By Joaquim P. Marques de Sá

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7. A sand dune, Atacama desert, Chile 3 Granular Material Flows 50 Fig. 8. Dunes, Death Valley, California 3 Granular Material Flows 51 Fig. 9. Footprints in a sand dune, Death Valley, California. Just one stable conﬁguration, out of many possible ones… Fig. 10. Pattern of wind ripples, Death Valley California 3 Granular Material Flows 52 3 Granular Material Flows 53 Fig. 11. A granular (pattern) equilibrium state in a Zen garden in Kyoto, Japan Fig. 12. A stable pile of small pebbles in a Zen garden in Kyoto, Japan.

9. Footprints in a sand dune, Death Valley, California. Just one stable conﬁguration, out of many possible ones… Fig. 10. Pattern of wind ripples, Death Valley California 3 Granular Material Flows 52 3 Granular Material Flows 53 Fig. 11. A granular (pattern) equilibrium state in a Zen garden in Kyoto, Japan Fig. 12. A stable pile of small pebbles in a Zen garden in Kyoto, Japan. For the modeling of the growth, collapse and stability of piles of granular materials, in the context of the Monge– Kantorovich mass transportation theory, using p-Laplace equations we refer to the survey of L.

A presentation of the corresponding model hierarchy, the connections of the different PDE models in the hierarchy and a collection of references on the mathematical analysis of kinetic and macroscopic chemotaxis models can be found in [3]. The scaling limit of a phase space chemotaxis model leading to the Keller–Segel model was rigorously analysed in [1]. 4) (after appropriate rescaling). 4) are posed on Rn , n = 1, 2 or 3 and look for solutions such that r decays to 0 as |x| tends to inﬁnity.