• Previous Article
    On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations
  • CPAA Home
  • This Issue
  • Next Article
    Well-posedness for a non-isothermal flow of two viscous incompressible fluids
November 2018, 17(6): 2441-2454. doi: 10.3934/cpaa.2018116

$ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds

School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, China

Received  September 2017 Revised  February 2018 Published  June 2018

Fund Project: The author is supported by the National Science Foundation of China(NSFC, 11701347)

In this paper, we prove Perelman type $ \mathcal{W}$-entropy formulae and global differential Harnack estimates for positive solutions to porous medium equation on the closed Riemannian manifolds with Ricci curvature bounded below. As applications, we derive Harnack inequalities and Laplacian estimates.

Citation: Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116
References:
[1]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Springer, 2014.

[2]

A. Besse, Einstein Manifolds, Springer, Berlin, 1987.

[3]

B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow, Science press, 2006.

[4]

R. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom., 1 (1993), 113-126.

[5]

G. Y. HuangZ. J. Huang and H. Z. Li, Gradient estimates for the porous medium equations on Riemannian manifolds, J. Geom. Anal., 23 (2013), 1851-1875.

[6]

G. Y. Huang and H. Z. Li, Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian, Pacific J. Math., 268 (2014), 47-78.

[7]

B. Kotschwar and L. Ni, Gradient estimate for $p$-harmonic functions, $ 1/H$ flow and an entropy formula, Ann. Sci. éc. Norm. Supér., 42 (2009), 1-36.

[8]

J. F. Li and X. Xu, Differential Harnack inequalities on Riemannian manifolds Ⅰ: linear heat equation, Adv. Math., 226 (2011), 4456-4491.

[9]

P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201.

[10]

S. Li and X.-D. Li, $ W$-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials, Pacific J. Math., 278 (2015), 173-199.

[11]

S. Li and X.-D. Li, Harnack inequalities and $ W$-entropy formula for Witten Laplacian on manifolds with the $ K$-super Perelman Ricci flow, arXiv: 1412.7034v1.

[12]

S. Li and X.-D. Li, $ W$-entropy formulas on super Ricci flow and Langevin deformation on Wasserstein spaces over Riemannian manifolds, Science China Mathematics, https://doi.org/10.1007/s11425-017-9227-7.

[13]

S. Li and X.-D. Li, Hamilton differential Harnack inequality and $ W$-entropy for Witten Laplacian on Riemannian manifolds, J. Funct. Anal., 274 (2018), 3263-3290.

[14]

S. Li and X.-D. Li, On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows, Asian J. Math., (2017), in press, Special Issue, in honor of Prof. N. Moks 60th birthday, arXiv: 1706.05304.

[15]

S. Li and X.-D. Li, $W$-entropy, super Perelman Ricci flows and $ (K, m)$-Ricci solitons, arXiv: 1706.07040.

[16]

X.-D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl., 84 (2005), 1295-1361.

[17]

X.-D. Li, Perelman's entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature, Math. Ann., 353 (2012), 403-437.

[18]

X.-D. Li, Hamilton's Harnack inequality and the W-entropy formula on complete Riemannian manifolds, Stochastic Process. Appl., 126 (2016), 1264-1283.

[19]

P. LuL. NiJ. L. Vazquez and C. Villani, Local Aronson-Benilan esitmates and entropy formulae for porous medium and fast diffusion equations on manifolds, J.Math.Pures.Appl., 91 (2009), 1-19.

[20]

L. Ni, Monotonicity and Li-Yau-Hamilton Inequalities, Surv. Differ. Geom., 12, Geometric flows, (2008), 251–301.

[21]

L. Ni, The entropy formula for linear equation, J. Geom. Anal., 14 (2004), 87-100.

[22]

L. Ni, A note on Perelman's LYH inequality, Comm. Anal. Geom., 14 (2006), 883-905.

[23]

G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, arXiv.org/abs/maths0211159.

[24]

B. Qian, Remarks on differential Harnack inequalities, J. Math. Anal. Appl., 409 (2014), 556-566.

[25]

G. F. Wei and W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Diff. Geom., 83 (2009), 377-405.

[26]

Y. -Z. Wang and W. Y. Chen, Gradient estimates for weighted diffusion equations on smooth metric measure spaces, Journal of Mathematics(PRC), 33 (2013), 248-258.

[27]

Y. -Z. Wang and W. Y. Chen, Gradient estimates and entropy formula for doubly nonlinear diffusion equations on Riemannian manifolds, Math. Meth. Appl. Sci., 37 (2014), 2772-2781.

[28]

Y. -Z. WangJ. Yang and W. Y. Chen, Gradient estimates and entropy formulae for weighted $ p$-heat equations on smooth metric measure spaces, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 963-974.

[29]

Y. -Z. Wang, Differential Harnack estimates and entropy formulae for weighted $ p$-heat equations, Results Math., 71 (2017), 1499-1520.

show all references

References:
[1]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Springer, 2014.

[2]

A. Besse, Einstein Manifolds, Springer, Berlin, 1987.

[3]

B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow, Science press, 2006.

[4]

R. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom., 1 (1993), 113-126.

[5]

G. Y. HuangZ. J. Huang and H. Z. Li, Gradient estimates for the porous medium equations on Riemannian manifolds, J. Geom. Anal., 23 (2013), 1851-1875.

