Multibody and Macroscopic Impact Laws: A Convex Analysis Standpoint

Abstract

These lecture notes address mathematical issues related to the modeling of impact laws for systems of rigid spheres and their macroscopic counterpart. We analyze the so-called Moreau's approach to define multibody impact laws at the mircroscopic level, and we analyze the formal macroscopic extensions of these laws, where the non-overlapping constraint is replaced by a barrier-type constraint on the local density. We detail the formal analogies between the two settings, and also their deep discrepancies, detailing how the macroscopic impact laws, natural ingredient in the so-called Pressureless Euler Equations with a Maximal Density Constraint, are in some way irrelevant to describe the global motion of a collection of inertial hard spheres. We propose some preliminary steps in the direction of designing macroscopic impact models more respectful of the underlying microscopic structure, in particular we establish micro-macro convergence results under strong assumptions on the microscopic structure.

Defective Laplacians and paradoxical phenomena in crowd motion modeling

“…This problem fits into the abstract framework of Proposition 32 in Section 5, in the infinite dimensional situation. Well-posedness comes from that fact that B is surjective, which is a straight consequence of Poincaré inequality (see [21,Proposition 3] for details).…”

“…This is a straight consequence of the Maximum Principle. We give here a formal proof of this property, and we refer to [21] for a more detailed proof. Since U is concentrating, the contraint ∇ • u ≥ 0 is saturated over Ω, so that ∇ • u is identically 0.…”

“…Let us assume that the desired velocity U is tangent to the wall. Let β → J (β) the flux functional defined by (21). We assume that desired velocities are not modified at the exit, i.e.…”

“…where the boundary integral over Γ can be replaced by an integral over Γ 0 = Γ out ∪ Γ up , thanks to the boundary condition on Γ w . Now, since we assume that desired velocities are not modified at the exit, and since we are interested in variations of the flux only, we may drop the first term of (21) in the definition of J :…”

“…Now consider a situation where the saturated zone is a smooth domain (like in the evacuation situation represented in Figure 4). As detailed in [21], if the desired velocity field in concentrating, i.e. ∇ • U < 0, then p cannot vanish on an open subset on Ω (it would rule out the constraint in this subdomain).…”