In this paper, we introduce a notion of barycenter in the Wasser-stein space which generalizes McCann's interpolation to the case of more than two measures. We provide existence, uniqueness, characterizations and regularity of the barycenter, and relate it to the multi-marginal optimal transport problem considered by Gangbo and´Swi¸ech in . We also… (More)
The Cucker-Smale model for flocking or swarming of birds or insects is generalized to scenarios where a typical bird will be subject to a) a friction force term driving it to fly at optimal speed, b) a repulsive short range force to avoid collisions, c) an attractive " flocking " force computed from the birds seen by each bird inside its vision cone, and d)… (More)
We study the long-time asymptotics of the doubly nonlinear diffusion equation ρ t = div(|∇ρ m | p−2 ∇ρ m) in R n , in the range n−p n(p−1) < m < n−p+1 n(p−1) and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L 1-algebraic decay of the… (More)
A general inequality –established in – relating the relative total energy of probability densities, their Wasserstein distance and their production entropy functional, is shown to easily imply most known geometrical –Gaussian and Euclidean– inequalities. Some of the implications are known but included here for pedagogical reasons.
We obtain new a priori estimates for spatially inhomogeneous solutions of a kinetic equation for granular media, as first proposed in  and, more recently, studied in . In particular, we show that a family of convex functionals on the phase space is non-increasing along the flow of such equations, and we deduce consequences on the asymptotic behaviour… (More)
We study a linear fractional Fokker–Planck equation that models non-local diffusion in the presence of a potential field. The non-locality is due to the appearance of the 'fractional Laplacian' in the corresponding PDE, in place of the classical Laplacian which distinguishes the case of regular diffusion. We prove existence of weak solutions by combining a… (More)
We show that a smooth, small enough Cauchy datum launches a unique classical solution of the relativistic Vlasov-Darwin (RVD) system globally in time. A similar result is claimed in  following the work in . Our proof does not require estimates derived from the conservation of the total energy, nor those previously given on the transverse component… (More)