Non-local heat flows and gradient estimates on closed manifolds

TL;DR

This paper investigates two types of non-local heat flows on closed manifolds that preserve the L^2 norm, analyzing their global behavior and deriving gradient estimates for their positive solutions.

Contribution

It introduces and analyzes two novel L^2 norm preserved non-local heat flows, providing results on their global existence, stability, asymptotic behavior, and gradient estimates.

Findings

Global existence and stability of the flows

Asymptotic behavior characterization

Gradient estimates for positive solutions

Abstract

In this paper, we study two kind of L^2 norm preserved non-local heat flows on closed manifolds. We first study the global existence, stability and asymptotic behavior to such non-local heat flows. Next we give the gradient estimates of positive solutions to these heat flows.