On the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on Riemannian manifolds

Abstract

In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or CD(0, m)-condition and on compact manifolds equipped with time dependent metrics and potentials. Our results can be regarded as natural extensions of a result due to Savaré and Toscani [39] on the concavity of the Renyi entropy for NLDE on Euclidean spaces. Moreover, we prove that the rigidity models for the Renyi entropy power are the Einstein or quasi-Einstein manifolds and a special (K, m)-Ricci flow with Hessian solitons. Inspired by Lu-Ni-Vazquez-Villani [34], we prove the Aronson-Benilan estimates for NLDE on compact Riemannian manifolds with CD(0, m)-condition. We also prove the NIW formula which indicates an intrinsic relationship between the second order derivative of the Renyi entropy power N p , the p-th Fisher information I p and the time derivative of the W -entropy associated with NLDE. Finally, we prove the entropy isoperimetric inequality for the Renyi entropy power and the Gagliardo-Nirenberg-Sobolev inequality on complete Riemannian manifolds with non-negative Ricci curvature or CD(0, m)-condition and maximal volume growth condition.

On the p-Renyi entropy power for nonlinear diffusion equations on Riemannian manifolds

“…2). When p = 2, we recover the concavity of Renyi entropy power of porous medium equation and fast diffusion equation in [6] and [4].…”

“…On the other hand, S. Li and X.-D Li [3,4] proved the concavity of Shannon entropy power for the Witten Laplacian heat equation ∂ t u = Lu := ∆u − ∇φ • ∇u and the concavity of Rényi entropy power for the porous medium equation ∂ t u = Lu γ with γ > 1 on weighted Riemannian manifolds with CD(0, m) or CD(−K, m) conditions, and also on (0, m) or (−K, m) super Ricci flows.…”

“…Motivated by the works of [8,6,3,4], in [11] and [10], the first author and coauthors studied the concavities of the p-Shannon entropy power and the p-Rényi entropy power for positive solutions to the p-heat equations ∂ t u p−1 = (p − 1) p−1 ∆ p u and the doubly nonlinear diffusion equations on Riemannian manifolds with nonnegative Ricci curvature, respectively. More precisely, let u be a positive solution to the doubly nonlinear diffusion equation…”

“…In [4], they also obtained the NIW formula for porous medium equation and fast diffusion equation ∂ t u = ∆u γ on compact Riemannian manifold,…”

Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry

“…Namely, (M, g(t)) t∈I is called super Ricci flow if ∂ t g ≥ −2 Ric, which has been introduced by McCann-Topping [33] from the viewpoint of optimal transport theory. Recently, the super Ricci flow has begun to be investigated from various perspectives, especially metric measure geometry (see e.g., [3], [4], [16], [19], [20], [21], [25], [26], [27], [28], [29], [30], [39]). A Ricci flow (M, g(t)) t∈I is said to be ancient when I = (−∞, 0], which is one of the crucial concepts in singular analysis of Ricci flow.…”