논문KCI 등재
Perelman type entropy formulae and differential Harnack estimates for weighted doubly nonlinear diffusion equations under curvature dimension condition
대한수학회 (The Korean Mathematical Society)|2021.11|대한수학회보 (Bulletin of the KMS)|제58권 제6호|pp. 1539-1561|DOI 10.4134/BKMS.b210087
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초록
We prove Perelman type $\mathcal{W}$-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 $K\ge0$ and $m\ge n$, which are also new for the non-weighted case. As applications, we derive some Harnack inequalities.
Abstract
We prove Perelman type $\mathcal{W}$-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 $K\ge0$ and $m\ge n$, which are also new for the non-weighted case. As applications, we derive some Harnack inequalities.
참고문헌 (33)
- 84, "S´eminaire de probabilit´es, XIX, 1983", D. Bakry
- D. Bakry, "Analysis and geometry of Markov diffusion operators" (2014), Springer
- Daguang Chen, "Gradient estimates for doubly nonlinear diffusion equations", Nonlinear Analysis: Theory, Methods & Applications, 112, pp.156~164 (2015), Elsevier BV[DOI]
- F. Fang, "Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature", Ann. Inst. Fourier(Grenoble), 59(2), pp.563~573 (2009)
- Akito Futaki, "On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons", Annals of Global Analysis and Geometry, 44(2), pp.105~114 (2013), Springer Science and Business Media LLC[DOI]
- Richard S. Hamilton, "Matrix Harnack estimate for the heat equation", Communications in Analysis and Geometry, 1(1), pp.113~126 (1993), International Press of Boston[DOI]
- Guangyue Huang, "Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds", Journal of Geometric Analysis, 23(4), pp.1851~1875 (2013), Springer Science and Business Media LLC[DOI]
- Guangyue Huang, "Gradient estimates and entropy formulae of porous medium and fast diffusion equations for the Witten Laplacian", Pacific Journal of Mathematics, 268(1), pp.47~78 (2014), Mathematical Sciences Publishers[DOI]
- Brett Kotschwar, "Local gradient estimates of $p$-harmonic functions, $1", H$-flow, and an entropy formula, 2009, pp.42(1), 1~36[DOI]
- X. -D. Li, "Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds", J. Math. Pures Appl, 84(10), pp.1295~1361 (2005)[DOI]
- Xiang-Dong Li, "Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature", Mathematische Annalen, 353(2), pp.403~437 (2012), Springer Science and Business Media LLC[DOI]
- Songzi Li, "TheW-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials", Pacific Journal of Mathematics, 278(1), pp.173~199 (2015), Mathematical Sciences Publishers[DOI]
- S. Li, "Harnack inequalities and W-entropy formula for Witten Laplacian on manifolds with the K-super Perelman Ricci flow", arXiv:1412.7034v1, ~
- Songzi Li, "W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds", Science China Mathematics, 61(8), pp.1385~1406 (2018), Springer Science and Business Media LLC[DOI]
- Songzi Li, "Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds", Journal of Functional Analysis, 274(11), pp.3263~3290 (2018), Elsevier BV[DOI]
- Songzi Li, "On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows", Asian Journal of Mathematics, 22(3), pp.577~598 (2018), International Press of Boston[DOI]
- Songzi Li, "W-Entropy, Super Perelman Ricci Flows, and (K, m)-Ricci Solitons", The Journal of Geometric Analysis, 30(3), pp.3149~3180 (2020), Springer Science and Business Media LLC[DOI]
- S. Li, "W-entropy formula and Langevin deformation of flows on Wasserstein space over Riemannian manifolds", arXiv:1604.02596, ~
- Junfang Li, "Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation", Advances in Mathematics, 226(5), pp.4456~4491 (2011), Elsevier BV[DOI]
- Peter Li, "On the parabolic kernel of the Schrödinger operator", Acta Mathematica, 156(0), pp.153~201 (1986), International Press of Boston[DOI]
- P. Lu, "Local Aronson-B´enilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds", J. Math. Pures Appl, 91(1), pp.1~19 (2009)[DOI]
- Lei Ni, "The entropy formula for linear heat equation", Journal of Geometric Analysis, 14(1), pp.87~100 (2004), Springer Science and Business Media LLC[DOI]
- Lei Ni, "Addenda to “The entropy formula for linear heat equation”", Journal of Geometric Analysis, 14(2), pp.369~374 (2004), Springer Science and Business Media LLC[DOI]
- G. Perelman, "The entropy formula for the Ricci flow and its geometric applications", arXiv.org, maths0211159, ~
- Bin Qian, "Remarks on differential Harnack inequalities", Journal of Mathematical Analysis and Applications, 409(1), pp.556~566 (2014), Elsevier BV[DOI]
- J. L. V´azquez, "The porous medium equation, Oxford Mathematical Monographs" (2007), Clarendon Press, Oxford University Press
- C. Villani, "Optimal transport" (2009), Springer-Verlag
- Yu-Zhao Wang, "Differential Harnack estimates and entropy formulae for weighted p-heat equations", Results in Mathematics, 71(3-4), pp.1499~1520 (2017), Springer Science and Business Media LLC[DOI]
- "W-entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds", Commun. Pure Appl. Anal, 17(6), pp.2441~2454 (2018), American Institute of Mathematical Sciences (AIMS)[DOI]
- Yuzhao Wang, "Gradient estimates and entropy monotonicity formula for doubly nonlinear diffusion equations on Riemannian manifolds", Mathematical Methods in the Applied Sciences, 37(17), pp.2772~2781 (2014), Wiley[DOI]
- Yuzhao WANG, "Gradient Estimates and Entropy Formulae for Weighted p-Heat Equations on Smooth Metric Measure Spaces", Acta Mathematica Scientia, 33(4), pp.963~974 (2013), Elsevier BV[DOI]
- Guofang Wei, "Comparison geometry for the Bakry-Emery Ricci tensor", Journal of Differential Geometry, 83(2), pp.377~405 (2009), International Press of Boston[DOI]
- Shan Yan, "Elliptic gradient estimates for the doubly nonlinear diffusion equation", Nonlinear Analysis, 176, pp.20~35 (2018), Elsevier BV[DOI]
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