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

TL;DR

This paper develops probabilistic martingale-based methods to derive local and global gradient estimates for solutions to porous medium and fast diffusion equations, advancing analytical tools in nonlinear PDE analysis.

Contribution

It introduces a novel probabilistic approach using martingale techniques to obtain gradient estimates for PMEs and FDEs, which was not previously explored.

Findings

Established new local and global gradient estimates for PMEs and FDEs.

Demonstrated the effectiveness of martingale methods in nonlinear PDE analysis.

Provided a probabilistic framework for future research in nonlinear diffusion equations.

Abstract

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