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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. Huang, Z. 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. Lu, L. Ni, J. 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. Wang, J. 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.

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##### 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. Huang, Z. 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. Lu, L. Ni, J. 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. Wang, J. 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.
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