Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds

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• 作者：Guangyue Huang ; Zhijie Huang ; Haizhong Li
• 关键词：Porous medium equation ; Li–Yau type estimate ; Harnack inequality ; 35B45 ; 35K55
• 刊名：Journal of Geometric Analysis
• 出版年：2013
• 出版时间：October 2013
• 年：2013
• 卷：23
• 期：4
• 页码：1851-1875
• 全文大小：349KB
• 参考文献：1. Aronson, D.G., Bénilan, P.: Régularité des solutions de?l’équation des milieux poreux dans ?sup class="a-plus-plus"> / n . C.?R.?Acad. Sci. Paris Sér. A-B 288, 103-05 (1979)
  2. Bakry, D., Qian, Z.M.: Harnack inequalities on a manifold with positive or negative Ricci curvature. Rev. Matem. Iberoam. 15(1) (1999)
  3. Chen, L., Chen, W.Y.: Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds. Ann. Glob. Anal. Geom. 35, 397-04 (2009) CrossRef
  4. Davies, E.B.: Heat kernels and spectral theory. In: Cambridge Tracts in Math, vol. 92. Camb. Univ. Press, Cambridge (1989)
  5. Hamilton, R.: A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1, 113-25 (1993)
  6. Huang, G.Y., Ma, B.Q.: Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Arch. Math. 94, 265-75 (2010) CrossRef
  7. Li, X.D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl. 84, 1295-361 (2005)
  8. Li, J.F., Xu, X.J.: Differential Harnack inequalities on Riemannian manifolds I: linear heat equation. Adv. Math. 226, 4456-491 (2011) CrossRef
  9. Li, P., Yau, S.-T.: On the parabolic kernel of the Schr?dinger operator. Acta Math. 156, 153-01 (1986) CrossRef
  10. Lu, P., Ni, L., Vázquez, J., Villani, C.: Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds. J. Math. Pures Appl. 91, 1-9 (2009)
  11. Ma, L.: Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds. J. Funct. Anal. 241, 374-82 (2006) CrossRef
  12. Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. International Press, Sanerville (1994)
  13. Vázquez, J.: The Porous Medium Equation, Oxford Mathematical Monographs. Clarendon Press, Oxford Univ. Press, Oxford (2007)
  14. Yang, Y.Y.: Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Proc. Am. Math. Soc. 136, 4095-102 (2008) CrossRef
  15. Yau, S.-T.: On the Harnack inequalities for partial differential equations. Commun. Anal. Geom. 2, 431-50 (1994)
  16. Yau, S.-T.: Harnack inequality for non-self-adjoint evolution equations. Math. Res. Lett. 2, 387-99 (1995) CrossRef
  
• 作者单位：Guangyue Huang (1)
  Zhijie Huang (1)
  Haizhong Li (1)
  
  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China
  
• ISSN：1559-002X

文摘

In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=\Delta u^m$$ where m>1, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li–Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results improve the ones of Lu, Ni, Vázquez, and Villani (in J. Math. Pures Appl. 91:1-9, 2009). Moreover, our results recover the ones of Davies (in Cambridge Tracts Math vol. 92, 1989), Hamilton (in Comm. Anal. Geom. 1:113-25, 1993) and Li and Xu (in Adv. Math. 226:4456-491, 2011).