Preprint
KurdykaŁojasiewiczSimon 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
KurdykaŁojasiewicz 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., 8789, Paris, 1963] and Kurdyka [Ann. Inst.
Fourier, 48 (3), 769783, 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 KurdykaŁojasiewicz
inequality in a neighbourhood of an equilibrium of
\(\mathcal{E}\).
 Construction of a talweg
curve yielding the validity of a KurdykaŁojasiewicz
inequality with optimal growth function \(\theta\) and
characterisation of the validity of KurdykaŁojasiewicz
inequality.
 Characterisation of Lyapunov stable
equilibrium points of energy functionals satisfying a
KurdykaŁojasiewicz inequality near such points.

The equivalence between the KurdykaŁojasiewicz inequality,
the classical entropyentropy 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 LottVillani [Ann. Math. (2),169(3):903991,
2009] and Sturm [Acta Math., 196(1):133177, 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
LogSobolev 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, KurdykaŁojasiewiczSimon inequality, Wasserstein distances, logarithmic Sobolev inequality, Talagrand's entropytransportation inequality.
AMS Subject Classification:
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.
