A deterministic particle approximation for a fourth-order equation

TL;DR

This paper introduces the first deterministic particle approximation for a fourth-order PDE related to cell adhesion, using a limit from nonlocal equations and Wasserstein gradient flow stability, supported by numerical simulations.

Contribution

It presents a novel deterministic particle approximation for a fourth-order PDE, derived via nonlocal equations and Wasserstein gradient flow analysis.

Findings

First deterministic particle approximation for a fourth-order PDE.

Validation through numerical simulations.

Connection established between nonlocal equations and the PDE.

Abstract

We provide a deterministic particle approximation to a fourth order equation with applications in cell-cell adhesion. In order to do that, first we show that the equation can be asymptotically obtained as a limit from a class of well-posed nonlocal partial differential equations. These latter have the advantage that the particles' empirical measure naturally satisfies the equation. Afterwards, we obtain stability of the 2-Wasserstein gradient flow of this family of nonlocal equations that we use in order to recover a deterministic particle approximation of the fourth order equation. Up to our knowledge, in this manuscript we derive the first deterministic particle approximation for a fourth-order partial differential equation. Finally, we give some numerical simulations of the model at the particles level.