Nonlinear diffusion: Geodesic Convexity is equivalent to Wasserstein Contraction

It is well known that nonlinear diffusion equations can be interpreted as a gradient flow in the space of probability measures equipped with the Euclidean Wasserstein distance. Under suitable convexity conditions on the nonlinearity, due to R. J. McCann [10], the associated entropy is geodesically convex, which implies a contraction type property between all solutions with respect to this distance. In this note, we give a simple straightforward proof of the equivalence between this contraction type property and this convexity condition, without even resorting to the entropy and the gradient flow structure. We consider the nonlinear diffusion equation ∂ut ∂t = ∆f(ut), t > 0, x ∈ R d (1) where f(r) is an increasing continuous function on r ∈ [0,+∞) and C2 smooth for r > 0 such that f(0) = 0. The well-posedness theory in L1(Rd) for this equation is a classical matter in the nonlinear parabolic PDEs theory developed in the last 40 years, see [14] and the references therein. Following [14, Chap. 9], by a solution we mean a map u = (ut)t≥0 ∈ C([0,∞), L1(Rd)), with ut ≥ 0 and mass ∫ ut(x)dx =M for all t, such that i) for all T > 0, the function ∇(f ◦ ut) ∈ L 2((0, T ) × Rd), ii) u weakly satisfies equation (1), i.e., it satisfies the identity