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November  2017, 37(11): 5541-5560. doi: 10.3934/dcds.2017241

## A discrete Bakry-Emery method and its application to the porous-medium equation

 Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria

* Corresponding author: A. Jüngel

Received  February 2017 Revised  June 2017 Published  July 2017

Fund Project: The authors acknowledge partial support from the Austrian Science Fund (FWF), grants P22108, P24304, and W1245, and the Austrian-French Program of the Austrian Exchange Service (ÖAD)

The exponential decay of the relative entropy associated to a fully discrete porous-medium equation in one space dimension is shown by means of a discrete Bakry-Emery approach. The first ingredient of the proof is an abstract discrete Bakry-Emery method, which states conditions on a sequence under which the exponential decay of the discrete entropy follows. The second ingredient is a new nonlinear summation-by-parts formula which is inspired by systematic integration by parts developed by Matthes and the first author. Numerical simulations illustrate the exponential decay of the entropy for various time and space step sizes.

Citation: Ansgar Jüngel, Stefan Schuchnigg. A discrete Bakry-Emery method and its application to the porous-medium equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5541-5560. doi: 10.3934/dcds.2017241
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##### References:
Admissible region $S$ for $\varepsilon =1/4$ (left) and $\varepsilon =1/100$ (right). The set $S_c$, defined by $-1<\alpha-\beta<2$, is shown in light blue for comparison; it contains the dark blue region $S$
The regions of admissible $(A,B)$ such that $T(X,Y)\ge 0$ for all $X$, $Y\ge 0$ using $c$ as in (19) with $\kappa_c=\kappa$ and $\kappa=A/4$ (left), $\kappa=A/100$ (right). The set $R$ is depicted in dark blue, $R_c\supset R$ in light blue
Level sets $(X^A+Y^A-2)(X+Y-2)=0$ and $(X^A+Y^A-2)(X+Y-2)=1$ for $A=0.6$, $B=4$ (left) and $A=1.6$, $B=2.5$ (right). We have chosen $\kappa=\kappa_0=A/200$ and $c$ as in (19)
Evolution of the total mass for two test scenarios (left: $\alpha=2$, $\beta=0.5$, right: $\alpha=3$, $\beta=4$)
Evolution of the relative entropy for two test scenarios in the admissible region (left: $\alpha=2$, $\beta=0.5$, right: $\alpha=3$, $\beta=4$)
Evolution of the relative entropies for $(\alpha,\beta)$ outside of the admissible region
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