Local Aronson–Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds

By P. Lu, Lei Ni, Juan Luis V'azquez, Cédric Villani. 2008

104 citations recorded by the source corpus.

Abstract In this work we derive local gradient and Laplacian estimates of the Aronson–Benilan and Li–Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discover some new entropy formulae for these equations.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/6645ee53f3490e5c3f7231c4dbc528b014277598

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “• Work on the Porous Medium Equation on manifolds was done in the 2000s, like [43], [244]; fast diffusion was treated in [44]; general Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds were obtained in [187].”
Laplacian cut-offs, porous and fast diffusion on manifolds and other applications (2016, background): “2) which is called Porous Medium Equation (PME) when the exponent m > 1 and Fast Diffusion Equation (FDE) when 0 < m < 1, has been widely studied in the Euclidean setting (see [35] and [36] for detailed surveys), and, in recent years, several papers studied the properties of the solutions of those e”
Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds (2011, background): “In [10], Lu, Ni, Vázquez, and Villani studied gradient estimates of (1.”
Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold (2008, background): “In a recent paper [33] Lu, Ni, Villani and one of the authors investigate Harnack inequalities and entropies for porous medium and fast diffusion equations on static manifolds that are closely related to Yau, Hamilton and Perelman’s work, and on the other hand are close to the subject of this paper.”
The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour (2016, methodology): “We are not aware of a direct analogue of such inequality in the Riemannian context till the paper [20], in which some local Aronson-Bénilan formulas are proved by a clever use of Li-Yau type techniques (see also [14] for some improvements).”
Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature (2012, background): “Since Perelman’s preprint [33] was published on Arxiv in 2002, many people have extended the monotonicity of the W-entropy to other type geometric heat flows on Riemannian manifolds [13,20, 25 ,29,30].”