{
  "id": "1707.03129",
  "version": "v1",
  "published": "2017-07-11T05:00:53.000Z",
  "updated": "2017-07-11T05:00:53.000Z",
  "title": "Kurdyka-Łojasiewicz-Simon inequality for gradient flows in metric spaces",
  "authors": [
    "Daniel Hauer",
    "José Mazón"
  ],
  "comment": "submitted - Keywords: Gradient flows in metric spaces, Kurdyka-{\\L}ojasiewicz-Simon inequality, Wasserstein distances, Log-Sobolev inequality, Talagrand's entropy transportation inequality",
  "categories": [
    "math.AP"
  ],
  "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, and to establish decay rates. Our main results are. - Introduction of the Kurdyka-{\\L}ojasiewicz gradient inequality in the metric space framework, which in the Euclidean space $\\mathbb{R}^{N}$ is due to {\\L}ojasiewicz [\\'Editions du C.N.R.S., 1963] and Kurdyka [Ann. Inst. Fourier, 1998]. - Proof of the trend to equilibrium in the entropy sense and the metric sense with decay rates of gradient flows generated by a functional $\\mathcal{E}$ satisfying a Kurdyka-{\\L}ojasiewicz inequality in a neighbourhood of an equilibrium of $\\mathcal{E}$. - Construction of a \\emph{talweg curve} yielding the validity of a Kurdyka-{\\L}ojasiewicz inequality with optimal growth function and characterisation of the validity of Kurdyka-{\\L}ojasiewicz inequality. - Characterisation of Lyapunov stable equilibrium points of energy functionals satisfying a Kurdyka-{\\L}ojasiewicz inequality near such points. - The equivalence between the Kurdyka-{\\L}ojasiewicz 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 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., 2009]and Sturm [Acta Math., 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-11T05:00:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J52",
      "49Q20",
      "53B21",
      "35B40",
      "58J35",
      "35K90"
    ],
    "keywords": [
      "gradient flows",
      "kurdyka-łojasiewicz-simon inequality",
      "ricci curvature",
      "metric sense",
      "decay rates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}