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
;
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, 34(4), 1355–1374. 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 34, no. 4 (2014): 1355–74. 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, 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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