• Previous Article
    Well-posedness for a non-isothermal flow of two viscous incompressible fluids
  • CPAA Home
  • This Issue
  • Next Article
    On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations
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. Google Scholar

[2]

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

[3]

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

[4]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[8]

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

[9]

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

[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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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. Google Scholar

[15]

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

[16]

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

[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.   Google Scholar

[18]

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

[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.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[29]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[8]

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

[9]

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

[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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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. Google Scholar

[15]

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

[16]

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

[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.   Google Scholar

[18]

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

[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.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[29]

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

[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]

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

[15]

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

[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

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (50)
  • HTML views (114)
  • Cited by (0)

Other articles
by authors

[Back to Top]