Kurdyka–Łojasiewicz–Simon inequality for gradient flows in metric spaces

DOI10.1090/tran/7801zbMath1425.49023arXiv1707.03129OpenAlexW2964220228MaRDI QIDQ5234511

Daniel Hauer, José M. Mazón Ruiz

Publication date: 26 September 2019

Published in: Transactions of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1707.03129

zbMATH Keywords

Wasserstein distancesgradient flows in metric spaceslogarithmic SobolevKurdyka-Łojasiewicz-Simon inequalityTalagrand's entropy-transportation inequality

Mathematics Subject Classification ID

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

Related Items (7)

Sparse optimization on measures with over-parameterized gradient descent ⋮ Mean Field Analysis of Deep Neural Networks ⋮ Asymptotic convergence of solutions to the forest kinematic model ⋮ Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type ⋮ Decay Estimates in Time for Classical and Anomalous Diffusion ⋮ Convergence of elastic flows of curves into manifolds ⋮ Nonlinear spectral decompositions by gradient flows of one-homogeneous functionals

Cites Work

• Scale-invariant extinction time estimates for some singular diffusion equations
• Global attractors for gradient flows in metric spaces
• Total variation flow and sign fast diffusion in one dimension
• Asymptotics for a class of non-linear evolution equations, with applications to geometric problems
• Contractions in the 2-Wasserstein length space and thermalization of granular media
• Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations
• Functional analysis, Sobolev spaces and partial differential equations
• Best constant in Sobolev inequality
• Un exemple concernant le comportement asymptotique de la solution du problème \(du/dt+\partial\varphi(\mu)\ni=0\)
• On gradients of functions definable in o-minimal structures
• The geometry of optimal transportation
• A simple unified approach to some convergence theorems of L. Simon
• Optimal transportation and applications. Lectures given at the C. I. M. E. summer school, Martina Franca, Italy, September 2--8, 2001
• On the Łojasiewicz--Simon gradient inequality.
• Asymptotic behavior for doubly degenerate parabolic equations
• Rate of decay to equilibrium in some semilinear parabolic equations.
• Parabolic quasilinear equations minimizing linear growth functionals
• Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality
• Some qualitative properties for the total variation flow
• A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions
• Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates
• Entropies and equilibria of many-particle systems: an essay on recent research
• Geometric inequalities via a general comparison principle for interacting gases
• Optimal transport for applied mathematicians. Calculus of variations, PDEs, and modeling
• Ricci curvature for metric-measure spaces via optimal transport
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Entropy methods for the Boltzmann equation. Lectures from a special semester at the Centre Émil Borel, Instititut H. Poincaré, Paris 2001.
• On the geometry of metric measure spaces. I
• On the geometry of metric measure spaces. II
• Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory
• On the stable equilibrium points of gradient systems
• THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION
• The Convergence Problem for Dissipative Autonomous Systems
• The Kurdyka–Łojasiewicz–Simon inequality and stabilisation in nonsmooth infinite-dimensional gradient systems
• Lecture Notes on Gradient Flows and Optimal Transport
• Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity
• Eulerian Calculus for the Displacement Convexity in the Wasserstein Distance
• Polar factorization and monotone rearrangement of vector‐valued functions
• The Variational Formulation of the Fokker--Planck Equation
• The Łojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems
• Optimal Transport
• The Dirichlet problem for the total variation flow
• Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities
• Nonlinear semigroups generated by \(j\)-elliptic functionals
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item

This page was built for publication: Kurdyka–Łojasiewicz–Simon inequality for gradient flows in metric spaces