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

Abstract:

This paper is dedicated to providing new tools and methods for studying the trend to equilibrium of gradient flows in metric spaces $(\mathfrak {M},d)$ in the entropy and metric sense, to establish decay rates and finite time of extinction, and to characterize Lyapunov stable equilibrium points. More precisely, our main results are as follows:

Introduction of a gradient inequality in the metric space framework, which in the Euclidean space ${\mathbb {R}}^{N}$ was obtained by Łojasiewicz [Éditions du Centre National de la Recherche Scientifique, Paris, 1963, pp. 87–89], later improved by Kurdyka [Ann. Inst. Fourier 48 (1998), no. 3, 769–783], and generalized to the Hilbert space framework by Simon [Ann. of Math. (2) 118 (1983), no. 3, 525—571].

Obtainment of the trend to equilibrium in the entropy and metric sense of gradient flows generated by a function ${\mathcal {E}} : \mathfrak {M}\to (-\infty ,+\infty ]$ satisfying a Kurdyka–Łojasiewicz–Simon inequality in a neighborhood of an equilibrium point of ${\mathcal {E}}$. Sufficient conditions are given implying decay rates and finite time of extinction of gradient flows.

Construction of a talweg curve in $\mathfrak {M}$ with an optimal growth function yielding the validity of a Kurdyka–Łojasiewicz–Simon inequality.

Characterization of Lyapunov stable equilibrium points of ${\mathcal {E}}$ satisfying a Kurdyka–Łojasiewicz–Simon inequality near such points.

Characterization of the entropy-entropy production inequality with the Kurdyka–Łojasiewicz–Simon inequality.

As an application of these results, the following properties are established.

New upper bounds on the extinction time of gradient flows associated with the total variational flow.

If the metric space $\mathfrak {M}$ is the $p$-Wasserstein space $\mathcal {P}_{p}({\mathbb {R}}^{N})$, $1<p<\infty$, then new HWI-, Talagrand, and logarithmic Sobolev inequalities are obtained for functions ${\mathcal {E}}$ associated with nonlinear diffusion problems modeling drift, potential, and interaction phenomena. It is shown that these inequalities are equivalent to the Kurdyka–Łojasiewicz–Simon inequality, and hence they imply a trend to equilibrium of the gradient flows of ${\mathcal {E}}$ with decay rates or arrival in finite time.