Gradient flow structures for discrete porous medium equations Journal Article

Author(s): Erbar, Matthias; Maas, Jan
Article Title: Gradient flow structures for discrete porous medium equations
Abstract: We consider discrete porous medium equations of the form ∂tρt=Δϕ(ρt), where Δ is the generator of a reversible continuous time Markov chain on a finite set χ, and ϕ is 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.
Keywords: Non-local transportation metrics; Rényi entropy; porous medium equations
Journal Title: Discrete and Continuous Dynamical Systems- Series A
Volume: 34
Issue 4
ISSN: 1078-0947
Publisher: Southwest Missouri State University  
Date Published: 2014-04-01
Start Page: 1355
End Page: 1374
DOI: 10.3934/dcds.2014.34.1355
Open access: yes (repository)