On the Dynamics of a Fluid-Particle Interaction Model: The Bubbling Regime

TL;DR

This paper investigates the global existence and long-term behavior of solutions in a fluid-particle interaction model within the bubbling regime, demonstrating stability towards a unique stationary state under certain conditions.

Contribution

It establishes the global-in-time existence and asymptotic stability of solutions for a coupled fluid-particle model in unbounded domains, a novel analysis in this regime.

Findings

Solutions exist globally in time under physical assumptions.

The system stabilizes to a unique stationary state.

The analysis applies to unbounded physical domains.

Abstract

This article deals with the issues of global-in-time existence and asymptotic analysis of a fluid-particle interaction model in the so-called bubbling regime. The mixture occupies the physical space $\Omega \subset \mathbb{R}^3$ which may be unbounded. The system under investigation describes the evolution of particles dispersed in a viscous compressible fluid and is expressed by the conservation of fluid mass, the balance of momentum and the balance of particle density often referred as the Smoluchowski equation. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and the particles exert mutually by action-reaction principle. We show that solutions exist globally in time under reasonable physical assumptions on the initial data, the physical domain, and the external potential. Furthermore, we prove the large-time stabilization of the system towards a unique stationary state fully determined by the masses of the initial density of particles and fluid and the external potential.