Unconditional alignment of solutions to the Fokker-Planck-Navier-Stokes system with locally averaged Brinkman force

TL;DR

This paper proves unconditional velocity alignment and synchronization in a coupled particle-fluid system using a novel entropy and hypocoercivity approach, applicable to various nonlocal alignment models.

Contribution

It introduces a new entropy method and hypocoercivity framework to establish unconditional alignment for the FPNS system and develops well-posedness theory for weak and strong solutions.

Findings

Unconditional alignment of particle and fluid velocities for all weak solutions.

Quantitative decay estimates preventing density concentration.

Applicability to a broad class of nonlocal alignment protocols.

Abstract

We study a coupled Fokker-Planck--Navier-Stokes (FPNS) system modeling the dynamics of interacting particles suspended in a viscous incompressible fluid, where the coupling occurs through a locally averaged Brinkman drag force. Our main result is the unconditional alignment and synchronization of particle and fluid velocities for all weak solutions, in any dimension, on the periodic domain. The proof leverages a new entropy method and a hypocoercivity framework, which together yield quantitative decay estimates and prevent density concentration, a key obstacle in previous analyses. Our approach applies to a broad class of nonlocal alignment protocols, including the Cucker-Smale model. We also prove develop well-posedness theory for global weak solutions with hypoelliptic regularization in any dimension, and global strong solutions in two dimensions.