# American Institute of Mathematical Sciences

• Previous Article
Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials
• DCDS Home
• This Issue
• Next Article
The 3D liquid crystal system with Cannone type initial data and large vertical velocity
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, 2017, 37 (11) : 5541-5560. doi: 10.3934/dcds.2017241
##### References:

show all references

##### 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
 [1] Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15 [2] Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195 [3] Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic & Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013 [4] Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003 [5] Xavier Ros-Oton, Joaquim Serra. Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2131-2150. doi: 10.3934/dcds.2015.35.2131 [6] Thabet Abdeljawad. Fractional operators with boundary points dependent kernels and integration by parts. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 351-375. doi: 10.3934/dcdss.2020020 [7] Zhong Tan, Yong Wang, Fanhui Xu. Large-time behavior of the full compressible Euler-Poisson system without the temperature damping. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1583-1601. doi: 10.3934/dcds.2016.36.1583 [8] Youshan Tao, Lihe Wang, Zhi-An Wang. Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 821-845. doi: 10.3934/dcdsb.2013.18.821 [9] Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152 [10] Ken Shirakawa, Hiroshi Watanabe. Large-time behavior for a PDE model of isothermal grain boundary motion with a constraint. Conference Publications, 2015, 2015 (special) : 1009-1018. doi: 10.3934/proc.2015.1009 [11] Robert M. Strain, Keya Zhu. Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$. Kinetic & Related Models, 2012, 5 (2) : 383-415. doi: 10.3934/krm.2012.5.383 [12] Jishan Fan, Fei Jiang. Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions. Communications on Pure & Applied Analysis, 2016, 15 (1) : 73-90. doi: 10.3934/cpaa.2016.15.73 [13] Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211 [14] Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096 [15] Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22 [16] Colette Guillopé, Samer Israwi, Raafat Talhouk. Large-time existence for one-dimensional Green-Naghdi equations with vorticity. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021040 [17] Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701 [18] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 [19] Sylvain Ervedoza, Enrique Zuazua. A systematic method for building smooth controls for smooth data. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1375-1401. doi: 10.3934/dcdsb.2010.14.1375 [20] Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021080

2019 Impact Factor: 1.338