Heat equation and Poisson equation in matrix geometry

TL;DR

This paper investigates the behavior of the Poisson and heat equations within a matrix geometry framework, demonstrating existence, positivity, and entropy stability of solutions over time.

Contribution

It introduces analysis of these equations in matrix geometry, establishing global existence, positivity, and entropy stability results that are novel in this context.

Findings

Solutions exist for all time and remain positive definite

Heat equation solutions exhibit entropy stability

Provides insights into matrix geometric PDEs

Abstract

In this paper, we study the Poisson equation and heat equation in a model matrix geometry $M_n$. Our main results are about the Poisson equation and global behavior of the heat equation on $M_n$. We can show that if $c_0$ is the initial positive definite matrix in $M_n$, then $c(t)$ exists for all time and is positive definite too. We can also show the entropy stability of the solutions to the heat equation.