Gradient flow structures for discrete porous medium equations

2012/12/05 by Matthias Erbar, Jan Maas, Erbar, Matthias +1
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Probability (math.PR) #math.CA #math.FA #math.MG #math.PR

paper · pdf · doi:10.48550/arxiv.1212.1129

19 pages

arxiv created 2012/12/05 · arxiv updated 2012/12/06

Abstract

We consider discrete porous medium equations of the form ∂_t ρ_t = Δϕ(ρ_t), where Δis the generator of a reversible continuous time Markov chain on a finite set X, and ϕis an increasing function. We show that these equations arise as gradient flows of certain entropy functionals with respect to suitable non-local transportation metrics. This may be seen as a discrete analogue of the Wasserstein gradient flow structure for porous medium equations in Rⁿ discovered by Otto. We present a one-dimensional counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence to the Wasserstein metric.

Related

• A gradient flow for the Porous Medium Equations with Dirichlet boundary conditions
• Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation
• Gradient flows of the entropy for finite Markov chains
• Nonlinear Fokker--Planck--Kolmogorov equations as gradient flows on the space of probability measures
• Diffusive transport on the real line: semi-contractive gradient flows and their discretization