Gradient estimates for porous medium equations under the Ricci flow. (English) Zbl 1374.53096
Summary: A local gradient estimate for positive solutions of porous medium equations on complete noncompact Riemannian manifolds under the Ricci flow is derived. Moreover, a global gradient estimate for such equations on compact Riemannian manifolds is also obtained.
MSC:
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
References:
| [1] | M Bailesteanu, XD Cao, A Pulemotov. Gradient estimates for the heat equation under the Ricci flow, J Funct Anal, 2010, 258: 3517–3542. · Zbl 1193.53139 · doi:10.1016/j.jfa.2009.12.003 |
| [2] | H D Cao, M Zhu. Aronson-Bénilan estimates for the porous medium equation under the Ricci flow, J Math Pures Appl, 2015, 104: 729–748. · Zbl 1323.53070 · doi:10.1016/j.matpur.2015.05.001 |
| [3] | Y Hu, ZM Qian, ZC Zhang. Gradient estimates for porous medium and fast diffusion equations via FBSDE approach, arXiv: 1206.1394, 2012. |
| [4] | G Y Huang, Z J Huang, HZ Li. Gradient estimates for the porous medium equations on Riemannian manifolds, J Geom Anal, 2013, 23: 1851–1875. · Zbl 1278.35037 · doi:10.1007/s12220-012-9310-8 |
| [5] | J F Li, XJ Xu. Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv Math, 2011, 226: 4456–4491. · Zbl 1226.58009 · doi:10.1016/j.aim.2010.12.009 |
| [6] | P Li, S-T Yau. On the parabolic kernel of the Schrödinger operator, Acta Math, 1986, 156: 153–201. · Zbl 0611.58045 · doi:10.1007/BF02399203 |
| [7] | S P Liu. Gradient estimates for solutions of the heat equation under Ricci flow, Pacific J Math, 2009, 243: 165–180. · Zbl 1180.58017 · doi:10.2140/pjm.2009.243.165 |
| [8] | P Lu, L Ni, J-L Vázquez, C Villani. Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds, J Math Pures Appl, 2009, 91: 1–19. · Zbl 1156.58015 · doi:10.1016/j.matpur.2008.09.001 |
| [9] | B Q Ma, J Li. Gradient estimates of porous medium equations under the Ricci flow, J Geom, 2014, 105: 313–325. · Zbl 1297.53047 · doi:10.1007/s00022-013-0209-8 |
| [10] | Z M Qian, ZC Zhang. Local estimates for positive solutions of porous medium equations, arXiv: 1403.1821, 2014. |
| [11] | P Souplet, Q Zhang. Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull Lond Math Soc, 2006, 38: 1045–1053. · Zbl 1109.58025 · doi:10.1112/S0024609306018947 |
| [12] | J-L Vázquez. Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Ser Math Appl, Vol 33, Oxford University Press, 2006. |
| [13] | J-L Vázquez. The Porous Medium Equation, Oxford Math Monogr, Clarendon Press, Oxford, 2007. |
| [14] | X B Zhu. Hamilton’s gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds, J Math Anal Appl, 2013, 402: 201–206. · Zbl 1331.35354 · doi:10.1016/j.jmaa.2013.01.018 |
| [15] | X B Zhu. Local Aronson-Bénilan estimates for porous medium equations under Ricci flow, J Partial Differential Equations, 2011, 24: 324–333. · Zbl 1265.35041 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
