scholarly journals Gradient estimates for positive solutions of the heat equation under geometric flow

2011 ◽  
Vol 253 (2) ◽  
pp. 489-510 ◽  
Author(s):  
Jun Sun
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Gradient Estimates ◽  
Geometric Flow

scholarly journals Gradient estimates for the heat equation under the Ricci-harmonic map flow

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Mihai Bailesteanu
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Ricci Flow ◽  
Harmonic Map ◽  
Gradient Estimates ◽  
Complete Manifold ◽  
Harnack Inequalities ◽  
Harmonic Map Flow

AbstractThe paper establishes a series of gradient estimates for positive solutions to the heat equation on a manifold M evolving under the Ricci flow, coupled with the harmonic map flow between M and a second manifold N. We prove Li-Yau type Harnack inequalities and we consider the cases when M is a complete manifold without boundary and when M is compact without boundary.

scholarly journals Aronson-B\’{e}nilan estimates for weighted porous medium equations under the geometric flow

2021 ◽  
Author(s):  
shahroud azami
Keyword(s):  
Porous Medium ◽  
Positive Solutions ◽  
Measure Space ◽  
Gradient Estimates ◽  
Geometric Flow ◽  
Harnack Inequalities ◽  
Porous Medium Equations

In this paper, we study Aronson-B\’{e}nilan gradient estimates for positive solutions of weighted porous medium equations $$\partial_{t}u(x,t)=\Delta_{\phi}u^{p}(x,t),\,\,\,\,(x,t)\in M\times[0,T]$$ coupled with the geometric flow $\frac{\partial g}{\partial t}=2h(t),\,\,\,\frac{\partial \phi}{\partial t}=\Delta \phi$ on a complete measure space $(M^{n},g,e^{-\phi}dv)$. As an application, by integrating the gradient estimates, we derive the corresponding Harnack inequalities.

Gradient estimates for a nonlinear heat equation under the Finsler-Ricci flow

2019 ◽  
Vol 69 (2) ◽  
pp. 409-424
Author(s):  
Fanqi Zeng ◽  
Qun He
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Ricci Flow ◽  
Finsler Manifold ◽  
Nonlinear Heat Equation ◽  
Gradient Estimates ◽  
Harnack Inequalities

Abstract This paper considers a compact Finsler manifold (Mn, F(t), m) evolving under the Finsler-Ricci flow and establishes global gradient estimates for positive solutions of the following nonlinear heat equation: $$\begin{array}{} \partial_{t}u=\Delta_{m} u, \end{array} $$ where Δm is the Finsler-Laplacian. As applications, several Harnack inequalities are obtained.

Gradient and Harnack-type estimates for PageRank

2020 ◽  
pp. 1-19
Author(s):  
Paul Horn ◽  
Lauren M. Nelsen
Keyword(s):  
Heat Equation ◽  
Positive Solutions ◽  
Gradient Estimate ◽  
Personalized Pagerank ◽  
Geometric Properties ◽  
Algorithmic Design

Abstract Personalized PageRank has found many uses in not only the ranking of webpages, but also algorithmic design, due to its ability to capture certain geometric properties of networks. In this paper, we study the diffusion of PageRank: how varying the jumping (or teleportation) constant affects PageRank values. To this end, we prove a gradient estimate for PageRank, akin to the Li–Yau inequality for positive solutions to the heat equation (for manifolds, with later versions adapted to graphs).

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