Related papers: Neumann heat flow and gradient flow for the entrop…

For the Neumann heat flow on nonconvex Riemannian domains $D\subset M$, we provide sharp gradient estimates and transport estimates with a novel $\sqrt t$-dependence, for instance, $$\text{Lip}( P^D_tf)\le e^{2S \,…

Let $X$ be a vector field and $Y$ be a co-vector field on a smooth manifold $M$. Does there exist a smooth Riemannian metric $g_{\alpha \beta}$ on $M$ such that $Y_\beta = g_{\alpha \beta} X^\alpha$? The main result of this note gives…

We establish a representation of the heat flow with Wentzell boundary conditions on smooth domains as gradient descent dynamics for the entropy in a suitably extended Otto manifold of probability measures with additional boundary parts. Yet…

In this paper, we extend the Hamilton's gradient estimates \cite{har93} and a monotonicity formula of entropy \cite{ni04} for heat flows from smooth Riemannian manifolds to (non-smooth) metric measure spaces with appropriate Riemannian…

We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is…

We introduce notions of dynamic gradient flows on time-dependent metric spaces as well as on time-dependent Hilbert spaces. We prove existence of solutions for a class of time dependent energy functionals in both settings. In particular we…

We present two approaches to the heat flow on a Finsler manifold $(M,F)$: either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on the reverse $L^2$-Wasserstein space $\mathcal{P}_2(M)$ of probability measures on $M$ for…

A solution to the heat equation between Riemannian manifolds, where the domain is compact and possibly has boundary, will not leave a compact and locally convex set before the image of the boundary does.

We study both the local and global existence of a gradient flow of the Sinai-Ruelle-Bowen entropy functional on a Hilbert manifold of expanding maps of a circle equipped with a Sobolev norm in the tangent space of the manifold. We show…

Metric measure spaces satisfying the reduced curvature-dimension condition $CD^*(K,N)$ and where the heat flow is linear are called $RCD^*(K,N)$-spaces. This class of non smooth spaces contains Gromov-Hausdorff limits of Riemannian…

On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup is proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for $p_t(x,y)$…

Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a…

We prove a Li-Yau gradient estimate for positive solutions to the heat equation, with Neumann boundary conditions, on a compact Riemannian submanifold with boundary ${\bf M}^n\subseteq {\bf N}^n$, satisfying the integral Ricci curvature…

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…

The paper considers a manifold $M$ evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on $M$. Among other results, we prove Li-Yau-type inequalities in this context. We…

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of…

Based on the idea of a recent paper by Ambrosio-Gigli-Savar\'e in Invent. Math. (2013), we show that flow of the $q$-Cheeger energy, called $q$-heat flow, solves the gradient flow problem of the Renyi entropy functional in the…

We investigate the heat flow in an open, bounded set $D$ in $\mathbb{R}^2$ with polygonal boundary $\partial D$. We suppose that $D$ contains an open, bounded set $\widetilde{D}$ with polygonal boundary $\partial \widetilde{D}$. The initial…

We prove the following sharp upper bound for the gradient of the Neumann semigroup $P_t$ on a $d$-dimensional compact domain $\OO$ with boundary either $C^2$-smooth or convex: $$\|\nn P_t\|_{1\to \infty}\le \ff{c}{t^{(d+1)/2}},\ \ t>0,$$…

This paper is devoted to the investigation of gradient flows in asymmetric metric spaces (for example, irreversible Finsler manifolds and Minkowski normed spaces) by means of discrete approximation. We study basic properties of curves and…