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Kurdyka-Łojasiewicz-Simon inequality for gradient flows in metric spaces. (English) Zbl 1425.49023
Summary: 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:
$$\bullet$$
Introduction of a gradient inequality in the metric space framework, which in the Euclidean space $${\mathbb{R}}^N$$ was obtained by S. Łojasiewicz [in: Equ. Derivees Partielles, Paris 1962. Colloques internat. Centre Nat. Rech. Sci. 117, 87–89 (1963; Zbl 0234.57007)], later improved by K. Kurdyka [Ann. Inst. Fourier 48, No. 3, 769–783 (1998; Zbl 0934.32009)], and generalized to the Hilbert space framework by L. Simon [Ann. Math. (2) 118, 525–571 (1983; Zbl 0549.35071)].
$$\bullet$$
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.
$$\bullet$$
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.
$$\bullet$$
Characterization of the entropy-entropy production inequality with the Kurdyka-Łojasiewicz-Simon inequality.
As an application of these results, the following properties are established.
$$\bullet$$
New upper bounds on the extinction time of gradient flows associated with the total variational flow.
$$\bullet$$
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.

##### MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting 49J52 Nonsmooth analysis 39B62 Functional inequalities, including subadditivity, convexity, etc. 35K90 Abstract parabolic equations 58J35 Heat and other parabolic equation methods for PDEs on manifolds
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