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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 and Continuous Dynamical Systems, 2017, 37 (11) : 5541-5560. doi: 10.3934/dcds.2017241
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lect. Math., Birkhäuser, Basel, 2005.

[2]

D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités, } XIX, 1983/84, Lecture Notes in Mathmatics, 1123 (1985), 177-206.  doi: 10.1007/BFb0075847.

[3]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[4]

J.-D. BenamouG. CarlierQ. Mérigot and E. Oudet, Discretization of functionals involving the Monge-Ampére operator, Numer. Math., 134 (2016), 611-636.  doi: 10.1007/s00211-015-0781-y.

[5]

P. CaputoP. Dai Pra and G. Posta, Convex entropy decay via the Bochner-Bakry-Emery approach, Ann. Inst. H. Poincaré Prob. Stat., 45 (2009), 734-753.  doi: 10.1214/08-AIHP183.

[6]

J.A. CarrilloJ. DolbeaultI. Gentil and A. Jüngel, Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete Contin. Dyn. Syst. B, 6 (2006), 1027-1050.  doi: 10.3934/dcdsb.2006.6.1027.

[7]

J. A. Carrillo, A. Jüngel and M. C. Santos, Displacement convexity for the entropy in semidiscrete nonlinear Fokker-Planck equations, Submitted for publication, 2016. arXiv: 1611.04716.

[8]

C. Chainais-HillairetA. Jüngel and S. Schuchnigg, Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities, Math. Model. Numer. Anal., 50 (2016), 135-162.  doi: 10.1051/m2an/2015031.

[9]

C. Chainais-HillairetA. Jüngel and P. Shpartko, A finite-volume scheme for a spinorial matrix drift-diffusion model for semiconductors, Numer. Meth. Partial Diff. Eqs., 32 (2016), 819-846.  doi: 10.1002/num.22030.

[10]

S. ChowW. HuangY. Li and H. Zhou, Fokker-Planck equations for a free energy functional or Markov process on a graph, Arch. Rational Mech. Anal., 203 (2012), 969-1008.  doi: 10.1007/s00205-011-0471-6.

[11]

E. Emmrich, Variable time-step θ-scheme for nonlinear evolution equations governed by a monotone operator, Calcolo, 46 (2009), 187-210.  doi: 10.1007/s10092-009-0007-8.

[12]

M. Erbar and J. Maas, Gradient flow structures for discrete porous medium equations, Discrete Contin. Dyn. Sys., 34 (2014), 1355-1374. 

[13]

M. Fathi and J. Maas, Entropic Ricci curvature bounds for discrete interacting systems, Ann. Appl. Prob., 26 (2016), 1774-1806.  doi: 10.1214/15-AAP1133.

[14]

A. Glitzky, Exponential decay of the free energy for discretized electro-reaction-diffusion systems, Nonlinearity, 21 (2008), 1989-2009.  doi: 10.1088/0951-7715/21/9/003.

[15]

G. Hardy, J. Littlewood and G. Pólya, Inequalities, Second edition. Cambridge University Press, Cambridge, 1952.

[16]

A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659.  doi: 10.1088/0951-7715/19/3/006.

[17]

A. Jüngel and J.-P. Milišić, Entropy dissipative one-leg multistep time approximations of nonlinear diffusive equations, Numer. Meth. Part. Diff. Eqs., 31 (2015), 1119-1149.  doi: 10.1002/num.21938.

[18]

A. Jüngel and S. Schuchnigg, Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations, Commun. Math. Sci., 15 (2017), 27-53, arXiv: 1506.07040. doi: 10.4310/CMS.2017.v15.n1.a2.

[19]

A. Jüngel and W. Yue, Discrete Beckner inequalities via the Bochner-Bakry-Emery approach for Markov chains, To appear in Ann. Appl. Prob., 2017.

[20]

O. JungeD. Matthes and H. Osberger, A fully discrete variational schemefor solving nonlinear Fokker-Planck equations in multiple space dimensions, SIAM J. Numer. Anal., 55 (2017), 419-443.  doi: 10.1137/16M1056560.

[21]

J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292.  doi: 10.1016/j.jfa.2011.06.009.

[22]

J. Maas and D. Matthes, Long-time behavior of a finite volume discretization for a fourth order diffusion equation, Nonlinearity, 29 (2016), 1992-2023.  doi: 10.1088/0951-7715/29/7/1992.

[23]

A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Diff. Eqs., 48 (2013), 1-31.  doi: 10.1007/s00526-012-0538-8.

[24]

I. Schoenberg, The finite Fourier series and elementary geometry, Amer. Math. Monthly, 57 (1950), 390-404.  doi: 10.2307/2307639.

[25]

O. Shisha, On the discrete version of Wirtinger's inequality, Amer. Math. Monthly, 80 (1973), 755-760.  doi: 10.2307/2318162.

[26]

K.-T. Sturm, Transport inequalities, gradient estimates, entropy and Ricci curvature, Commun. Pure Appl. Math., 58 (2005), 923-940.  doi: 10.1002/cpa.20060.

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lect. Math., Birkhäuser, Basel, 2005.

[2]

D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités, } XIX, 1983/84, Lecture Notes in Mathmatics, 1123 (1985), 177-206.  doi: 10.1007/BFb0075847.

