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 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., 87-89, 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 entropy-entropy 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–Łojasiewicz-Simon inequality, Wasserstein distances, logarithmic Sobolev inequality, Talagrand's entropy-transportation inequality.

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