Gradient flows of interacting Laguerre cells as discrete porous media flows

TL;DR

This paper investigates how discrete particle systems based on Laguerre tessellations evolve over time and demonstrates their convergence to solutions of nonlinear porous medium PDEs in the high cell number limit.

Contribution

It introduces a novel discrete model based on Laguerre cells and proves its convergence to continuous porous medium equations as the number of cells increases.

Findings

Discrete models converge to porous medium PDE solutions

Use of modulated energy argument for convergence proof

Establishment of high cell limit behavior

Abstract

We study a class of discrete models in which a collection of particles evolves in time following the gradient flow of an energy depending on the cell areas of an associated Laguerre (i.e. a weighted Voronoi) tessellation. We consider the high number of cell limit of such systems and, using a modulated energy argument, we prove convergence towards smooth solutions of nonlinear diffusion PDEs of porous medium type.