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 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, finite time of extinction, and to
characterize Lyapunov stable equilibrium points. More
precisely, our main results are.
 Introduction of a gradient inequality in
the metric space framework, which in the Euclidean
space \(\mathbb{R}^{N}\) is due to Łojasiewicz [Éditions du C.N.R.S., 8789, Paris, 1963] and
Kurdyka [Ann. Inst. Fourier, 48 (3), 769–783, 1998].

Establish trend to equilibrium in the entropy and
metric sense of gradient flows generated by a functional
\(\mathcal{E} : \mathfrak{M}\to (\infty,+\infty]\) satisfying a
Kurdyka–Łojasiewicz inequality in a neighborhood of an
equilibrium point of \(\mathcal{E}\). In particular, 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 inequality.

Characterize Lyapunov stable equilibrium points of energy
functionals satisfying a Kurdyka–Łojasiewicz inequality near such
points.

Characterization of the entropyentropy production
inequality with the Kurdyka–Łojasiewicz inequality near
equilibrium points of \(\mathcal{E}\).
As an application of these results, the following results 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 functionals
\(\mathbb{E}\) associated with nonlinear diffusion problems modeling
drift, potential and interaction phenomena.
It is shown that these inequalities are equivalent to the
Kurdyka–Łojasiewicz inequality and so, imply trend to equilibrium
of the gradient flows of \(\mathbb{E}\) with decay rates or arrive in finite
time.
Keywords:
Gradient flows in metric spaces, Kurdyka–ŁojasiewiczSimon 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 (696kB) file.
It is also on the arXiv: arxiv.org/abs/1707.03129.
Wednesday, July 12, 2017. Revised Thursday, November 9, 2017. 
