Niveau: Supérieur, Doctorat, Bac+8
An Analytical Framework to Describe the Interactions Between Individuals and a Continuum Rinaldo M. Colombo1 Magali Lecureux-Mercier2 June 22, 2011 Abstract We consider a discrete set of individual agents interacting with a continuum. Examples might be a predator facing a huge group of preys, or a few shepherd dogs driving a herd of sheeps. Analytically, these situations can be described through a system of ordinary differential equations coupled with a scalar conservation law in several space dimensions. This paper provides a complete well posedness theory for the resulting Cauchy problem. A few applications are considered in detail and numerical integrations are provided. Keywords: Mixed P.D.E.–O.D.E. Problems, Conservation Laws, Ordinary Differential Equations 2010 MSC: 35L65, 34A12, 37N99 1 Introduction In various situations a small set of individuals interacts with a continuum. Below, we consider a predator (the individual) seeking to split a flock of preys (the continuum). An entirely different case is that of shepherd dogs (the individuals) confining, or steering, sheeps (the continuum). A very famous, albeit fabulous, example comes from the fairy tale [7] of the pied piper, where a musician (the individual) frees a city from rats (the continuum) using his magic flute. These are sample instances that all fit in the analytical framework developed below, but several other situations are conceivable.
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