[6]

G. Y. Huang and H. Z. Li, Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian, Pacific J. Math., 268 (2014), 47-78.

[7]

B. Kotschwar and L. Ni, Gradient estimate for $p$-harmonic functions, $ 1/H$ flow and an entropy formula, Ann. Sci. éc. Norm. Supér., 42 (2009), 1-36.

[8]

J. F. Li and X. Xu, Differential Harnack inequalities on Riemannian manifolds Ⅰ: linear heat equation, Adv. Math., 226 (2011), 4456-4491.

[9]

P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201.

[10]

S. Li and X.-D. Li, $ W$-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials, Pacific J. Math., 278 (2015), 173-199.

[11]

S. Li and X.-D. Li, Harnack inequalities and $ W$-entropy formula for Witten Laplacian on manifolds with the $ K$-super Perelman Ricci flow, arXiv: 1412.7034v1.

[12]

S. Li and X.-D. Li, $ W$-entropy formulas on super Ricci flow and Langevin deformation on Wasserstein spaces over Riemannian manifolds, Science China Mathematics, https://doi.org/10.1007/s11425-017-9227-7.

[13]

S. Li and X.-D. Li, Hamilton differential Harnack inequality and $ W$-entropy for Witten Laplacian on Riemannian manifolds, J. Funct. Anal., 274 (2018), 3263-3290.

[14]

S. Li and X.-D. Li, On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows, Asian J. Math., (2017), in press, Special Issue, in honor of Prof. N. Moks 60th birthday, arXiv: 1706.05304.

[15]

S. Li and X.-D. Li, $W$-entropy, super Perelman Ricci flows and $ (K, m)$-Ricci solitons, arXiv: 1706.07040.

[16]

X.-D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl., 84 (2005), 1295-1361.

[17]

X.-D. Li, Perelman's entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature, Math. Ann., 353 (2012), 403-437.

[18]

X.-D. Li, Hamilton's Harnack inequality and the W-entropy formula on complete Riemannian manifolds, Stochastic Process. Appl., 126 (2016), 1264-1283.

[19]

P. LuL. NiJ. L. Vazquez and C. Villani, Local Aronson-Benilan esitmates and entropy formulae for porous medium and fast diffusion equations on manifolds, J.Math.Pures.Appl., 91 (2009), 1-19.

[20]

L. Ni, Monotonicity and Li-Yau-Hamilton Inequalities, Surv. Differ. Geom., 12, Geometric flows, (2008), 251–301.

[21]

L. Ni, The entropy formula for linear equation, J. Geom. Anal., 14 (2004), 87-100.

[22]

L. Ni, A note on Perelman's LYH inequality, Comm. Anal. Geom., 14 (2006), 883-905.

[23]

G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, arXiv.org/abs/maths0211159.

[24]

B. Qian, Remarks on differential Harnack inequalities, J. Math. Anal. Appl., 409 (2014), 556-566.

[25]

G. F. Wei and W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Diff. Geom., 83 (2009), 377-405.

[26]

Y. -Z. Wang and W. Y. Chen, Gradient estimates for weighted diffusion equations on smooth metric measure spaces, Journal of Mathematics(PRC), 33 (2013), 248-258.

[27]

Y. -Z. Wang and W. Y. Chen, Gradient estimates and entropy formula for doubly nonlinear diffusion equations on Riemannian manifolds, Math. Meth. Appl. Sci., 37 (2014), 2772-2781.

[28]

Y. -Z. WangJ. Yang and W. Y. Chen, Gradient estimates and entropy formulae for weighted $ p$-heat equations on smooth metric measure spaces, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 963-974.

[29]

Y. -Z. Wang, Differential Harnack estimates and entropy formulae for weighted $ p$-heat equations, Results Math., 71 (2017), 1499-1520.

[1]

Wen Wang, Dapeng Xie, Hui Zhou. Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093

[2]

Paul W. Y. Lee, Chengbo Li, Igor Zelenko. Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 303-321. doi: 10.3934/dcds.2016.36.303

[3]

Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783

[4]

Giuseppe Savaré. Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1641-1661. doi: 10.3934/dcds.2014.34.1641

[5]

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

[6]

Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013

[7]

Alberto Farina, Enrico Valdinoci. A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1139-1144. doi: 10.3934/dcds.2011.30.1139

[8]

Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665

[9]

Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011

[10]

Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337

[11]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393

[12]

Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793

[13]

María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations & Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001

[14]

Andrei Agrachev, Ugo Boscain, Mario Sigalotti. A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 801-822. doi: 10.3934/dcds.2008.20.801

[15]

Fatma Gamze Düzgün, Ugo Gianazza, Vincenzo Vespri. $1$-dimensional Harnack estimates. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 675-685. doi: 10.3934/dcdss.2016021

[16]

Jing Li, Yifu Wang, Jingxue Yin. Non-sharp travelling waves for a dual porous medium equation. Communications on Pure & Applied Analysis, 2016, 15 (2) : 623-636. doi: 10.3934/cpaa.2016.15.623

[17]

Xinfu Chen, Jong-Shenq Guo, Bei Hu. Dead-core rates for the porous medium equation with a strong absorption. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1761-1774. doi: 10.3934/dcdsb.2012.17.1761

[18]

Sofía Nieto, Guillermo Reyes. Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1123-1139. doi: 10.3934/cpaa.2013.12.1123

[19]

Gabriele Grillo, Matteo Muratori, Fabio Punzo. On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5927-5962. doi: 10.3934/dcds.2015.35.5927

[20]

Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (39)
  • HTML views (103)
  • Cited by (0)

Other articles
by authors

[Back to Top]