%0 Book Section
%D 2014
%T Lecture notes on gradient flows and optimal transport
%A Sara Daneri
%A Giuseppe Savarè
%X We present a short overview on the strongest variational formulation for gradient flows of geodesically λ-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009.
%I Cambridge University Press
%G en
%U http://urania.sissa.it/xmlui/handle/1963/35093
%1 35348
%2 Mathematics
%4 1
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%R 10.1017/CBO9781107297296
%0 Journal Article
%J SIAM J. Math. Anal. 40 (2008) 1104-1122
%D 2008
%T Eulerian calculus for the displacement convexity in the Wasserstein distance
%A Sara Daneri
%A Giuseppe Savarè
%X In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by Otto and Westdickenberg [SIAM J. Math. Anal., 37 (2005), pp. 1227-1255] and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.
%B SIAM J. Math. Anal. 40 (2008) 1104-1122
%I SIAM
%G en_US
%U http://hdl.handle.net/1963/3413
%1 922
%2 Mathematics
%3 Functional Analysis and Applications
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%R 10.1137/08071346X