Congestion-driven dendritic growth

Bertrand Maury; Aude Roudneff-Chupin; Filippo Santambrogio

doi:10.3934/dcds.2014.34.1575

Article Contents

2014, Volume 34, Issue 4: 1575-1604. Doi: 10.3934/dcds.2014.34.1575

This issue Previous Article A survey of the Schrödinger problem and some of its connections with optimal transport Next Article A glimpse into the differential topology and geometry of optimal transport

Congestion-driven dendritic growth

1.
Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex
2.
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, 91405 Orsay cedex

Received: September 30, 2012

Revised: February 28, 2013

Published: March 2014

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Abstract

In order to observe growth phenomena in biology where dendritic shapes appear, we propose a simple model where a given population evolves feeded by a diffusing nutriment, but is subject to a density constraint. The particles (e.g., cells) of the population spontaneously stay passive at rest, and only move in order to satisfy the constraint $\rho\leq 1$, by choosing the minimal correction velocity so as to prevent overcongestion. We treat this constraint by means of projections in the space of densities endowed with the Wasserstein distance $W_2$, defined through optimal transport. This allows to provide an existence result and suggests some numerical computations, in the same spirit of what the authors did for crowd motion (but with extra difficulties, essentially due to the fact that the total mass may increase). The numerical simulations show, according to the values of the parameter and in particular of the diffusion coefficient of the nutriment, the formation of dendritic patterns in the space occupied by cells.

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Mathematics Subject Classification: 35K57, 49J45, 49M25, 92B05.

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