Browse > Article
http://dx.doi.org/10.4134/BKMS.b210087

PERELMAN TYPE ENTROPY FORMULAE AND DIFFERENTIAL HARNACK ESTIMATES FOR WEIGHTED DOUBLY NONLINEAR DIFFUSION EQUATIONS UNDER CURVATURE DIMENSION CONDITION  

Wang, Yu-Zhao (School of Mathematical Sciences Shanxi University)
Publication Information
Bulletin of the Korean Mathematical Society / v.58, no.6, 2021 , pp. 1539-1561 More about this Journal
Abstract
We prove Perelman type 𝒲-entropy formulae and differential Harnack estimates for positive solutions to weighed doubly nonlinear diffusion equation on weighted Riemannian manifolds with CD(-K, m) condition for some K ≥ 0 and m ≥ n, which are also new for the non-weighted case. As applications, we derive some Harnack inequalities.
Keywords
Weighted doubly nonlinear diffusion equations; Perelman type entropy formula; differential Harnack estimates; Bakry-Emery Ricci curvature; curvature dimension condition;
Citations & Related Records
연도 인용수 순위
  • Reference
1 D. Chen and C. Xiong, Gradient estimates for doubly nonlinear diffusion equations, Nonlinear Anal. 112 (2015), 156-164. https://doi.org/10.1016/j.na.2014.08.017   DOI
2 F. Fang, X.-D. Li, and Z. Zhang, Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 2, 563-573.   DOI
3 A. Futaki, H. Li, and X.-D. Li, On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking solitons, Ann. Global Anal. Geom. 44 (2013), no. 2, 105-114. https://doi.org/10.1007/s10455-012-9358-5   DOI
4 R. S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993), no. 1, 113-126. https://doi.org/10.4310/CAG.1993.v1.n1.a6   DOI
5 G. Huang, Z. Huang, and H. Li, Gradient estimates for the porous medium equations on Riemannian manifolds, J. Geom. Anal. 23 (2013), no. 4, 1851-1875. https://doi.org/10.1007/s12220-012-9310-8   DOI
6 G. Huang and H. Li, Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian, Pacific J. Math. 268 (2014), no. 1, 47-78. https://doi.org/10.2140/pjm.2014.268.47   DOI
7 X.-D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pures Appl. (9) 84 (2005), no. 10, 1295-1361. https://doi.org/10.1016/j.matpur.2005.04.002   DOI
8 X.-D. Li, Perelman's entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature, Math. Ann. 353 (2012), no. 2, 403-437. https://doi.org/10.1007/s00208-011-0691-y   DOI
9 D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, 348, Springer, Cham, 2014. https://doi.org/10.1007/978-3-319-00227-9
10 Y. Wang and W. Chen, Gradient estimates and entropy monotonicity formula for doubly nonlinear diffusion equations on Riemannian manifolds, Math. Methods Appl. Sci. 37 (2014), no. 17, 2772-2781. https://doi.org/10.1002/mma.3016   DOI
11 S. Yan and L. F. Wang, Elliptic gradient estimates for the doubly nonlinear diffusion equation, Nonlinear Anal. 176 (2018), 20-35. https://doi.org/10.1016/j.na.2018.06.004   DOI
12 Y. Wang, J. Yang, and W. 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), no. 4, 963-974. https://doi.org/10.1016/S0252-9602(13)60055-7   DOI
13 B. Qian, Remarks on differential Harnack inequalities, J. Math. Anal. Appl. 409 (2014), no. 1, 556-566. https://doi.org/10.1016/j.jmaa.2013.07.043   DOI
14 B. Kotschwar and L. Ni, Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula, Ann. Sci. Ec. Norm. Super. (4) 42 (2009), no. 1, 1-36. https://doi.org/10.24033/asens.2089   DOI
15 S. Li and X.-D. Li, The W-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials, Pacific J. Math. 278 (2015), no. 1, 173-199. https://doi.org/10.2140/pjm.2015.278.173   DOI
16 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.
17 C. Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009. https://doi.org/10.1007/978-3-540-71050-9
18 Y.-Z. Wang, Differential Harnack estimates and entropy formulae for weighted p-heat equations, Results Math. 71 (2017), no. 3-4, 1499-1520. https://doi.org/10.1007/s00025-017-0675-7   DOI
19 Y.-Z. Wang, $\mathcal{W}$-entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds, Commun. Pure Appl. Anal. 17 (2018), no. 6, 2441-2454. https://doi.org/10.3934/cpaa.2018116   DOI
20 S. Li and X.-D. Li, W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds, Sci. China Math. 61 (2018), no. 8, 1385-1406. https://doi.org/10.1007/s11425-017-9227-7   DOI
21 P. Lu, L. Ni, J. Vazquez, and C. Villani, Local Aronson-Benilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds, J. Math. Pures Appl. (9) 91 (2009), no. 1, 1-19. https://doi.org/10.1016/j.matpur.2008.09.001   DOI
22 G. Wei and W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377-405. https://doi.org/10.4310/jdg/1261495336   DOI
23 J. L. Vazquez, The porous medium equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.
24 S. Li and X.-D. Li, Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds, J. Funct. Anal. 274 (2018), no. 11, 3263-3290. https://doi.org/10.1016/j.jfa.2017.09.017   DOI
25 S. Li and X.-D. Li, On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows, Asian J. Math. 22 (2018), no. 3, 577-597. https://doi.org/10.4310/AJM.2018.v22.n3.a10   DOI
26 S. Li and X.-D. Li, W-entropy, super Perelman Ricci flows, and (K, m)-Ricci solitons, J. Geom. Anal. 30 (2020), no. 3, 3149-3180. https://doi.org/10.1007/s12220-019-00193-4   DOI
27 J. Li and X. Xu, Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv. Math. 226 (2011), no. 5, 4456-4491. https://doi.org/10.1016/j.aim.2010.12.009   DOI
28 P. Li and S.-T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Math. 156 (1986), no. 3-4, 153-201. https://doi.org/10.1007/BF02399203   DOI
29 L. Ni, The entropy formula for linear heat equation, J. Geom. Anal. 14 (2004), no. 1, 87-100. https://doi.org/10.1007/BF02921867   DOI
30 L. Ni, Addenda to: "The entropy formula for linear heat equation , J. Geom. Anal. 14 (2004), no. 2, 369-374. https://doi.org/10.1007/BF02922078   DOI
31 D. Bakry and M. Emery, Diffusions hypercontractives, in Seminaire de probabilites, XIX, 1983/84, 177-206, Lecture Notes in Math., 1123, Springer, Berlin. https://doi.org/10.1007/BFb0075847
32 S. Li and X.-D. Li, W-entropy formula and Langevin deformation of flows on Wasserstein space over Riemannian manifolds, arXiv:1604.02596.
33 G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv.org/abs/maths0211159.