Kinetic and Related Models doi:10.3934/krm.2015.8.201 c American Institute of Mathematical Sciences Volume 8, Number 2, June 2015 pp. 201–214 REMARKS ON A CLASS OF KINETIC MODELS OF GRANULAR MEDIA: ASYMPTOTICS AND ENTROPY BOUNDS Martial Agueh Department of Mathematics and Statistics University of Victoria, PO BOX 1700 STN CSC Victoria, BC, V8W 2Y2, Canada Guillaume Carlier CEREMADE, UMR CNRS 7534 Universit´ e Paris IX Dauphine, Pl. de Lattre de Tassigny 75775 Paris Cedex 16, France Reinhard Illner Department of Mathematics and Statistics University of Victoria, PO BOX 1700 STN CSC Victoria, BC, V8W 2Y2, Canada (Communicated by Jose Antonio Carrillo) Abstract. We obtain new a priori estimates for spatially inhomogeneous so- lutions of a kinetic equation for granular media, as first proposed in [3] and, more recently, studied in [1]. In particular, we show that a family of con- vex functionals on the phase space is non-increasing along the flow of such equations, and we deduce consequences on the asymptotic behaviour of solu- tions. Furthermore, using an additional assumption on the interaction kernel and a “potential for interaction”, we prove a global entropy estimate in the one-dimensional case. 1. Introduction. We are concerned with kinetic models of granular media as de- rived in [3, 4, 1]. More precisely, let d ≥ 1 be an integer, and consider a system of N identical particles (e.g., grains) moving in R d . Assume that the particles move freely up to an instant when two of them occupy the same position; then they collide (inelastically) at this position according to an interaction rule to be defined later. After collision, they acquire new velocities, and then continue to move freely until another collision occurs. Let x i (t) ∈ R d and v i (t) ∈ R d denote the respective posi- tion and velocity of particle i ∈{1, 2, ··· ,N } at time t ∈ [0, ∞), and let (x 0 i ,v 0 i ) be its initial position and velocity. Then (very formally) the motion of the N particles 2010 Mathematics Subject Classification. Primary: 82C21, 82C22, 82C70. Key words and phrases. Kinetic granular media, global in time estimates, asymptotic behavior, entropy bounds. Martial Agueh and Guillaume Carlier warmly thank the hospitality of the Fields Institute (Toronto, Canada), where part of the present research was conducted during the Thematic Semes- ter on Variational Problems in Physics, Economics and Geometry. 201