
  "updated": "2018-12-21T06:37:14.000Z",
  "title": "Particle dynamics subject to impenetrable boundaries: existence and uniqueness of mild solutions",
  "authors": [
    "M. Kimura",
    "P. van Meurs",
    "Z. X. Yang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "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 $\\Omega$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-21T06:37:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mild solutions",
      "particle dynamics subject",
      "impenetrable boundaries",
      "uniqueness",
      "weak solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}