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

### Daniel Hauer and José Mazón

#### Abstract

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, to establish decay rates, and to characterise Lyapunov stable equilibrium points. Our main results are.

• Introduction of the Kurdyka-Łojasiewicz gradient inequality in the metric space framework, which in the Euclidean space $$\mathbb{R}^{N}$$ is due to Łojasie\-wicz [Éditions du C.N.R.S., 87-89, Paris, 1963] and Kurdyka [Ann. Inst. Fourier, 48 (3), 769-783, 1998].
• Proof of the trend to equilibrium in the entropy sense and the metric sense with decay rates of gradient flows generated by an energy functional $$\mathcal{E} : \mathfrak{M}\to (-\infty,+\infty]$$ satisfying a Kurdyka-Łojasiewicz inequality in a neighbourhood of an equilibrium of $$\mathcal{E}$$.
• Construction of a talweg curve yielding the validity of a Kurdyka-Łojasiewicz inequality with optimal growth function $$\theta$$ and characterisation of the validity of Kurdyka-Łojasiewicz inequality.
• Characterisation of Lyapunov stable equilibrium points of energy functionals satisfying a Kurdyka-Łojasiewicz inequality near such points.
• The equivalence between the Kurdyka-Łojasiewicz 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 (compact) 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. (2),169(3):903-991, 2009] and Sturm [Acta Math., 196(1):133-177, 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 for energy functionals associated with nonlinear diffusion problems modelling drift, potential and interaction. In particular, we show that every gradient flow of these problems tends to an equilibrium in $$\mathcal{P}_{p,d}(X)$$ and give decay rates.

Keywords: Gradient flows in metric spaces, Kurdyka-Łojasiewicz-Simon inequality, Wasserstein distances, logarithmic Sobolev inequality, Talagrand's entropy-transportation inequality.

: Primary 49J52; secondary - 49Q20 - 53B21 - 35B40 - 58J35 - 35K90.

This paper is available as a pdf (1024kB) file. It is also on the arXiv: arxiv.org/abs/1707.03129.

 Wednesday, July 12, 2017