Gradient flow structures for discrete porous medium equations

M. Erbar, J. Maas, Discrete and Continuous Dynamical Systems- Series A 34 (2014) 1355–1374.


Journal Article | Published
Author
Erbar, Matthias; Maas, JanIST Austria
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.
Publishing Year
Date Published
2014-04-01
Journal Title
Discrete and Continuous Dynamical Systems- Series A
Volume
34
Issue
4
Page
1355 - 1374
IST-REx-ID

Cite this

Erbar M, Maas J. Gradient flow structures for discrete porous medium equations. Discrete and Continuous Dynamical Systems- Series A. 2014;34(4):1355-1374. doi:10.3934/dcds.2014.34.1355 
Erbar, M., & Maas, J. (2014). Gradient flow structures for discrete porous medium equations. Discrete and Continuous Dynamical Systems- Series A. Southwest Missouri State University. https://doi.org/10.3934/dcds.2014.34.1355 
Erbar, Matthias, and Jan Maas. “Gradient Flow Structures for Discrete Porous Medium Equations.” Discrete and Continuous Dynamical Systems- Series A. Southwest Missouri State University, 2014. https://doi.org/10.3934/dcds.2014.34.1355  .
M. Erbar and J. Maas, “Gradient flow structures for discrete porous medium equations,” Discrete and Continuous Dynamical Systems- Series A, vol. 34, no. 4. Southwest Missouri State University, pp. 1355–1374, 2014.
Erbar M, Maas J. 2014. Gradient flow structures for discrete porous medium equations. Discrete and Continuous Dynamical Systems- Series A. 34(4), 1355–1374.
Erbar, Matthias, and Jan Maas. “Gradient Flow Structures for Discrete Porous Medium Equations.” Discrete and Continuous Dynamical Systems- Series A, vol. 34, no. 4, Southwest Missouri State University, 2014, pp. 1355–74, doi:10.3934/dcds.2014.34.1355  .
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