Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Gradient estimates for porous medium and fast diffusion equations by martingale method

Ying Hu, Zhongmin Qian, and Zichen Zhang

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In this paper, we establish several local and global gradient estimates for positive solutions of Porous Medium Equations (PMEs) and Fast Diffusion Equations (FDEs). Our proof is probabilistic and uses martingale techniques.


Dans cet article, nous établissons plusieurs estimées (locales et globales) des gradients des solutions positives des équations aux milieux poreux et des équations de la diffusion rapide. Notre preuve est probabiliste et utilise des techniques de martingales.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1793-1820.

Received: 10 April 2015
Revised: 1 June 2016
Accepted: 1 June 2016
First available in Project Euclid: 27 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)

Gradient estimate Porous medium equation Fast diffusion equation Martingale technique


Hu, Ying; Qian, Zhongmin; Zhang, Zichen. Gradient estimates for porous medium and fast diffusion equations by martingale method. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1793--1820. doi:10.1214/16-AIHP771.

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  • [1] M. Arnaudon and A. Thalmaier. Li–Yau type gradient estimates and Harnack inequalities by stochastic analysis. In Probabilistic Approach to Geometry 29–48. Adv. Stud. Pure Math. 57. Math. Soc. Japan, Tokyo, 2010.
  • [2] D. G. Aronson and P. Bénilan. Régularité des solutions de l’équation des milieux poreux dans $\mathbb{R}^{N}$. C. R. Acad. Sci. Paris Sér. A–B 288 (2) (1979) A103–A105.
  • [3] D. G. Aronson and L. A. Caffarelli. The initial trace of a solution of the porous medium equation. Trans. Amer. Math. Soc. 280 (1) (1983) 351–366.
  • [4] P. Bénilan and M. G. Crandall. The continuous dependence on $\varphi$ of solutions of $u_{t}-\Delta\varphi(u)=0$. Indiana Univ. Math. J. 30 (2) (1981) 161–177.
  • [5] P. Bénilan, M. G. Crandall and M. Pierre. Solutions of the porous medium equation in $\mathbb{R}^{N}$ under optimal conditions on initial values. Indiana Univ. Math. J. 33 (1) (1984) 51–87.
  • [6] G. Bernard. Existence theorems for fast diffusion equations. Nonlinear Anal. 43 (5) (2001) 575–590.
  • [7] M. Bertsch and M. Ughi. Positivity properties of viscosity solutions of a degenerate parabolic equation. Nonlinear Anal. 14 (7) (1990) 571–592.
  • [8] M. Bonforte and J. L. Vazquez. Global positivity estimates and Harnack inequalities for the fast diffusion equation. J. Funct. Anal. 240 (2) (2006) 399–428.
  • [9] M. Bonforte and J. L. Vázquez. Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations. Adv. Math. 223 (2) (2010) 529–578.
  • [10] L. A. Caffarelli and A. Friedman. Regularity of the free boundary of a gas flow in an $n$-dimensional porous medium. Indiana Univ. Math. J. 29 (3) (1980) 361–391.
  • [11] L. A. Caffarelli, J. L. Vázquez and N. I. Wolanski. Lipschitz continuity of solutions and interfaces of the $N$-dimensional porous medium equation. Indiana Univ. Math. J. 36 (2) (1987) 373–401.
  • [12] E. Chasseigne and J. L. Vazquez. Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities. Arch. Ration. Mech. Anal. 164 (2) (2002) 133–187.
  • [13] B. E. J. Dahlberg and C. E. Kenig. Nonnegative solutions of generalized porous medium equations. Rev. Mat. Iberoam. 2 (3) (1986) 267–305.
  • [14] J. L. Doob. Stochastic Processes. Wiley, New York; Chapman & Hall, London, 1953.
  • [15] R. S. Hamilton. A matrix Harnack estimate for the heat equation. Comm. Anal. Geom. 1 (1) (1993) 113–126.
  • [16] M. A. Herrero and M. Pierre. The Cauchy problem for $u_{t}=\Delta u^{m}$ when $0<m<1$. Trans. Amer. Math. Soc. 291 (1) (1985) 145–158.
  • [17] Y. Hu and Z. Qian. BMO martingales and positive solutions of heat equations. Math. Control Relat. Fields 5 (3) (2015) 453–473.
  • [18] G. Huang, Z. Huang and H. Li. Gradient estimates for the porous medium equations on Riemannian manifolds. J. Geom. Anal. 23 (4) (2013) 1851–1875.
  • [19] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 113. Springer-Verlag, New York, 1991.
  • [20] B. L. Kotschwar. Hamilton’s gradient estimate for the heat kernel on complete manifolds. Proc. Amer. Math. Soc. 135 (9) (2007) 3013–3019.
  • [21] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural’ceva. Linear and Quasi-Linear Equations of Parabolic Type. Izdat. “Nauka”, Moscow, 1967.
  • [22] P. Li and S. T. Yau. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (3–4) (1986) 153–201.
  • [23] P. Lu, L. Ni, J. L. Vázquez and C. Villani. Local Aronson–Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds. J. Math. Pures Appl. (9) 91 (1) (2009) 1–19.
  • [24] L. Ma, L. Zhao and X. Song. Gradient estimate for the degenerate parabolic equation $u_{t}=\Delta F(u)+H(u)$ on manifolds. J. Differential Equations 244 (5) (2008) 1157–1177.
  • [25] J. Picard. Gradient estimates for some diffusion semigroups. Probab. Theory Related Fields 122 (4) (2002) 593–612.
  • [26] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer-Verlag, Berlin, 1999.
  • [27] P. Souplet and Q. S. Zhang. Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. Lond. Math. Soc. 38 (6) (2006) 1045–1053.
  • [28] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer-Verlag, Berlin, 1979.
  • [29] J. L. Vázquez. Smoothing and decay estimates for nonlinear parabolic equations of porous medium type, 2005.
  • [30] J. L. Vázquez. Symmetrization and mass comparison for degenerate nonlinear parabolic and related elliptic equations. Adv. Nonlinear Stud. 5 (1) (2005) 87–131.
  • [31] J. L. Vázquez. Smoothing and Decay Estimates for Nonlinear Diffusion Equations Equations of Porous Medium Type. Oxford Lecture Series in Mathematics and Its Applications 33. Oxford University Press, Oxford, 2006.
  • [32] J. L. Vázquez. The Porous Medium Equation. Mathematical Theory. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.
  • [33] X. Xu. Gradient estimates for $u_{t}=\Delta F(u)$ on manifolds and some Liouville-type theorems. J. Differential Equations 252 (2) (2012) 1403–1420.
  • [34] S. T. Yau. On the Harnack inequalities of partial differential equations. Comm. Anal. Geom. 2 (3) (1994) 431–450.
  • [35] X. Zhu. Hamilton’s gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds. Proc. Amer. Math. Soc. 139 (5) (2011) 1637–1644.
  • [36] X. Zhu. Hamilton’s gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds. J. Math. Anal. Appl. 402 (1) (2013) 201–206.