Recent progress in the mathematical analysis of active suspensions

Gérard-Varet, David

Journées équations aux dérivées partielles (2023), Talk no. 5, 12 p.

This note is based on the recent work [10], which analyzes the collective behaviour of suspensions of self-propelled particles in a fluid flow. The underlying model is a coupled Stokes-kinetic system of PDE’s, describing the fluid velocity and the distribution of particles in space and orientation. The stability analysis of the isotropic distribution relies on a careful study of the mixing and enhanced diffusion properties of the system. Mathematically, the interest of this study comes from the orientation variable $p\in {𝕊}^2$, which substitutes to the usual velocity variable $v\in {ℝ}^3$ from more standard models, and is responsible for new phenomena and difficulties.

Cette note s’appuie sur le travail récent [10], dédié à l’analyse du comportement collectif de particules auto-propulsées en suspension dans un écoulement fluide. Le modèle sous-jacent est un système d’EDP de type fluide/cinétique, qui décrit la vitesse du fluide et la distribution des particules en espace et en orientation. L’analyse de stabilité de la distribution isotrope repose sur une étude fine des propriétés de mélange et de diffusion accélérée du système. Mathématiquement, l’intérêt de l’étude vient de la variable d’orientation $p\in {𝕊}^2$, qui se substitue à la variable de vitesse $v\in {ℝ}^3$ des modèles plus classiques, et est à l’origine de difficultés et phénomènes nouveaux.

Classification: 35Q30, 35Q35
Keywords: Active suspensions, stability, mixing, enhanced dissipation
Mots-clés : Suspensions actives, stabilité, mélange

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