[3]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[4]

J.-D. BenamouG. CarlierQ. Mérigot and E. Oudet, Discretization of functionals involving the Monge-Ampére operator, Numer. Math., 134 (2016), 611-636.  doi: 10.1007/s00211-015-0781-y.

[5]

P. CaputoP. Dai Pra and G. Posta, Convex entropy decay via the Bochner-Bakry-Emery approach, Ann. Inst. H. Poincaré Prob. Stat., 45 (2009), 734-753.  doi: 10.1214/08-AIHP183.

[6]

J.A. CarrilloJ. DolbeaultI. Gentil and A. Jüngel, Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete Contin. Dyn. Syst. B, 6 (2006), 1027-1050.  doi: 10.3934/dcdsb.2006.6.1027.

[7]

J. A. Carrillo, A. Jüngel and M. C. Santos, Displacement convexity for the entropy in semidiscrete nonlinear Fokker-Planck equations, Submitted for publication, 2016. arXiv: 1611.04716.

[8]

C. Chainais-HillairetA. Jüngel and S. Schuchnigg, Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities, Math. Model. Numer. Anal., 50 (2016), 135-162.  doi: 10.1051/m2an/2015031.

[9]

C. Chainais-HillairetA. Jüngel and P. Shpartko, A finite-volume scheme for a spinorial matrix drift-diffusion model for semiconductors, Numer. Meth. Partial Diff. Eqs., 32 (2016), 819-846.  doi: 10.1002/num.22030.

[10]

S. ChowW. HuangY. Li and H. Zhou, Fokker-Planck equations for a free energy functional or Markov process on a graph, Arch. Rational Mech. Anal., 203 (2012), 969-1008.  doi: 10.1007/s00205-011-0471-6.

[11]

E. Emmrich, Variable time-step θ-scheme for nonlinear evolution equations governed by a monotone operator, Calcolo, 46 (2009), 187-210.  doi: 10.1007/s10092-009-0007-8.

[12]

M. Erbar and J. Maas, Gradient flow structures for discrete porous medium equations, Discrete Contin. Dyn. Sys., 34 (2014), 1355-1374. 

[13]

M. Fathi and J. Maas, Entropic Ricci curvature bounds for discrete interacting systems, Ann. Appl. Prob., 26 (2016), 1774-1806.  doi: 10.1214/15-AAP1133.

[14]

A. Glitzky, Exponential decay of the free energy for discretized electro-reaction-diffusion systems, Nonlinearity, 21 (2008), 1989-2009.  doi: 10.1088/0951-7715/21/9/003.

[15]

G. Hardy, J. Littlewood and G. Pólya, Inequalities, Second edition. Cambridge University Press, Cambridge, 1952.

[16]

A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659.  doi: 10.1088/0951-7715/19/3/006.

[17]

A. Jüngel and J.-P. Milišić, Entropy dissipative one-leg multistep time approximations of nonlinear diffusive equations, Numer. Meth. Part. Diff. Eqs., 31 (2015), 1119-1149.  doi: 10.1002/num.21938.

[18]

A. Jüngel and S. Schuchnigg, Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations, Commun. Math. Sci., 15 (2017), 27-53, arXiv: 1506.07040. doi: 10.4310/CMS.2017.v15.n1.a2.

[19]

A. Jüngel and W. Yue, Discrete Beckner inequalities via the Bochner-Bakry-Emery approach for Markov chains, To appear in Ann. Appl. Prob., 2017.

[20]

O. JungeD. Matthes and H. Osberger, A fully discrete variational schemefor solving nonlinear Fokker-Planck equations in multiple space dimensions, SIAM J. Numer. Anal., 55 (2017), 419-443.  doi: 10.1137/16M1056560.

[21]

J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292.  doi: 10.1016/j.jfa.2011.06.009.

[22]

J. Maas and D. Matthes, Long-time behavior of a finite volume discretization for a fourth order diffusion equation, Nonlinearity, 29 (2016), 1992-2023.  doi: 10.1088/0951-7715/29/7/1992.

[23]

A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Diff. Eqs., 48 (2013), 1-31.  doi: 10.1007/s00526-012-0538-8.

[24]

I. Schoenberg, The finite Fourier series and elementary geometry, Amer. Math. Monthly, 57 (1950), 390-404.  doi: 10.2307/2307639.

[25]

O. Shisha, On the discrete version of Wirtinger's inequality, Amer. Math. Monthly, 80 (1973), 755-760.  doi: 10.2307/2318162.

[26]

K.-T. Sturm, Transport inequalities, gradient estimates, entropy and Ricci curvature, Commun. Pure Appl. Math., 58 (2005), 923-940.  doi: 10.1002/cpa.20060.

Figure 1.  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 $S$
Figure 2.  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
Figure 3.  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)
Figure 4.  Evolution of the total mass for two test scenarios (left: $\alpha=2$, $\beta=0.5$, right: $\alpha=3$, $\beta=4$)
Figure 5.  Evolution of the relative entropy for two test scenarios in the admissible region (left: $\alpha=2$, $\beta=0.5$, right: $\alpha=3$, $\beta=4$)
Figure 6.  Evolution of the relative entropies for $(\alpha,\beta)$ outside of the admissible region
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