The Geometry of Dissipative Evolution Equations: The Porous Medium Equation

Example of a displacement convex functional of first order

“…Geodesic convexity has important consequences for the existence and uniqueness of gradient flows in the space of probability measures as recognized by Otto [20] and developed by Ambrosio, Gigli, and Savaré [2], and others. Furthermore, uniform geodesic convexity provides quantitative information on asymptotics of gradient flows [20].…”

“…Furthermore, uniform geodesic convexity provides quantitative information on asymptotics of gradient flows [20]. Moreover it implies a number of important functional inequalities, see [26,Ch.…”

Optimal transport and Perelman’s reduced volume

“…The Eulerian approach of Benamou-Brenier considers smooth maps c : [t 0 , t 1 ] → P ∞ (M) that minimize an action E(c), among all such curves with the same endpoints [3]. In the associated Otto calculus, one considers P ∞ (M) to be an infinite-dimensional Riemannian manifold and E(c) to be the corresponding energy of the curve c, so the Euler-Lagrange equation for E becomes the geodesic equation on P ∞ (M) [18]. Otto and Villani used this approach to compute the time-derivatives of the entropy function E along the curve c [19].…”

Optimal Shapes and Masses, and Optimal Transportation Problems

“…Then ψ : R + → R + is a positive increasing function. In the original formulation Monge considered ψ(t) = t, while in economical applications it is reasonable to expect that ψ is concave (see for instance [106]); on the other hand, strictly convex functions ψ are used to study certain classes of differential equations (see for instance [12], [140], [154]). Then a general formulation of the Monge problem is:…”