Constructing a relativistic heat flow by transport time steps
Nonlinear diffusion from a delocalized source : affine self-similarity, time reversal, & nonradial focusing geometries
Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates.
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 homogencous 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,...
Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
We investigate Prékopa-Leindler type inequalities on a Riemannian manifold equipped with a measure with density where the potential and the Ricci curvature satisfy for all , with some . As in our earlier work [], the argument uses optimal mass transport on , but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to...
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