Collapse of Inelastic Hard Spheres in Dimension $$d \ge 2$$

Abstract We study the collapse of three inelastic particles in dimension $$d \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . We obtain general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles. In the first case, we recover that the particles arrange in a nearly linear chain, already studied by Zhou and Kadanoff, and in the second case, we obtain that the particles arrange in an equilateral triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct an initial configuration leading to a nearly linear collapse, stable under perturbations, and such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision.