A new set of inequalities is introduced, based on a novel but natural interpolation between Borel probability measures on R d . Using these estimates in lieu of convexity or rearrangement inequalities, the existence and uniqueness problems are solved for a family of attracting gas models. In these models, the gas interacts with itself through a force which increases with distance and is governed by an equation of state P=P(*) relating pressure to density. P(*)Â* (d&1)Âd is assumed non-decreasing for a d-dimensional gas. By showing that the internal and potential energies for the system are convex functions of the interpolation parameter, an energy minimizing state unique up to translation is proven to exist. The concavity established foras a function of t # [0, 1] generalizes the Brunn Minkowski inequality from sets to measures.

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The long-time asymptotics of certain nonlinear, nonlocal, diffusive equations with a gradient flow structure are analyzed. In particular, a result of Benedetto, Caglioti, Carrillo and Pulvirenti [4] guaranteeing eventual relaxation to equilibrium velocities in a spatially homogeneous model of granular flow is extended and quantified by computing explicit relaxation rates. Our arguments rely on establishing generalizations of logarithmic Sobolev inequalities and mass transportation inequalities, via either the Bakry-Emery method or the abstract approach of Otto and Villani [28].