Discrete Morse flow for Ricci flow and Porous Media equation

TL;DR

This paper develops a discrete Morse flow approach to approximate Ricci flow on a 2-sphere with punctures and the porous media equation on planar domains, providing weak solutions under certain initial conditions.

Contribution

It introduces a novel discrete Morse flow method for Ricci flow on punctured spheres and porous media equations, enabling weak solution approximations.

Findings

Weak approximated discrete Morse flow exists for Ricci flow on football surfaces.

Weak solutions for porous media equations are obtained on planar domains.

The method applies under specific initial metric assumptions.

Abstract

In this paper, we study the discrete Morse flow for the Ricci flow on football, which is the 2-sphere with removed north and south poles and with the metric $g_0$ of constant scalar curvature, and and for Porous media equation on a bounded regular domain in the plane. We show that with a suitable assumption about $g(0)$ we have a weak approximated discrete Morse flow for the approximated Ricci flow and Porous media equation on any time intervals.