Kurdyka-Łojasiewicz-Simon inequality for gradient flows in metric spaces

Daniel Hauer, José Mazón

Published 2017-07-11Version 1

This paper is dedicated to providing new tools and methods in the study of the trend to equilibrium of gradient flows in metric spaces $(\mathfrak{M},d)$ in the entropy and metric sense, and to establish decay rates. Our main results are. - Introduction of the Kurdyka-{\L}ojasiewicz gradient inequality in the metric space framework, which in the Euclidean space $\mathbb{R}^{N}$ is due to {\L}ojasiewicz [\'Editions du C.N.R.S., 1963] and Kurdyka [Ann. Inst. Fourier, 1998]. - Proof of the trend to equilibrium in the entropy sense and the metric sense with decay rates of gradient flows generated by a functional $\mathcal{E}$ satisfying a Kurdyka-{\L}ojasiewicz inequality in a neighbourhood of an equilibrium of $\mathcal{E}$. - Construction of a \emph{talweg curve} yielding the validity of a Kurdyka-{\L}ojasiewicz inequality with optimal growth function and characterisation of the validity of Kurdyka-{\L}ojasiewicz inequality. - Characterisation of Lyapunov stable equilibrium points of energy functionals satisfying a Kurdyka-{\L}ojasiewicz inequality near such points. - The equivalence between the Kurdyka-{\L}ojasiewicz inequality, the classical entropy-entropy production inequality, (Talagrand's) entropy transportation inequality, and logarithmic Sobolev inequality on the $p$-Wasserstein space $\mathcal{P}_{p}(\mathbb{R}^{N})$ and on $\mathcal{P}_{p,d}(X)$, where $(X,d,\nu)$ is a measure length spaces satisfying a $(p,\infty)$-Ricci curvature bounded from below by $K\in \mathbb{R}$. Our notion of Ricci curvature is consistent in the case $p=2$ with the one introduced by Lott-Villani [Ann. Math., 2009]and Sturm [Acta Math., 2006]. As an application of these results, we establish new upper bounds on the extinction time of gradient flows associated with the total variational flow, new HWI-, Talagrand-, and Log-Sobolev inequalities.